Theory of Finite Simple Groups II: Commentary on the Classification Problems 0521764912, 9780521764919

Table of contents :
GERHARD MICHLER, Theory of Finite Simple Groups II -- Commentary on the Classification Problems
Contents
Acknowledgements
Introduction
1. Simple groups and indecomposable subgroups of GL_n(2)
1.1 Two alternative views on the classification problem
1.2 Simple groups are of infinite representation type, p = 2
1.3 The algorithm
1.4 Documentation of experimental results
1.5 Constructing projective irreducible modular representations
1.6 Thompson's group order formula revisited
2. Dickson group G_2(3) and related simple groups
2.1 Involution centralizers of Dickson's groups G_2(q), q odd
2.2 Fusion and conjugacy classes of even order
2.3 The 3-singular conjugacy classes
2.4 Janko's characterization of G_2(3)
2.5 Representatives of conjugacy classes
2.5.1 Conjugacy classes of W=
2.5.2 Conjugacy classes of X_1=
2.5.3 Conjugacy classes of X_2=
2.5.4 Conjugacy classes of N_G(d_1) = X_3 =
2.6 Character tables of local subgroups of G_2(3)
2.6.1 Character table of N_G(3_A) \cong N_G(3_B)
2.6.2 Character table of N_G(3_D) \cong N_G(3_E)
2.6.3 Character table of N_G(7_A)
2.6.4 Character table of N_G(13_A) \cong N_G(13_B)
3. Conway's simple group Co_3
3.1 Construction of the involution centralizer
3.2 Construction of a simple group of Co_3-type
3.3 Uniqueness proof
3.4 Representatives of conjugacy classes
3.4.1 Conjugacy classes of H=
3.4.2 Conjugacy classes of E=
3.4.3 Conjugacy classes of D=
3.4.4 Conjugacy classes of N_1 = N_G(r_1) =
3.4.5 Conjugacy classes of N_2 = N_G(r_2) =
3.4.6 Conjugacy classes of N_3 = N_G(r_3) =
3.4.7 Conjugacy classes of N_5 = N_G(f_1) =
3.4.8 Conjugacy classes of N_6 = N_G(f_2) =
3.5 Character tables of local subgroups
3.5.1 Character table of E=
3.5.2 Character table of D=
3.5.3 Character table of H=
3.5.4 Character table of U=C_G(u) \cong \times Q
3.5.5 Character table of N_1 = N_G(r_1) =
3.5.6 Character table of N_2 = N_G(r_2) =
3.5.7 Character table of N_3 = N_G(r_3) =
3.5.8 Character table of N_5 = N_G(f_1) =
3.5.9 Character table of N_6 = N_G(f_2) =
4. Conway's simple group Co_2
4.1 Extensions of the Mathieu group M_{22} and Aut(M_{22})
4.2 Construction of the 2-central involution centralizer
4.3 Construction of Conway's simple group Co_2
4.4 On the uniqueness of Co_2
4.5 Representatives of conjugacy classes
4.5.1 Conjugacy classes of H(Co_2) =
4.5.2 Conjugacy classes of D(Co_2) =
4.5.3 Conjugacy classes of E(Co_2) =
4.6 Character tables of local subgroups of Co_2
4.6.1 Character table of E_3 = E(Co_2) =
4.6.2 Character table of H(Co_2) =
5. Fischer's simple group Fi_{22}
5.1 Construction of the 2-central involution centralizers
5.2 Construction of Fischer's simple group Fi_{22}
5.3 Sketch of a uniqueness proof
5.4 The remaining cases E_1, E_4 and E_5
5.5 Representatives of conjugacy classes
5.5.1 Conjugacy classes of H(Fi_{22}) =
5.5.2 Conjugacy classes of D(Fi_{22}) =
5.5.3 Conjugacy classes of E(Fi_{22}) =
5.6 Character tables of local subgroups of Fi_{22}
5.6.1 Character table of E_2 = E(Fi_{22}) =
5.6.2 Character table of H(Fi_{22}) =
6. Fischer's simple group Fi_{23}
6.1 Extensions of the Mathieu group M_{23}
6.2 Construction of a 2-central involution centralizer
6.3 Construction of Fischer's simple group Fi_{23}
6.4 On the uniqueness of Fi_{23}
6.5 Representatives of conjugacy classes
6.5.1 Conjugacy classes of E(Fi_{23}) =
6.5.2 Conjugacy classes of H(Fi_{23}) =
6.5.3 Conjugacy classes of D(Fi_{23}) =
6.5.4 Conjugacy classes of Fi_{23} =
6.6 Character tables of local subgroups of Fi_{23}
6.6.1 Character table of E = E(Fi_{23}) =
6.6.2 Character table of D(Fi_{23}) =
6.6.3 Character table of H(Fi_{23}) =
7. Conway's simple group Co_1
7.1 Extensions of the Mathieu group M_{24}
7.2 Construction of the 2-central involution centralizer of Co_1
7.3 Construction of Conway's simple group Co_1
7.4 On the uniqueness of Co_1
7.5 Representatives of conjugacy classes
7.5.1 Conjugacy classes of E(Co_1) =
7.5.2 Conjugacy classes of H(Co_1) =
7.6 Character tables of local subgroups of Co_1
7.6.1 Character table of E(Co_1) =
7.6.2 Character table of H(Co_1) =
8. Janko's group J_4
8.1 Structure of the given centralizer
8.2 Conjugacy classes and group order
8.3 Existence and uniqueness proofs
8.4 Other constructions in GL_{1333}(11) and GL_{112}(2)
8.5 Representatives of conjugacy classes
8.5.1 Conjugacy classes of H = C_G(z) =
8.5.2 Conjugacy classes of E = N_G(A) =
8.6 Character tables of local subgroups
8.6.1 Character table of H(J_4) =
8.6.2 Character table of E = N_G(A) =
9. Fischer's simple group Fi'_{24}
9.1 The 2-fold cover of the automorphism group Aut(Fi_{22})
9.2 A semi-simple representation of Fi_{23} in GL_{8671}(13)
9.3 Construction of the irreducible subgroup G of GL_{8671}(13)
9.4 G is isomorphic to Fischer's simple group Fi'_{24}
9.5 Presentation of 2-central involution centralizer
9.6 On the uniqueness of Fi'_{24}
9.7 Representatives of conjugacy classes
9.7.1 Conjugacy classes of A_1 = 2Aut(Fi_{22}) =
9.7.2 Conjugacy classes of E(Fi_{24}) =
9.7.3 Conjugacy classes of H(Fi'_{24}) =
9.8 Character tables of local subgroups
9.8.1 Character table of A_1 = 2Aut(Fi_{22}) =
9.8.2 Character table of H(Fi'_{24}) =
9.8.3 Character table of E(Fi'_{24}) =
10. Tits' group ^2F_4(2)'
10.1 Construction of the 2-central involution centralizer
10.2 Fusion
10.3 Existence proof of Tits' simple group inside GL_{26}(73)
10.4 Group order
10.5 The 3-, 5- and 13-singular conjugacy classes
10.6 Uniqueness proof
10.7 Representatives of conjugacy classes
10.7.1 Conjugacy classes of H =
10.7.2 Conjugacy classes of N_G(S_5) =
10.7.3 Conjugacy classes of D =
10.7.4 Conjugacy classes of E =
10.7.5 Conjugacy classes of U =
10.7.6 Conjugacy classes of N_3 =
10.7.7 Conjugacy classes of N_5 =
10.8 Character tables of local subgroups
10.8.1 Character table of H = C_G(z)
10.8.2 Character table of D = N_H(A)
10.8.3 Character table of E = N_G(A)
10.8.4 Character table of U = C_G(u)
10.8.5 Character table of N_3
10.8.6 Character table of N_5
10.8.7 Character table of NS_5 = N_G(S_5)
11. McLaughlin's group McL
11.1 Construction of the 2-central involution centralizer
11.2 Structure of the given centralizer H = 2A_8
11.3 Existence and uniqueness proof
11.4 Representatives of conjugacy classes
11.4.1 Conjugacy classes of E =
11.4.2 Conjugacy classes of H =
11.4.3 Conjugacy classes of D =
11.4.4 Conjugacy classes of G =
11.5 Character tables of local subgroups
11.5.1 Character table of E =
11.5.2 Character table of D =
11.5.3 Character table of H =
12. Rudvalis' group Ru
12.1 Construction of the 2-central involution centralizer
12.2 Construction of a simple group of Ru-type
12.3 Fusion
12.4 Uniqueness proof
12.5 Representatives of conjugacy classes
12.5.1 Conjugacy classes of H =
12.5.2 Conjugacy classes of D =
12.5.3 Conjugacy classes of E =
12.5.4 Conjugacy classes of M =
12.5.5 Conjugacy classes of G =
12.6 Character tables of local subgroups
12.6.1 Character table of H =
12.6.2 Character table of E=
12.6.3 Character table of D=
12.6.4 Character table of N_G(3_A) \cong 3Aut(A_6)
12.6.5 Character table of M = N_G(R) = (d_1i, d_2)
12.6.6 Character table of N_G(5_A) \cong 5^{1+2} : (Q_8 \times 4)
12.6.7 Character table of N_G(5_B) \cong 5 : 4 \times A_5
12.6.8 Character table of G=
13. Lyons' group Ly
13.1 Structure of the given centralizer
13.2 Conjugacy classes of elements of even order
13.3 Conjugacy classes of p-singular elements
13.4 Group order
13.5 Existence and uniqueness proofs
13.6 Representatives of conjugacy classes
13.6.1 Conjugacy classes of H =
13.6.2 Conjugacy classes of D = N_H(A) =
13.6.3 Conjugacy classes of N = N_G (A)=
13.6.4 Conjugacy classes of E = N_G(3_A) = \cong 3McL : 2
13,6.5 Conjugacy classes of R = N_G(f) =
13.6.6 Conjugacy classes of L = N_R(V) =
13.6. 7 Conjugacy classes of M = N_G(V) =
13.7 Character tables of local subgroups
13.7.1 Character table of H = \cong 2A_{11}
13.7.2 Character table of D =
13.7.3 Character table of N =
13.7.4 Character table of E = N_G(3_A) \cong 3McL : 2
13.7.5 Character table of R = N_G(f) \cong 5^{1+4} : 4S_6
13.7.6 Character table of M = N_G(V) \cong 5^3.L_3(5)
13.7.7 Character table of L = N_R(V) \cong 5^3. (5^2 : GL_2(5))
14. Suzuki's group Suz
14.1 The centralizer of a 2-central involution
14.2 Even conjugacy classes and group order
14.3 Existence proof of Suz inside GL_{143}(13)
14.4 Uniqueness proof
14.5 Representatives of conjugacy classes
14.5.1 Conjugacy classes of H =
14.5.2 Conjugacy classes of E = N_G(A) =
14.5.3 Conjugacy classes of D =
14.5.4 Conjugacy classes of W =
14.5.5 Conjugacy classes of M = N_G(V) =
14.5.6 Conjugacy classes of C_G(u) =

Citation preview

Theory of Finite Simple Groups II Commentary on the Classification Problems GERHARD MICHLER Institute of Experimental Mathematics University of Duisburg-Essen and Cornell University

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www .cambridge.org Information on this title: www.cambridge.org/9780521764919

© G.

Michler 2010

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this publication is available from the British Library ISBN 978-0-521-76491-9 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Theory of Finite Simple Groups II Commentary on the Classification Problems This second volume provides a coherent explanation for the existence of the known 26 sporadic simple groups originally arising out of many unrelated contexts. Chapter 1 presents a new algorithm: constructing centralizers of 2-central involutions of finite simple groups from indecomposable subgroups of the general linear groups GLn (2) using the representation theoretic and algorithmic methods developed in the first volume. It is shown that 25 sporadic simple groups can be constructed by this algorithm. The smallest Mathieu group M11 can be omitted for theoretical reasons. The algorithm is not restricted to sporadic simple groups as is shown explicitly in Chapters 2 and 10. Here the author describes the constructions of Conway's groups Co3, Co2, Co1, Fischer's groups Fi22, Fi23, Fi~ 4 , Janko's group J4, McLaughlin's group McL, Rudvalis' group Ru, Lyons' group Ly, Suzuki's group Suz, and O'Nan's group ON. Their uniqueness is proved whenever possible. The computational existence proofs are documented in the accompanying DVD. Chapter 16 outlines such theoretically possible constructions for the baby monster and the monster from well determined indecomposable subgroups of GLg (2) and GL10 (2), respectively. The other sporadic groups were constructed in volume I. The present mathematical literature does not contain an accessible proof of the announced classification theorem asserting that there are exactly 26 sporadic simple groups. On the other hand the literature has not paid much attention to R. Brauer's warning (published in 1979) that there may be iI).finitely many sporadic groups. Therefore the author describes Brauer's ideas on a general classification scheme in Chapter 1 and states several related open problems in Chapter 16. Some require new experiments with the author's algorithm. GERHARD MICHLER is an Emeritus Professor of the Institute of Experimental Mathematics at the University of Duisburg-Essen and Adjunct Professor at Cornell University.

NEW MATHEMATICAL MONOGRAPHS

Editorial Board Bela Bollobas William Fulton Anatole Katok Frances Kirwan Peter Sarnak Barry Simon Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/ uk/ series/ sSeries.asp? code=NMM 1 M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups 2 J. B. Garnett and D. E. Marshall Harmonic Measure 3 P. Cohn Free Ideal Rings and Localization in General Rings 4 E. Bombieri and W. Gubler Heights in Diophantine Geometry 5 Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs 6 S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures 7 A. Shlapentokh Hilbert's Tenth Problem 8 G. Michler Theory of Finite Simple Groups I 9 A. Baker and G. Wiistholz Logarithmic Forms and Diophantine Geometry 10 P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds 11 B. Bekka, P. de la Harpe and A. Valette Kazhdan's Property (T) 12 J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory 13 M. Grandis Directed Algebraic Topology

In Memory of

Ruth I. Michler 1967 (Ithaca, NY)-2000 (Boston, MA)

Contents

page xi

Acknowledgements

1

Introduction

1

Simple groups and indecomposable subgroups 12 of GLn(2) Two alternative views on the classification problem 13 1.1 1.2 Simple groups are of infinite representation type, p = 2 17 The algorithm 19 1.3 Documentation of experimental results 28 1.4 1.5 Constructing projective irreducible modular representations 32 Thompson's group order formula revisited 35 1.6

2

Dickson group G2 (3) and related simple groups 38 2.1 Involution centralizers of Dickson's groups G2(q), q odd 39 Fusion and conjugacy classes of even order 44 2.2 The 3-singular conjugacy classes 47 2.3 Janka's characterization of G2(3) 65 2.4 Representatives of conjugacy classes 2.5 73 2.6 Character tables of local subgroups of G2(3) 76

3

Conway's simple group Co 3 3.1 Construction of the involution centralizer Construction of a simple group of Co3 -type 3.2 3.3 Uniqueness proof 3.4 Representatives of conjugacy classes Character tables of local subgroups 3.5

78 80 89 95 125 131

4

Conway's simple group Co 2 4.1 Extensions of the Mathieu group M22 and Aut(M22)

140 141

vii

Contents

viii 4.2 4.3 4.4 4.5 4.6

Construction of the 2-central involution centralizer Construction of Conway's simple group Co 2 On the uniqueness of Co 2 Representatives of conjugacy classes Character tables of local subgroups of Co 2

149 160 167 168 175

5

Fischer's simple group Fi 22 Construction of the 2-central involution centralizers 5.1 Construction of Fischer's simple group Fi 22 5.2 Sketch of a uniqueness proof 5.3 The remaining cases E 1, E 4 and E5 5.4 5.5 Representatives of conjugacy classes Character tables of local subgroups of Fi22 5.6

184 185 195 202 204 206 212

6

Fischer's simple group Fi 23 6.1 Extensions of the Mathieu group M23 6.2 Construction of a 2-central involution centralizer 6.3 Construction of Fischer's simple group Fi23 On the uniqueness of Fi 23 6.4 Representatives of conjugacy classes 6.5 6.6 Character tables of local subgroups of Fi23

222 223 228 245 250 252 259

7

Conway's simple group Co 1 7.1 Extensions of the Mathieu group M24 Construction of the 2-central involution centralizer 7.2 of Co1 Construction of Conway's simple group Co1 7.3 On the uniqueness of Co1 7.4 Representatives of conjugacy classes 7.5 Character tables of local subgroups of Co1 7.6

273 275 284 307 312 315 320

8

Janko's group J4 8.1 Structure of the given centralizer Conjugacy classes and group order 8.2 8.3 Existence and uniqueness proofs 8.4 Other constructions in GL1333(1l) and GL112(2) 8.5 Representatives of conjugacy classes 8.6 Character tables of local subgroups

338 340 343 349 354 356 361

9

Fischer's simple group Fi~ 4 The 2-fold cover of the automorphism group 9.1 Aut(Fi22) 9.2 A semi-simple representation of Fi23 in GLs671 (13)

372 374 377

Contents 9.3

IX

9.4 9.5 9.6 9.7 9.8

Construction of the irreducible subgroup 18 of GLs671(13) 18 is isomorphic to Fischer's simple group Fi; 4 Presentation of 2-central involution centralizer On the uniqueness of Fi; 4 Representatives of conjugacy classes Character tables of local subgroups

392 398 407 416 420 428

10

Tits' 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

group 2F 4 (2)' Construction of the 2-central involution centralizer Fusion Existence proof of Tits' simple group inside GL 26 (73) Group order The 3-, 5- and 13-singular conjugacy classes Uniqueness proof Representatives of conjugacy classes Character tables of local subgroups

461 462 471 475 478 482 487 490 494

11

McLaughlin's group Mel 11.1 Construction of the 2-central involution centralizer 11.2 Structure of the given centralizer H = 2A8 11.3 Existence and uniqueness proof 11.4 Representatives of conjugacy classes 11.5 Character tables of local subgroups

500 501 505 508 515 517

12

Rudvalis' group Ru 12.1 Construction of the 2-central involution centralizer 12.2 Construction of a simple group of Ru-type 12.3 Fusion 12.4 Uniqueness proof 12.5 Representatives of conjugacy classes 12.6 Character tables of local subgroups

519 520 527 531 541 545 550

13

Lyons' group Ly 13.1 Structure of the given centralizer 13.2 Conjugacy classes of elements of even order 13.3 Conjugacy classes of p-singular elements, p E {3,5,7,11} 13.4 Group order 13.5 Existence and uniqueness proofs 13.6 Representatives of conjugacy classes 13.7 Character tables of local subgroups

559 560 564 567 580 583 591 598

X

Contents

14

Suzuki's group Suz 14.1 The centralizer of a 2-central involution 14.2 Even conjugacy classes and group order 14.3 Existence proof of Suz inside GL143 (13) 14.4 Uniqueness proof 14.5 Representatives of conjugacy classes 14.6 Character tables of local subgroups

611 612 615 632 636 652 660

15

O'Nan's group ON 15.1 The centralizer of a 2-central involution 15.2 Fusion 15.3 3-singular classes 15.4 Embedding Janko's group J 1 into ON-type groups 15.5 Existence and uniqueness proof 15.6 Local subgroups, fusion and character table 15.7 Representatives of conjugacy classes 15.8 Character tables of local subgroups

672 674 675 679 682 685 689 692 696

16

Concluding remarks and open problems 16.1 On the monster and the baby monster 16.2 Uniqueness problems 16.3 Is there a 27th sporadic simple group? 16.4 Is there a general classification scheme?

700 701 709 713 715

Appendix: Table of contents of the accompanying DVD A.1 Folder DVD.l: Pdf files of quoted tables A.2 Folder DVD.2: MAGMA files of generating matrices and permutations References Index

720 720 722 725 732

Acknowledgements

The computational work described in this book was performed at the Institute of Experimental Mathematics of Essen University between 1989 and 2003 and at Cornell University since 2004. The recent research with H. Kim on the large Conway and Fischer groups described in this book has been supported by the grant NFS/SCREMS DMS-0532/06. The author owes special thanks to his coauthors of various joint articles on some sporadic simple groups dealt with in this book: H. Kim (Yale University), M. Kratzer (Bayerische Staatsbibliothek, Munich), W. Lempken (Essen), A. Previtali (Como), R. Staszewski (Essen), K. Waki (Hirasaki), L. Wang (Peking University) and M. Weller (Essen). The author also thanks his former secretary, B. Hasel at Essen University, for typing some parts of the manuscript. The complex tables in some of the chapters were printed automatically by using a special program developed by M. Weller, P. Young and H. Kim. The layout of the entire manuscript and the construction of the table of contents, the references and index were carried out by B. Chan and Professor K. Dennis of Cornell University. The accompanying DVD was produced with the help of H. Kim and M. Weller. The author is greatly indebted to all of them for their technical support. The author thanks the Department of Mathematics of Cornell University for its great hospitality during his stay as a visiting and adjunct professor since 2003. Finally, he would like to thank Roger Astley at Cambridge University Press for his constant assistance and the copyeditor ·for many suggestions that improved the presentation of the book. The entire work carried out in Germany and in the United States benefitted greatly from the strong support and understanding of my wife, Waltraud.

xi

Introduction

This is the second and last volume of the author's introduction to the representation theoretic and algorithmic theory of abstract finite simple groups. In particular, it yields the theoretical and algorithmic background for uniform existence and uniform uniqueness proofs of the (known) sporadic simple groups. A finite simple group G is called sporadic if it is not isomorphic to an alternating group An or a finite simple group of Lie type described in R. W. Carter's book [13]. The theoretical results and algorithms presented in the first volume [92] and here hold in general. They are not restricted to sporadic groups at all as will be shown again in this volume. Many results on abstract finite simple groups are inspired by the celebrated Brauer-Fowler Theorem. It asserts that there are only finitely many simple groups G which possess an involution z # l such that its centralizer Cc (z) is isomorphic to a given group H of even order. But it does not give any hint of how to find such a group H without knowing at least one of the simple groups G. In particular, most of the known sporadic simple groups G were not discovered by a construction from a given centralizer H. In his survey article [37], p. 71, D. Gorenstein wrote in 1979: "Much of the excitement generated by the developments in simple group theory over the last 20 years can be directly attributed to the discovery of over 20 sporadic new simple groups ... The existence of these strange objects ... revealed the richness of the subject and lent an air of mystery to the nature of simple groups." After a brief survey about the "presently known" 26 sporadic simple groups he remarks: "There you have the 26 beautiful enigmatic sporadic groups with Janka's fourth group and the Fischer monster still waiting to be born. Arising out of so many unrelated contexts, is it yet possible that there is a single, coherent explanation for their existence? If so, it will require some new 1

2

Introduction

vision, seemingly beyond the capabilities of the present generation, to discover it." It is one purpose of this second volume to answer Gorenstein's question using the representation theoretic and algorithmic methods developed in the first volume [92]. This solution is achieved by means of the author's Algorithm 1.3.8 and its iterated version 1.3.15 described in Chapter 1. They construct centralizers H of 2-central involutions z of certain finite simple groups G from indecomposable subgroups T of the general linear groups GLn (2). By Definition 1.3.5 such a subgroup T is called indecomposable if the natural vector space V = Fn is an indecomposable FT-module over the prime field F = GF(2) of characteristic 2. Irreducible subgroups T of GLn (2) are defined similarly. In general His not uniquely determined by T. Several steps of Algorithm 7.4.8 of [92] have been incorporated into Algorithm 1.3.8. Thus it is possible to build the simple target groups G from a previously constructed centralizer H provided all its conditions can be satisfied in the course of the construction. This algorithm constructs (in theory) all simple groups G whose Sylow 2-subgroups S have a non-cyclic elementary abelian normal subgroup A of order 2n such that Na(A)/Ca(A) ~ T:::; GLn(2) and the centralizer Ca(A) is either A or an iterated FT-module extension. In view of Theorem 4.8.5 of [92] and Remark 1.3.14, these requirements are not serious restrictions. It is shown in this book that these conditions are satisfied by all known sporadic groups. By Kondo's work [77] they hold for all alternating groups A4k+r with k ~ 2 and O : 0. Therefore we prove in this section a classical theorem due to D. G. Higman [50] asserting that FG has only finitely many non-isomorphic indecomposable modules if and only if the Sylow p-subgroups of G are cyclic. In particular, if p = 2 then Corollary 1.4.18 of [92] implies that the group algebra FG of a finite simple group G is of infinite representation type in the sense of the following definition. Definition 1.2.1 The finite-dimensional algebra A over the field K is of finite representation type if there are only finitely many non-isomorphic indecomposable A-modules. Otherwise, A is said to be of infinite representation type.

':'lie proof of the following subsidiary result is due to Assem et al. [6], pp. 176-8. In [50] Higman proves a similar result for odd-dimensional modules. Lemma 1.2.2 Let F be a field of characteristic p > 0. Let P be an elementary abelian p-group of order p 2 . Then for each natural number d > 0 the group algebra F P has a 2d-dimensional indecomposable F Pmodule. In particular, F P is of infinite representation type.

Proof Let P = (x, y), X = (x) and Y = (y). Then FX ~ FY ~ F[T]/(TP), where F[T] denotes the polynomial ring in the indeterminate

18

Simple groups and indecomposable subgroups of GLn (2)

T, and (TP) denotes the ideal of F[T] generated by TP. This follows at once from the fact that F X is a local uniserial algebra in the sense of Definition 3.17.1 of [92] with Jacobson radical J(FX) = (1 - x)FX having index of nilpotency p. The map ¢ : 1- x ---. T induces an isomorphism between FX and F[T]/(TP). Let T1 = 1-x and T2 = 1-y. Then FP ~ F[T1,T2 ]/(Tf ,1]'), where B = F[T1,T2] denotes the polynomial ring in the two commuting variables T1 and T2, and W = (Tf, T!) is its ideal generated by Tf and Tf. Clearly, W :S; V = (T1, T2 )2. Hence there is an epimorphism T : B---. A= B/V. Thus each A-module is a B-module annihilated by V. For each integer d > 0 let Rd = F[T]/(TP) and let Wd be the direct sum of two copies of the F[T]-module Rd. So Wd = {(n1,n2) I ni E Rd, i = 1, 2}. On Wd define an A-module action by (n1, n2)T1 = (0, n1T) and (n 1, n2)T2 = (0, n1). This action is well defined because T[ T:J and T1T2 annihilate Wd. Hence Wd is an A-module of dimension 2d. Each endomorphism a E EndF[T] (Wd) is represented by a 2 x 2-matrix given by

where all ai,j E EndF[T] (Rd)- Using the defined A-module structure on Wd it is easy to verify that a E EndA (Wd) if and only if

M a = ( a11 0 ) a2:1 a1,1 · Hence a 2 = a if and, only if M; = Ma, which is equivalent to ar,1 = a1,1 E Endp[r](Rd) ~ F[T]/(TP). Since F[T]/(TP) is a local ring, its identity element is the unique non-zero idempotent. Hence Wd is an indecomposable A-module. Thus it is also an indecomposable F Pmodule. D

Theorem 1.2.3 (D. G. Higman) Let F be a field of characteristic p > 0. The group algebra FG of the finite group G is of finite representation type if and only if the Sylow p-subgroups of G are cyclic. Proof Let P be a Sylow p-subgroup of G and N = Na(P). Suppose that P is not cyclic. Let (P) be the Frattini subgroup of P. Then P/(P) is elementary abelian and non-cyclic. Thus it has an epimorphic image E which is elementary abelian of order p 2. Hence E, and therefore P, have infinitely many non-isomorphic indecomposable F Pmodules ~ with strictly increasing dimensions dimp~ for i = 1, 2, ...

1. 3 The algorithm

19

by Lemma 1.2.2. Since each indecomposable FG-module is P-projective by Lemma 3.10.3(b) of [92] it follows from the Krull~Schmidt Theorem and Theorem 3.5.11 of [92] that each induced FG-module Rf contains an indecomposable direct summand Mi such that R,;, is isomorphic to a direct sum.mand of Mi JP. Thus G is of infinite representation type. Suppose that P is cyclic. The number of projective indecomposable FG-modules equals the number of p-regular conjugacy classes of G. Theorem 3.10.5 of [92] due to J. A. Green states that each non-projective indecomposable FG-module M has a vertex Vanda source S which is uniquely determined up to Ne (V)-conjugation. As F P is uniserial FV is also uniserial. Thus it has exactly IVI indecomposable FV-modules. Hence dimp(S) :S: IVI :S: IPI implies that dimp(M) :S: IVIIG: VI. So G is of finite representation type. Further important results on finite groups of infinite representation type are due to Crawley-Boevey [15] and Erdmann [25]. D

1.3 The algorithm The main purpose of this section is to describe an algorithm which constructs the centralizer H = Cc (z) of a 2-central involution z of some finite simple groups G from a given indecomposable subgroup U of some general linear group GLn(2), provided all its conditions can be satisfied in the course of the construction. In this book only finite simple groups G are considered whose Sylow 2-subgroups S have a non-cyclic elementary abelian characteristic subgroup A. In view of the classification of the simple groups with dihedral [36] and semi-dihedral [2] Sylow 2-subgroups and the Bender-Suzuki Theorem [7], [126], this is not a serious restriction by Theorem 4.8.5 of [92]. It asserts that a fixed Sylow 2-subgroup S of any other simple group G contains a maximal non-cyclic characteristic elementary abelian subgroup A such that

(*)

G = (Nc(A), Cc(x) Ix E J(N)),

where I(N) denotes a set of representatives of the conjugacy classes of involutions of N = N c (A). This result provides the theoretical background for Algorithm 7.4.6 of [92] constructing simple groups G having a 2-central involution z with a given centralizer H = Cc (z). It constructs the subgroup G 1 = (H,Nc(A)) of G from the given centralizer H.

20

Simple groups and indecomposable subgroups of GLn (2)

Often G 1 itself turns out to be a simple group. However, G1 may be a proper subgroup of a larger simple subgroup of G; see Theorem 8.6.6 of [92]. In fact, the following theorem of M. Suzuki [123] provides an infinite family of finite simple groups G having a unique conjugacy class of involutions z such that S = Cc(z) is a Sylow 2-subgroup whose non-cyclic center Z(S) = A is the unique maximal characteristic subgroup of S and there is another Sylow 2-subgroup S1 of G such that G1 = (Nc(A),S1 ) = Nc(B) < G, where B > A is a maximal elementary abelian normal subgroup of S. In particular, G 1 is not simple.

Theorem 1.3.1 (M. Suzuki) Let G be a finite simple group such that its Sylow 2-subgroups S have non-cyclic centers Z(S) and are not disjoint from their conjugates in G. If the centralizer Cc (x) of each involution x E G has a normal Sylow 2-subgroup then G "" PSL 3 (2n), n?. 2.

Proof The assertion follows from M. Suzuki's Theorem 8 [123].

D

In many examples dealt with in [92] and in this book a maximal characteristic subgroup A is a maximal elementary abelian normal subgroup of S. It need not be the only maximal elementary abelian normal subgroup of S; see Lemma 11.1.2 of [92]. Furthermore, a maximal characteristic elementary abelian subgroup A of a Sylow 2-subgroup S of a finite simple group G is not necessarily a maximal elementary abelian normal subgroup of S, as the above example PSL 3 (4) shows. Other examples are provided by the simple groups studied in the following two chapters. Therefore we prove an analog of (*) also for this situation.

Lemma 1.3.2 Let G be a finite simple group having no strongly embedded subgroups. Suppose that A cl 1 is a maximal non-cyclic elementary abelian characteristic subgroup of the Sylow 2-subgroup S of the finite simple group G such that A is not a maximal elementary abelian normal subgroup of S. Then A is the intersection of finitely many maximal elementary abelian normal subgroups Bi of S which form an Aut(S)-orbit, i = 1,2, .. ,k. Let E = Nc(B1) be of maximal order of the normalizers of the Bi. If Nc(S)::; Ethen G = (Nc(B1),Cc(x) Ix E J(E)), where I(E) denotes a set of representatives of the conjugacy classes of involutions of E = N c (B1).

1.3 The algorithm

21

Proof Let B be a maximal elementary abelian normal subgroup of S containing A. Let D be the intersection of the orbit BAut(S), where Aut(S) denotes the automorphism group of S. Then Dis a characteristic elementary abelian subgroup of S containing the maximal characteristic elementary abelian normal subgroup A of S. Hence D = A. Let U = (E, Ca(x) I x E J(E)). Let a be any involution of U. Then there is a Sylow 2-subgroup T of U such that a E T. By Sylow's Theorem Tu = S :S E for some u E U. Hence there is an e E E such that aue = x E J(E). Thus Ca(a) = [Ca(x)jtueJ- 1 :SU. As Na(S) :SE, by hypothesis Theorem 4.8.4 of [92] implies that U = G, because G does not have any strongly embedded subgroup. D Remark 1.3.3 If a maximal non-cyclic elementary abelian normal subgroup of S is also a maximal elementary abelian normal subgroup of S then the condition Na (S) :S Na (A) is automatically satisfied. Otherwise it has to be checked. In the applications given in the following two chapters, S is self-normalizing in G. So the conditions of Lemma 1.3.2 hold again.

In view of the above remarks and Lemma 1.3.2, we introduce the following notation, which is kept throughout this section. Notation 1.3.4 Let A be a maximal non-cyclic elementary abelian normal subgroup of S of a finite simple group G without strongly embedded subgroups such that Na (A) is maximal among the normalizers of maximal elementary abelian normal subgroups of Sand Nc(S) :S Nc(A). Let IAI -= 2n, C = Cc(A), E = Nc(A), T = E/C and let F = GF(2) be the field with two elements. Then T is isomorphic to a subgroup of GLn (2). Furthermore, the FT-module structure of A is uniquely determined by the matrices of the generators t; of T with respect to a given basis B of the vector space A over F.

The FT-module structure of A= Fn varies. Definition 1.3.5 A subgroup T of GLn (2) is called indecomposable if the natural F-vector space V of dimension n is an indecomposable FTmodule. If V is an irreducible FT-module then T is called irreducible subgroup

There are many finite simple groups G having a unique non-cyclic elementary abelian normal subgroup A which is an irreducible FT-module,

22

Simple groups and indecomposable subgroups of GLn (2)

e.g. the alternating groups G = A 4 k, k 2= 3 by Proposition 8.6.8 of [92] and Kondo [77]. The following example of an irreducible subgroup of GL 3 (2) showed up in Chapter 9 of [92] dealing with Janka's sporadic simple group J1. Example 1.3.6 GL 3 (2) has an irreducible subgroup T of order 21. It is generated by the following two matrices: 010) e= ( 001 , 110

100) d= ( 001 . 011

There are finite simple groups G having a Sylow 2-subgroup S such that S has a maximal non-cyclic elementary abelian normal subgroup A which is an indecomposable but not an irreducible FT-module. One example is the Tits simple group G = 2F 4 (2)' dealt with in Chapter 10. Example 1.3. 7 Lemma 10.1.1 provides three generating matrices in GL 5 (2) generating an indecomposable subgroup T of order 26 · 3. From the generating matrices of T it can immediately be seen that V = F 5 is an indecomposable FT-module with three composition factors of dimensions 2, 1 and 2. The main purpose of this section is to describe an algorithm which constructs from a given indecomposable subgroup T of GLn (2) simple groups G of the form G = (H, E) having a Sylow 2-subgroup S with a maximal elementary abelian normal subgroup A of order 2n such that T ~ E/Cc(A) of GLn(2), where E = Nc(A) and H = Cc(z) for an involution in the center Z(S) of S. It is now stated for the important special case that Ca (A) = A. Algorithm 1.3.8 (Michler) Let F = GF(2). Let T be an indecomposable subgroup of GLn (2) acting on V = Fn by matrix multiplication. Step 1 Calculate a faithful permutation representation PT of T and a finite presentation T = (till :S: i :S: r) with set R(T) of defining relations. Step 2 Compute all FT-module extension groups E of T by V by means of Holt's Algorithm /52}. Determine a complete set 6 of nonisomorphic extension groups E by means of the Cannon-Holt Algorithm {12}.

23

1.3 The algorithm

Step 3 Let E E 6. From the given presentation of E determine a faithful permutation representation PE of E. Using it and Kratzer's Algorithm 5. 3.18 of /92] calculate a complete system of representatives of all the conjugacy classes of E. Step 4 Let z =/. 1 be a 2-central involution of E such that D = CE (z) < E. Check that CE (V) = V and that V is a maximal elementary abelian normal subgroup of any fixed Sylow 2-subgroup S of

E. If it is not maximal, the algorithm terminates. Step 5 Construct a group H > D with the following properties:

(a) z belongs to the center Z(H) of H; (b) the index IH: DI is odd; (c) the normalizer NH(V) = D = CE(z). If no such H exists the algorithm terminates. Otherwise, apply for each constructed group H the following steps of Algorithm 1.4.6 of /92]. By Step 5(c) it may be assumed from now on that D = H n E. Step 6 Compute the character tables of the groups D, H and E, and the fusion patterns of the conjugacy classes of D in H and in E, respectively. Using Kratzer's Algorithm 7.3.10 of /92] calculate the finite set of compatible pairs

E = {(11,w)

E

charc(H) x charc(E) I z;ID

= WID

E

fcharc(D)}.

Step 7 Let (11, w) E :E be a compatible pair of faithful characters of H and E of minimal degree v(l) = n not yet dealt with. Let F be a finite splitting field of characteristic p > 0 for the irreducible constituents of the characters of the compatible pair (11, w) such that all their irreducible constituents belong to p-blocks of defect zero. Then do the following steps:

(a) Construct the faithful semi-simple F H -module m corresponding to II and the faithful FE-module W corresponding tow.

(b) Identify H and E with their isomorphic images in GLn(F) under their representations afforded by m and W, respectively. Determine by means of Theorem 7.2.2 of /92] a double coset

24

Simple groups and indecomposable subgroups of GLn (2) decomposition

u s

CcLn(F)(D)

=

CcLn(F)(H)T;CcL,,(F)(E)

i=l

of the centralizer CcLn (F) (D) of D in GLn (F). For each double cos et representative T;, let G; = (H, T;- 1 ET;), 1 S: i S: s. Then Cai (z) 2= H for all i. (c) For each i E {1, 2, ... , s} compute the orders of some suitable elements of G; and use this information to check whether a Sylow 2-subgroup of G; may be isomorphic to S. If the Sylow 2-subgroups of all constructed groups G; are not isomorphic to S, then the algorithm ends. Otherwise, let G = (H, T- 1 ET) be any of the groups G; having fulfilled the Sylow 2-group test of Step 7(c). Then the canonical n-dimensional vector space M = pn is an irreducible PG-module with restriction MIH ~ IJJ.

Step 8 Since MIH has a proper non-zero F H -submodule U, one can construct a permutation representation 1r : G ---+ Sm into the symmetric group on m letters with stabilizer fI =2'. H and a strong base and generating set for 1r(G) by Theorem 6.2.1 and Remark 5.2.12 of /92}, respectively. Step 9 Check by means of Proposition 5.2.14 of /92} that 1r has stabilizer fI = H. If so, then 7r: G---+ Sm is faithful, and IGI = IHI· m. In particular, G has a 2-central involution t with Cc (t) ~ H and one can calculate a presentation of G by means of Theorem 5. 2.18 of /92}. Step 10 Using the faithful permutation representation 1r : G ---+ Sm of G and Kratzer's Algorithm 5.3.18 of {92} compute a complete set of representatives of all the conjugacy classes of G. Step 11 Compute a character table of G by means of MAGMA or the algorithms described in /92}. If no conjugacy class of G is in the kernel of some irreducible character then G is a simple group. Otherwise, the concrete character table of G provides matrix generators of a proper normal subgroup of G.

1. 3 The algorithm

25

Remark 1.3.9 Keep the notation of Algorithm 1.3.8. Sometimes it will not be possible to construct the permutation representation (lH )c for technical reasons. Suppose one is able to build a permutation representation (lu of another subgroup U of G having a smaller index IG: UI than H. Then one can calculate the exact order of H = Cc ( z) of the constructed matrix group G by means of Proposition 2.6.6 of [92] and the number of fixed points of the action of z on (lu )c. The construction of Thompson's sporadic group Th in Chapter 12 of [92] provides such an example.

f

The following example shows how to construct the centralizer H of a 2-central involution of Janko's smallest sporadic group J1 from the irreducible subgroup T of GL 3 (2) defined in Example 1.3.6. The group H is used in Chapter 9 of [92] to construct Janko's smallest sporadic group J1 by means of Steps 5-11 of Algorithm 1.3.8. Example 1.3.10 Let E be the split extension E = V : T of the irreducible subgroup T of GL 3 (2) defined in Example 1.3.6 by its natural vector space V of dimension 3 over F = GF(2). As T is isomorphic to the Frobenius group of order 21, the extension E has an involution z EV such that D = CE(z) ~ A4. Since the alternating group A4 has index 5 in A 5 , Step 5 of Algorithm 1.3.8 can be immediately performed after constructing D = CE(z). Thus we have the following diagram:

J1 can be constructed now as in Chapter 9 of [92].

Whenever a simple group G has been constructed from a given centralizer H = Cc ( z) of a 2-central involution by means of Algorithm 7 .4.8 of [92] and a Sylow 2-subgroup S of G has. a self-centralizing maximal elementary abelian normal subgroup A of order 2n then it can also be constructed from a given indecomposable subgroup T = Nc(A)/A E GLn (2) by means of Algorithm 1.3.8. The following remarks present a survey describing the sporadic groups dealt with in [92] which meet

26

Simple groups and indecomposable subgroups of GLn (2)

these requirements. Of course, the Mathieu group M11 can be neglected because its Sylow 2-subgroups are semi-dihedral. Remark 1.3.11 By Theorem 8.1.9 of [92] the Mathieu group G = M12 has a Sylow 2-subgroup S having a self-centralizing elementary abelian normal subgroup A such that T 12 = Nc(A)/A 9' S4 is an indecomposable subgroup of GL 3 (2). The Mathieu groups M2 2, M23 and M24 have Sylow 2-subgroups 822, 8 23 and 8 24 containing self-centralizing elementary abelian normal subgroups A 22 , A 23 and A 24 of dimensions 4, 4 and 6 over F = GF(2) such that their normalizer quotient groups T22 9' 8 5 , T23 9' (3 x A5): 2 and T24 9' 3S6 are irreducible subgroups of GL 4 (2), GL 4(2) and GL5 (2), respectively. All these statements follow from Proposition 8.2.4 of [92]. Hence these four Mathieu groups can be constructed by means of Algorithm 1.3.8 from the four subgroups Ti stated before. Remark 1.3.12 Janko's group J1 , Held's group He, Harada's group Ha and Thompson's group Th can be constructed by means of Algorithm 1.3.8 from well defined irreducible subgroups of GL3 (2), GL5 (2), GL 5 (2) and GL 5 (2) by Theorems 9.3.2, 8.3.2, 11.2.2 and 12.3.1 of [92], respectively. Definition 1.3.13 Let T be an indecomposable subgroup of a general linear group GLn (2) such that some step of Algorithm 1.3.8 cannot be performed by theoretical reasons. Hence there is no simple group G having a Sylow 2-subgroup 8 containing a self-centralizing non-cyclic maximal elementary abelian normal subgroup A such that Nc(A)/Cc(A) 9' T. In such a case, we say that the application of Algorithm 1.3.8 to T fails. Remark 1.3.14 If a Sylow 2-subgroup 8 of a finite simple group G has a unique maximal non-cyclic elementary abelian normal subgroup A of order 4 then A cannot be self-centralizing because 8 is not a dihedral group by Theorem 4.8.3 of [92]. The sporadic groups J 2, J3 of Janko have such a Sylow 2-subgroup. Let GE {J 2, J 3}, C = Cc(A) and E = N 0 (A). Then Propositions 8.5.1 and 8.5.2 of [92] assert that the center Z(C) = A, V = C/A is elementary abelian of order 16. Hence E/A is an FT-module extension of an indecomposable subgroup T = E/C E GL 4 (2) by V. Therefore Eis an iterated FT-module extension of E/A by A.

1.3 The algorithm

27

In all these cases Steps 2 and 4 of Algorithm 1.3.8 have to be modified by now allowing E to be an iterated FT-module extension. All other steps remain essentially unchanged. Algorithm 1.3.15 Let T be an indecomposable subgroup of GLm (2) acting faithfully on V = pm by matrix multiplication where F = GF(2). Step 1 Construct the split extension E1 of T by V and a faithful permutation representation P E1 of E 1 . Determine then a presentation

of E 1 in terms of generators and a set R(E1 ) of defining relations. Step 2 Let n be a positive integer for which there exists an indecom-

posable F E 1 -module A of dimension n on which the normal subgroup V of E 1 acts trivially, and the following constructions can be per-

formed. Using PE1 and R(E1) construct all FE1 -module extensions E of E 1 by A by means of Holt's Algorithm [54} and MAGMA. Determine a complete set 'I'n of non-isomorphic iterated extension groups E by means of the Cannon~Holt Algorithm [12}. For each E E 'I'n determine a faithful permutation representation PE and check the following conditions:

(a) A is an elementary abelian normal subgroup in any Sylow 2subgroup S of E and A contains a non-cyclic characteristic subgroup of S; (b) C = CE (A) is a maximal special or homocyclic normal subgroup of E with center Z(C) = A or O[Z(C)] = {c E C[c2 = 1} = A, respectively; (c) C / A ~ V and E / C is isomorphic to an indecomposable subgroup of GLn (2). If no such integer n exists the Algorithm terminates. Otherwise, let 11:n be the set of all extensions E E 'I'n satisfying these conditions. Step 3 Let E E 11:n. Let PE be its faithful permutation representation.

Using it and Kratzer's Algorithm 5.3.18 of {92} calculate a complete system of representatives of all the conjugacy classes of E. Check that E has a 2-central involution z -:/- 1 such that D = CE(z) < E. Let Jn be the set of all extension groups En for which such an involution exists. If Jn is empty the algorithm terminates.

28

Simple groups and indecomposable subgroups of GLn (2) Otherwise, apply Steps 5-11 of Algorithm 1. 3. 8. If they can all be successfully performed, then it constructs a finite simple group G having a Sylow 2-subgroup S with an elementary abelian normal subgroup A such that Ca(A)/A ~ V, Na(A)/Ca(A) ~ T and G = (Ca(z),Na(A)).

Remark 1.3.16 The sporadic group HS of Higman-Sims can be constructed by Algorithm 1.3.15 by Lemma 10.1.2 and Proposition 10.2.2 of [92]. In this example choose T = GL 3 (2), m = n = 3. Then V and A are isomorphic irreducible FT-modules. Furthermore, E is the split extension of C = CE (A) by T, C is a direct product of three copies of a cyclic group of order 4, and fl( C) = A. Remark 1.3.17 Keep the notation of Algorithm 1.3.15. If the Sylow 2-subgroup of the chosen indecomposable subgroup T of GLm (2) is not cyclic then by Theorem 1.2.3 there are infinitely many integers n for which there is an indecomposable subgroup T1 of GLn (2) which is an epimorphic image of T. In particular, it is not possible to determine computationally all integers n for which all conditions of Step 2 of Algorithm 1. 3 .15 can be verified.

1.4 Documentation of experimental results

In order to do specific calculations in a finite group G with a given set {gk I1 :::; k :; n} of generators, we always construct a faithful permutation representation PG of G with a stabilizer U = (uj I 1 :::; j :; m) and an isomorphism T : G ----+ PG. The generators Uj of the stabilizer U are stated as short words in the generators gi of G. In general, MAGMA uses random algorithms when it is applied to PG. Its results are often described by finitely many permutations 1r E PG. In order to document them we have to be able to write the elements w = T- 1 ( 7r) E G as short words in the given generators gk of G. For small groups G the MAGMA command InverseWordMap(PG) works well. But for the large groups studied in this book its output would often be so large that it cannot be printed. Therefore we now describe an algorithm which is due to the author's former collaborator R. Staszewski (Essen) and his student H. Kim (Cornell University), whose senior thesis [73] contains the improved version presented here. It has been used efficiently in all large-scale calculations described in [72], [74] and the following chapters. In particular, Kim's implementation can be loaded into any MAGMA session.

1.4 Documentation of experimental results

29

Let G be a finite group with faithful permutation representation T : G --+ PG. Let fo1, 92, ... , 9n} be a fixed set of generators 9k of G ordered by their indices. Let T( G) of G be the complete word tree of G; see Definition 5.3.10 of [92]. The length £(w) of a word w E T(G) is the length of the path in T(G) of the initial point 9k of w tow. Let W 0 = {le}, the identity element of G. For each integer i 2': 1 let W; = {w E T(G)j£(w) = i}. In particular, W 1 = {gkjl s; ks; n}. The isomorphism T : G --+ PG induces an evaluation map ¢ : T(G) --+ PG defined by rp(gk) = T(gk) for 1 s; k s; n and its multiplicative extension. For each i 2': 1 define

PV; = {rp(wYJw E W;, 1 s; rs; order[rp(w)] - l}. Algorithm 1.4.1 (Kim-Staszewski) Let G be a finite group with faithful permutation representation T : G --+ PG. Let fo1, 92, ... , 9n} be a fixed set of generators 9k of G ordered by their indices. Let T( G) be the complete word tree of G. Let PU be a subgroup of PG. The following algorithms construct a set of generators Uj of the subgroup U = T- 1 (PU) in G such that all its generators Uj are short words in the given generators 9k of G. Let i 2". 1 be an integer such that PU i- PU;= (PUnPV;). Step 1 Let i 2". 1 be an integer such that PU i- PU; = (PU n PV;). Using the faithful permutation representation PG and MAGMA determine a minimal generating set PR; of PU;. Step 2 Calculate the set W;+ 1 of all words w E T( G) of length£( w) = i + 1, P¼+ 1 and PUi+l · Using membership tests with MAGMA check whether PU = PU;+l · If so, the algorithm ends after having determined a minimal generating set PR;+1 of PU. Otherwise, go to Step 1. Since PU is finite, the algorithm terminates and

where the Uj are powers of short words in the given generators 9k ofG. In particular, U = (ujjl s; j s; u). Remark 1.4.2 H. Kim's implementation of Algorithm 1.4.1 is efficient because it avoids substantial memory problems by storing the words w E W; in the word tree T( G) of G as sequences of indices of the generators 9k in the order they occur in the words w and not as the

30

Simple groups and indecomposable subgroups of GLn (2)

actual permutations T(w) in PG, e.g. the word w = gf · g~ · g 1 is stored as the sequence [1, 1, 1, 2, 2, l]. Furthermore, doing the membership test cp( w Y E PUi for all powers of cp( w) of a new word w in l¼ at the same time speeds up the generation process of PU. In fact, the algorithm often finds generators of PU which are powers of cp( w) of a word w E Wi although cp(w) does not belong to PU.

Algorithm 1.4.3 (Kim-Staszewski) Applying Algorithm 1.4.1 to cyclic subgroups generated by a permutation it can sometimes find a short word of an element g of a finite permutation group G = (gk 11 :S k :S n) in the given generators gk of G. If it fails then often it is helpful to calculate generators Uj of suitable subgroups U of G containing g and then try to get a word for g in the new generators Uj of U by means of Algorithm 1.4.1. This application of Algorithm 1.4.1 is called LookupWord(G,g) in Kim's senior thesis /74}. It improves an earlier version due to R. Staszewski. Remark 1.4.4 Algorithm 1.4.1 is of great help when it is necessary to document the stabilizer U of a constructed permutation representation PG of a finite group G. The following example will be used in Section 1.5. Applying the MAGMA command LowindexSubgroups (PG, n) MAGMA returns all conjugacy classes of subgroups PT of PG of index IPG: PTI :Sn. For each of these subgroups T of G Algorithm 1.4.1 finds a finite set of generators Uj as short words in the given generators 9k of G. Thus all new calculations with the permutation representation (lr )c can be documented. Remark 1.4.5 (P. Young) In the course of the calculations described in this book we have to do a number of demanding coset enumerations: Since MAGMA's command CosetAction(G, U) is very restricted, Paul Young and H. Kim developed a command MyCosetAction(G, U: maxsize: = n), which is a batch of standard MAGMA functions for coset enumerations. It allows us to increase the size of the coset table to a large extent. The size is regulated by the parameter n. Step 2 of Algorithm 1.3.8 requires the construction of extensions of a given group T by an elementary abelian group V. Often this extension splits. In order to be able to quote all the relations in the given generators of T and Vin a fixed set R(E) of defining relations of E in a uniform

31

1.4 Documentation of experimental results

way, we restate now the following definitions and well known subsidiary results from H. Kim's and the author's article [72]. Definition 1.4.6 Let F be a prime field of characteristic p > 0. Let G be a subgroup of GLn (F). Let V = pn be the canonical n-dimensional vector space. Then V is a right FG-module of the group algebra FG of G over F with respect to the multiplication (v · g) defined by the product of the row vector v EV times the matrix g E G. For each matrix g E G let g* = [g- 1JI' be the transpose of the inverse matrix g- 1 of g. Then by Definition 3.4.4 of [92] V becomes a right FG-module under the multiplication [v,g] := [g* · (vT)jT, where· denotes the product of the matrix g* times column vector vT. This right FG-module V* is called the dual FG-module of V. The semidirect product V ~ G of V and G consists of all pairs (v, g), v E V, g E G. Its multiplication * is defined by

(v1,g1) * (v2,g2): = (v1 + [v2,gi],g1g2)

for vi EV and gi E G,

V*g= (v,l)*(O,g)=(v+[0,1],1*g)=(v,g)

for

VE V, g E G.

Lemma 1.4.7 Let G = (xill Si Sr) be a finitely generated subgroup of GLn(F) with the set of defining relations R(G) in the given generators Xi. Let V = pn be the canonical n-dimensional vector space with n }. standard basis {ej 11 j Let R1 (V ~ G) be the set of all relations:

s s

e1] Let R2 (V

~

and

= 1

ek * ej = ej * ek

for all

1

s j, k s n.

G) be the set of all relations:

where (a(i)ji,a(i)jz, ... ,a(i)jn) =[(xi)*· ((ejf)f. Then

R(V

~

G) = R1(V

~

G) U R2(V

~

G) U R(G).

Proof Clearly {xill Si S r},{ejll S j Sn} is a set of generators of the semidirect product V ~ G of V and G. By Definition 1.4.6 the multiplication * agrees in G = {(O, g) lg E G} with the given multiplication of G. Hence R(G) is a defining set of relations of the complement G = (xill s i s r) of Vin R(V ~ G). As Vis elementary abelian of order !Fin, R 1(V ~ G) yields a presentation of the normal subgroup V of

32

Simple groups and indecomposable subgroups of GLn (2)

R(V > 0 from its given character table. In this section we describe an algorithm for such a construction using Theorem 5.6.5 of [92] due to M. Weller and the author. It provides a formula for calculating all the character values Xs (g), g E G of the irreducible constituents Xs of the permutation character (lu )G G with stabilizer U. Here we apply the method of proof of that theorem to an irreducible constituent x of p-defect zero of a permutation character (lu )° of G and get the irreducible matrix representation of G corresponding to x. Based on Weller's stand-alone implementation of Algorithm 5.7.1 of [92], A. Previtali developed in 2004 an implementation of that algorithm which can be loaded into a MAGMA session. In [94] it has been used to find all the character values of the simple O'Nan group ON; see Chapter 14. In his senior thesis [74] H. Kim has modified Weller's and Previtali's programs to get a program for calculating irreducible p-modular matrix representations over suitable finite fields P of characteristic p > 0 corresponding to the irreducible constituents Xs of p-defect zero of a permutation character (lu )° of a finite group G with stabilizer U. It works well provided all values Xs (g), g E G, are rational integers. In that case P can be chosen to be the prime field GP(p). Kim's program uses Previtali's program calculating double coset decompositions of G with respect to its subgroup U and the corresponding adjacency matrices. It applies several steps of Algorithm 5.7.1 of [92].

1. 5 Constructing projective irreducible modular representations 33

In view of all its important applications in this book it is now stated in detail.

Algorithm 1.5.1 (Kim-Michler-Previtali-Weller) Let G be a finite group with faithful permutation representation T : G -----+ PG. Let {g1, 92, ... , Yn} be a fixed set of generators 9k of G ordered by their indices. Let p > 0 be a prime and let x be an irreducible character of G of p-defect zero. Let F be a splitting field of characteristic p for the p-modular irreducible representation V afjorded by x. The following algorithm constructs the FG-module V up to isomorphism provided all its steps can be performed. Step 1 Apply the MAGMA command LowindexSubgroups(PG,n) to PG for a suitable integer n > l and, for each constructed subgroup PUm of PG having index m :::; n, calculate the fusion of its classes in PG and the character table of PUm and the inner product (x, (lpum )°) until MAGMA returns a subgroup PUm such that this inner product is non-zero. If no such index can be found, the algorithm fails. Otherwise let n 0 be the smallest index m such that (X, (lpum )°) f=. 0. Let PU = PUn 0 • Step 2 Using Algorithm 1.4.1 determine a set {Uj I 1 ::; j :::; m} of short generators of U in terms of the generators 9k of G. Determine a right transversal T = {ga I 1 ::; a :::; r} of U in G. Step 3 Compute a complete set of double coset representatives Xk, d

1 ::; k :::; d, of G

= LJ

U Xk U.

k=l

Step 4 Compute the d intersection matrices Dk cosets UxkU of G.

= (p}j) of all double

Step 5 Determine a basis A = {Z 1 , Z2 , ••• , Zt} of the solution space Z(B) of the system of linear equations d

(U)

L (p}j

- PL)qk = 0

k=l

described in Theorem 5.5.6 of [92]. Thus each basis vector Zj = L~=l qjkDk for uniquely determined qjk E (Ql. Step 6 Compute the structure constants zlv of the center Z(B) of the intersection algebra B with respect to the basis A by solving the system

Simple groups and indecomposable subgroups of GLn (2)

34

of linear equations: t

Zu zj

=

L zlv Zv. v=l

To each basis element associate its matrix

Step 7 Compute a basis Ji, h, ... , ft of Z(B) consisting of common eigenvectors Ji of the t pairwise commuting matrices Uj. Then there are t uniquely determined Cis E (Ql such that fi = L~=l CisZs for l ~ i ~ t. For each eigenvector Ji compute a sequence hii, hi2, ... , hit of eigenvalues of U1, U2, ... , Ut, respectively, such that Uj Ji = Ji hij. Then { ei

=

;i

Ji

I1 ~ i ~

t}

is the complete set of all primitive idempotents

ei

of Z(B), where

= L~=l hisCis · In particular, there are uniquely determined aik CC such that

=

½L!=l Cisqik E

qi

d

ei

= LaikDk

for i

= l, 2, ... , t.

k=l

Step 8 For each i E {l, 2, ... , t} compute is the multiplicity of Xi in (lu )°.

mi

= Jdimc(eiB), which

Step 9 For the given g E G compute the integers for l

~

k

~

d.

Step 10 For i = l, 2, ... , t the character values of Xi are computed by the following formula:

l

Xi(g)

d

= -m ~ m kaik· ~ 9

' k=l Step 11 There is a unique index i in i = 1, 2, ... , t such that (x, Xi) -=/=- 0 and so X = Xi· Fork = l, 2, ... , d calculate the first row R1,k of the r x r adjacency matrix Ak corresponding to the intersection matrix Dk by means of Definition 5.1. 'l of /92}. Then d

V

= LaikR1,k k=l

for i

= l, 2, ... , t

1.6 Thompson's group order formula revisited

35

is the first row of the central primitive idempotent E; in the endomorphism ring End pc (W) corresponding to the irreducible constituent X = Xi of W = (luf 0 F. Hence V = E;(W) is the irreducible FGmodule corresponding to the p-modular character x of defect zero.

1.6 Thompson's group order formula revisited

Thompson's group order formula stated in Theorem 4.2.1 of [92] provides an efficient method to calculate the order IGI of a finite group G having at least two distinct conjugacy classes of involutions {uf I 1 :::'. i :::'. t}, t > 1. In fact, IGI can be computed from the character tables of the centralizers Cc (u;) and the fusion patterns of the conjugacy classes of their involutions in G. In all applications described in [92] these centralizers Cc (u;) were so small that the constants of the group order formula could be calculated from their permutation presentations directly without using the character tables. The calculations of the particular constants of Thompson's group order formula of the large sporadic groups existing in the literature are not uniform and lengthy. Therefore we prove a general character formula for these constants in this section. Lemma 1.6.1 Let G be any finite group and let Irrc( G) be the set of all its distinct irreducible characters. Let x, y, z be any three elements ofG. Then the numberp(x,y,z) of pairs {x1,Y1) E (xG xyG)lx 1y1 = z} is given by the following character formula:

IGI

p(x,y,z) = \Cc(x)\ · \Cc(y)\

L

'I/J(x)'I/J(y)'I/J(z-1 )'I/J(l)-1.

7/JEirrc(G)

Proof The assertion follows immediately from Theorem 2.4.13 of [92].

D Definition 1.6.2 Let G be a finite group with at least two distinct conjugacy classes uG, vG of involutions. An element w of G is called strongly (u, v )-real if there is a pair ( x, y) E ( uG x vG) such that xy = w. Remark 1.6.3 With the hypothesis of Definition 1.6.2, Lemma 1.6.1 implies that w E G is strongly (u, v )-real if and only if

p(u,v,w) =

IGI ICc(u)l · ICc(v)I

L 7/JEirrc(G)

'I/J(u)'I/J(v)'I/J(w- 1)'I/J(l)- 1 =/= 0.

36

Simple grnups and indecomposable subgroups of GLn(2)

In particular, all strongly (u, v )-real elements of Gare real by Lemma 2.2.12(a) of [92]. Hence all their character values are real numbers. In particular, the strongly (u, v )-real conjugacy classes can be determined from the character table of the group G. Theorem 1.6.4 Let G be a finite group with at least two distinct conjugacy classes uG, vG of involutions. Let u1 = u, u2 = v, u3, . .. , Ut be representatives of the different conjugacy classes of involutions of G with respective centralizers Ui = Cc ( ui). Suppose that the fusion of all involutions of all centralizers U; in G has been determined and that the character tables of all centralizers and the power maps of their conjugacy classes are known. For each fixed i E {1,2, ... ,t} let {a;,jll S:: j s; r;} and {bi,kll s; ks; s;} be sets of representatives of all conjugacy classes of U; fusing in G with u and v, respectively. In particular,

uc n Ui = af,1 u af,2· .. U af,~i and V

G n Ui == bui i,1

u bui i,2

.. .

u bui i,s·i'

Let {Wn I 1 s; n s; m} be a set of representatives of all real u; -special conjugacy classes of U;. For each index n let d (. k i

J,

, Wn

aui I . lbui I . lwui ) _ l i,j i,k n IUi I

I

'lj!(au )'lj!(bi,k)'lj!(wn)

'lji(l)

Then the following assertions hold.

(a) A conjugacy class wui of U; is strongly (a;,j, b;,k)-real if and only if aui I· lbui I . ( . k ) - l i,j i,k p, J' ' w IU; I

(b) The number r(u,v,u;) = l{(x,y) E UG calculated by the character formula r,

r(u,v,ui)

=

IGI =

t

"'r(u v i~

'

'

vGl(xyr(x,y)

s, m(j,k)

LL L j=l k=l

(c)

X

u·)ICc(u)IICc(v)I_ ' ICc(u;)I

n=l

di(j,k,wn).

= u;}I

is

1.6 Thompson's group order formula revisited

37

Proof Assertion (a) holds by Lemma 1.6.1. (b) Let ,Q = {(x, y) E [uG n Cc (ui)] X [vG n Cc(u;)]l(xyr(x,y) = u;}. Theorem 4.2.l(a) of [92] asserts that r(u, v, u;) = IDI. Let (x, y) E D. Then w = xy is ui-special and real by Remark 1.6.3. Both properties are inherited by all its conjugates in U;. Furthermore, there is exactly one pair (j, k) of indices such that (x, y) E af,1x bf,k. Thus w is strongly ( a;,j, b;,k )-real. Therefore (a) implies that wui = w~i for a unique index n E {1 s;; n s;; m}. Thus d;(j, k, w) = d;(j, k, wn) > 0. Conversely, if (x,y) E {(x,y) E (uc x vc)} has a product q = xy such that d; (x, y, wn) = lw~i I · Pi (x, y, Wn) c/ 0 for some u;-special class w~i then q is U;-special by Lemma 1.6.1. Hence (xyr(x,y) = Ui for some integer n(x, y) > 0 depending on the pair (x, y). Thus (x, y) ED. Therefore Ti

r(u,v,u;) =

IDI =

Si

m

LLLdi(j,k,wn). j=l k=l n=l

Assertion (c) follows from (b) and Theorem 4.2.l(b) of [92].

D

2

Dickson group G2 (3) and related simple groups

It is well known that for 3 :S n :S 5 each simple linear group Tn = GLn (3) operating on the vector space Vn = pn over the prime field F = GF(2) has a non-split extension En of Tn by Vn which is uniquely determined up to isomorphism; see [41]. Furthermore, CEn (Vn) = Vn. Hence Algorithm 1.3.8 can be applied to any of these extensions. Theorem 12.3.l of [92] implies that for n = 5 such an application yields Thompson's sporadic simple group Th of order 215 · 310 · 53 · 72 · 13 · 19 · 31. In Chapter 3 we show that for n = 4 an application of Algorithm 1.3.8 returns Conway's sporadic group Co3 . Both simple groups are uniquely determined up to isomorphism by their centralizers of 2-central involutions. In this chapter the case n = 3 is dealt with. Let E be the uniquely determined non-split extension of GL 3 (2) by V = ½- It follows from P. Fong's main theorem of his article [31] and Theorem 1.6.2 of [92], due to R. Brauer, M. Suzuki and G. E. Wall, that there are infinitely many simple groups G2 ( q) having a 2-central involution z such that Hq = CG 2 (q) (z) has a Sylow 2-subgroup Sq that is isomorphic to a Sylow 2-subgroup S of E because there are infinitely odd prime powers q = pk such that q = 3, 5 mod (8). Furthermore, each Sq has an elementary abelian normal subgroup Aq such that Dq = NG 2 (q)(Aq) ""CE(z), where z is assumed to be the generator of the center Z (S) of S. In particular, all indices IG2 ( q) : Hq I are odd. Hence all conditions of Step 5 of Algorithm 1.3.8 are satisfied for infinitely many groups Hg of even order. Thus it cannot be proved that Algorithm 1.3.8 constructs in general only finitely many simple groups. The infinite series of simple groups G2 (q) was discovered by L. E. Dickson in 1905 [24]. They are defined over all finite fields order q =pk, where p > 0 is any prime. For odd primes p and q > 3, Fong characterized any G2 ( q) by the structure of its centralizer of an involution [31]. Fong's 38

2.1 Involution centralizers of Dickson's groups G2 (q), q odd

39

uniqueness proof uses J. Tits' theory of (B, N)-pairs [131]. In [68] Z. Janko completed Fong's study for the remaining case p = q = 3 by verifying the hypothesis of J. G. Thompson's deep Theorem 8.1 of [128]. Janko's theorem was quoted in the uniqueness proof for Thompson's sporadic group Th in Chapter 12 of [92] without proof. Therefore a selfcontained proof for it is given in this chapter. In his recent article [5] M. Aschbacher gives another characterization of G2 (3) which is independent of [128] and contains Janko's theorem as a corollary. As q = p = 3 Algorithm 1.3.8 can be applied to the non-split extension group E to get a group H = H 3 of even order. Weller, Previtali and the author constructed in [134] a simple matrix subgroup 5IB-) eq 324} we found the element w = (k1k2k5k2k3k 2k3k1k~) 11 of order 18 in N. Let b6 = w 9 • It has been checked computationally that r~ 6 = r~ and that T = (b 1 , b2 ) is an extra-special Sylow 3-subgroup of Y of order 27 and exponent 9. Hence X = (r3,b5) ~ 83, the symmetric group of order 6. Applying now the MAGMA command HasComplement(N ,X) one finds a complement Y of X in N. It is generated by b1, b2 and the involution y = (k2k4k2k5k1k3k5k1) 3. Furthermore, N =Xx Y. (b) Let N3 = Nc((r 3 )) and M = Cc(r3). Since N contains a Sylow 3-subgroup of N 3 we have (IM: YI, 3) = 1. Therefore Theorem 1.4.15 of [92] implies that M = (r3 ) x W for some subgroup W of G containing Y. Thus N3 = X x W because the symmetric group 83 is complete. The orders of CN1 (b6 ) and CN1 (y) are 25 · 34 and 25 · 34, respectively. Hence the involutions b6 and y belong to the conjugacy classes 22 and 23 of N 1 listed in Table 3.4.4. Thus Lemma 3.3.4(g) implies that b6 and y are G-conjugate to z and u, respectively. In particular, r 3 is G-conjugate to an element d in U = Cc (u) of order 3. By Lemma 3.3.5(g) r3 is not G-conjugate to any 3-element in H. Therefore Proposition 3.3.3 and Theorem 8.l.9(c) of [92] imply that Cu(d) ~ (d) x ((s) x A4), wheres is a non-2-central involution of M 12 . Hence Cu (d) has order 23 · 32 , and all its involutions are G-conjugate to u. Thus IWI is a multiple of 23 · 33 . Since W commutes with b6 we may assume that W is a subgroup of H. Using its faithful permutation representation PH and the MAGMA command subs:=Subgroups(PH : OrderMultipleOf :=2-3*3-3) we see that H has 60 conjugacy classes of subgroups having such an order. However, among them there are only two conjugacy classes of subgroups having an extra-special Sylow 3-subgroup of order 27 and exponent 9. Their orders are 23 · 33 · 7 and 24 · 33 · 7. Furthermore, getting the composition factors of these two groups one sees that the

116

Conway's simple group

(03

first is isomorphic to W 1 = Aut(PSL(2, 8)) and the second is the direct product of W 1 with (z). Hence W ~ W1 because b6 belongs to X and

XnW=l. Using the permutation representation PH of H = (x, y, h) and Kim's program Get Short Gens (H, W_ 1) MAG MA returns two generators q = (x 2 hx 3hx 2 ) 3 and t = (xh 2 xhx 3h) 2 of W 1 . Another application of the MAGMA command W = FPGroup(W_l) provides the following set of defining relations R(W) of the finitely presented group W = (q, t): q3 = l,

t3 = 1,

(q-1r1 )6 = 1,

(qc1 )6 = 1,

q-1r1 q-1tq-1c1 q-1c1q-1tqc1 q-1tq-1t q-1 c1 qtq-1 c1 q-1c1 qtq-1tq-1c1 qt

Since N 3 = X x W and X hold trivially for r = r3 and s

~

=

= 1,

1.

8 3 the remaining relations of R(N3 )

= b5.

(b) N 3 ~ 8 3 xPSL(2,8) holds by the proofof (a). Since N 3 has a small order we first calculated the regular faithful permutation representation pN3 by means of MAGMA from the given presentation of N 3 . Using pN3 we determined a Sylow 3-subgroup 8 of N 3 inside the normalizer NN3 ((q, r) ). By means of the short generator command MAGMA we found the generators r, x 2 = tqtqt 2 qt 2 and x 3 = (tqt 2 qt 2 qtqt) 2 of 8. They have respective orders 3, 9 and 3. Clearly, 8 is the stabilizer of a faithful permutation representation P N 3 of N 3 of degree 112. (c) The system of representatives of the conjugacy classes of N 3 has been computed by means of the faithful permutation representation P N 3 given in (b), MAGMA and Kratzer's Algorithm 5.3.18 of [92]. (d) The character table of N 3 has been calculated by means of P N3 and MAGMA. D

Lemma 3.3. 7 Let G be a finite simple group of Co 3 -type with a 2central involution z such that Cc(z) = H = (x,y,h) as described in Lemma 3.2.1. Then the following statements hold.

(a) Ji = (y 2 h) 2 EH generates the center of an extra-special Sylow 5-subgroup of G of order 53 and exponent 5. Furthermore, it is a representative of the unique 5-central conjugay class of G of order 5.

3.3 Uniqueness proof

(b) N5 (q;

117

= N c ( U1))

11

is isomorphic to the finitely presented group N 5 = :s; i :s; 5) with the following set R(N2 ) of defining relations:

= Q38 = Q45 = Q55 = 1, (q1,q5) = (q2,q5) = (q4,q5) = (q2,q3) = 1, -1 2 -1 -1 -1 -1 -1 -1 1 Q3 q5q3q5 = ql Q2 Q1Q2 = ql Q3 ql q3 = , -1 -2 -2 -2 2 -2 -1 -1 2 -1 -1 -1 1 q2 ql Q2 = Q3 ql Q3 = ql Q4 ql Q4 ql Q5 = , Q1 Q4 1 Q2Q1 1 Q4 2 Q2 q5 = Q1 1 Q4 1 q:,;1 Q1 1 Q4 1 q3q4 = l. 4

ql

=

6 Q2

(c) N5 = (t; I 1 :s; i :s; 5) has a faithful permutation representation P N5 with stabilizer (q1 , q2 , q3) of degree 53 . (d) A system of representatives n; of the 26 conjugacy classes of N 5 and the corresponding centralizers orders ICN5 (n;)I are given in Table 3.4, 7. (e) The character table of N 5 is stated in Table 3. 5. 8. (f) The elements Ji = q5 and h = q4 of the normalizer N 5 represent all conjugacy classes of elements of order 5 in G. Furthermore, h is not G-conjugate to any element of H. Proof (b) Let PH be the faithful permutation representation of H = (x, y, h) constructed in Proposition 3.1.2(0). Table 3.4.1 states that f = (y 2 h) 4 is an element of H with order 5. Using PH and MAGMA one can easily see that its normalizer N = NH(!) in H has order 24 · 3 · 5 and that it is generated by the three elements y 1 = (hxhx 2 hxhxh 2 )2, y 2 = ( hxhxh 2 xhxhx ) 5 and y 3 = xh 2 xhxhxhx 2 h 2 xhx 3 • Applying now the MAGMA command HasComplement (N, sub) one obtains a complement K of the subgroup (!) in N. Using MAGMA and the command GetShortGens(N, K) it follows that K = (e 1 ,e2 ,e3 ), where e1 = Y2, e2 = Y§ and e3 = Y3Y1Y§. They have respective orders 4, 6 and 8, and f = (Y1Y§) 6 . Furthermore, 3 = f3. The restriction of PH to K yields a faithful permutation representation PK of K. Using it and the MAGMA command K = FPGroup(PK) one obtains the following set R(K) of defining relations of the group K= (e1,e2,e3) of order 48:

r

Let X be the largest normal subgroup of N 5 = Ne(!) having odd order. Let F = GF(5). Then by the argument given in Lemma 3.3.4

Conway's simple group

118

(03

it follows that X 1 = X/(f) is a faithful irreducible FK-module of dimension 2 over F. From the regular permutation representation PK of degree 48 of K one constructs all irreducible representations of K over Fusing the Meataxe algorithm and MAGMA. There are 16 irreducible representations. Four of them are faithful. The following three matrices Mei of the three generators ei of K describe a faithful irreducible representation A of degree 2 over F: Me1

= (~ g),

Me2

=

(H)

and

Me3

=

(H).

Let MK be the subgroup of GL 2 (5) generated by these three matrices. Since it is irreducible the following two MAGMA commands Q : = Getvecs (MK) and Semidir (MK, Q) construct the semidirect product of K by A as a finitely presented group K1 = (e1, e2, e3, e4) of order 24 · 3 · 52 with set R(K1 ) of defining relations consisting of R(K) and the following relations: e45 -1 - '

K has two additional faithful irreducible representations of degree 2. Applying the same methods as above one obtains two other extension groups K 2 and K3. Using isomorphism tests with MAGMA the author checked that K1 ~ K2 ~ K3. In order to get the presentation of the group N 5 = Na (f) we now extend K 1 by a linear module described by the action of e3 on f. As 3 = f 3 we assign to e3 the matrix M3 = [3] E GL1 (5) and the identity matrix Id in GL 1(5) to all other generators of K1. This module is implemented into MAGMA by the command

r

FEalg := MatrixA1gebra. Using the presentation of K 1 and its permutation representation PKl MAGMA constructs the cohomology module CM by means of the command CM : = GModule (PK1, FEalg). The dimension of the second cohomology group of K 1 with coefficients in CM is 1, as can easily be verified by an application of the MAGMA command CohomologicalDimension (PE1, CM, 2). Applying the MAGMA commands P := ExtensionProcess(PE1, CM, Ki) and K11 := Extension(P, [1])

3.3 Uniqueness proof

119

one obtains the presentation of N 5 = Kll ~ Ne(!) given in statement (b), where the letter e is replaced with q. N 5 is the unique (up to isomorphism) non-split extension of K 1 by the cyclic group (!). (c) This assertion is trivial because N5 is a split extension of K by a group of order 125. (d) The system of representatives of the conjugacy classes of N 5 has been computed by means of P N 5 , MAGMA and Kratzer's Algorithm 5.3.18 of [92]. (a) This follows at once from Table 3.4.7 and Proposition 3.3.3 because each Sylow 5-subgroup of N 5 is extra-special of order 125 and exponent 5. (e) The character table of N 5 has been calculated by means of P N 5 and MAGMA. (f) Proposition 3.3.3 asserts that a Sylow 5-subgroup of G has order 125 and that G has two conjugacy classes of involutions represented by z and u. Thus (a) implies that N 5 contains a Sylow 5-subgroup S of G. It has exponent 5 by Table 3.4.7. This table also states that N 5 has two conjugacy classes of elements of order 5, one of which is 5central. Since q6 is G-conjugate to an element f of order 5 in H it is centralized by an element of z 0 . The representative q4 of the second class 52 of N 5 has a centralizer CN 5 (q4 ) with a cyclic Sylow 2-subgroup of order 2 by Table 3.4.7. Let x be its generator. Then it has been checked computationally that x belongs to the class 2 1 of involutions of N 5 . That is not G-conjugate to z by Table 3.4.7 because the Sylow 5-subgroups of H are cyclic and all involutions in a Sylow 2-subgroup of CH(!) are G-conjugate to z. Hence x E u 0 . Thus G has two conjugacy classes of elements of order 5. This completes the proof. D Lemma 3.3.8 Let G be a finite simple group of Co 3 -type with a 2central involution z such that Cc (z) = H = (x, y, h) as described in Lemma 3. 2.1. Let N 5 = (q; 11 ::; i ~ 5) be the finitely presented group constructed in Lemma 3.3. 7. Using the representatives of its conjugacy classes given in Table 3.4. 7 the following statements hold.

(a) fz = q4

E N 5 is a representative of a non-5-central conjugay class

of G of elements of order 5. (b) Ne ( (h)) is uniquely determined up to isomorphism. It is isomorphic to the finitely presented group N5 = (x 1 , x2, x3, X4) with the

120

Conway's simple group

(03

following set R(N6) of defining relations:

(x1,x3) = (x1,x4) = (x2,x3) = (x2,x4) = 1, (x1X21 X11 X21 )2

= (x11 X2X11 X21 )2 = 1.

(c) A system of representatives ni of the 20 conjugacy classes of N6 and the corresponding centralizer orders ICN6 (ni)I are given in Table 3.4.8. (d) The character table of N5 is stated in Table 3.5.9. Proof (a) By Lemma 3.3.7(a) and (f) h = q4 is not conjugate to Ji = q5 in G. Furthermore, G has exctly these two conjugacy classes of 5elements.The Sylow 2-subgroups of CN 5 ( (h)) are cyclic of order 2 by Table 3.4.7. (b) Using the faithful permutation representation P N 5 of N 5 defined in Lemma 3.3.7(c) and MAGMA the reader can verify again that M = NN5 ((h)) is generated by m1 = (qrq4) 2, m2 = (q3q1q4)2, m3 = q2q1q§ and IMI = 23 · 52. These generators of M have respective orders 5, 4

J:;

2

2 = and 2. Furthermore, the following relations hold: h = (m§m3) 6 , ff Thus X = (h, m§) of M is a Frobenius group of order 20. The subgroup Y = (u1,m1) is dihedral of order 10 because u1 = (m§m3) 5 has order 2 and m¥1 = mi. Another application of MAGMA yields that M = X x Y. It also asserts that the involutions u 1 and z 1 = q1q§ are conjugate in N 5 • By Tables 3.4.7 and 3.4.1 z1 is G-conjugate to the 2-central involution z of G. Now Table 3.4.1 implies that h is Gconjugate to the 5-element j = (y 2h ) 4 of H = (x, y, h) whose centralizer CH (j) has order 22 · 3 · 5 and a non-cyclic abelian Sylow 2-subgroup of order 4. Therefore Cc (h) / (h) > Y has a strongly self-centralizing Sylow 2-subgroup. Hence Cc(h) = (h) x PSL2(5) by Theorem 1.6.2 of [92] due to Brauer, Suzuki and Wall. PSL2(5) = (x1,x2) has a set of defining relations consisting of all the relations of R(N6 ) given in the statement except the ones involving the generators X3 and X4 of N5. Letting x 3 = h and x4 = m§ the given presentation of N5 = Nc(h) holds. (c) The system of representatives of the conjugacy classes of N5 has been computed by means of the regular permutation representation P N 6 of N 6 , MAGMA and Kratzer's Algorithm 5.3.18 of [92]. (d) The character table of N 5 has been calculated by means of P N 6 and MAGMA. This completes the proof. D

3.3 Uniqueness proof

121

Lemma 3.3.9 (Fendel) Let G be any finite simple group of Co3 -type with a 2-central involution z such that Ca(z) = H(x,y,h). Then the following assertions hold.

(a) The Sylow 7-normalizer N1 is a direct product of the symmetric group 83 and the F'robenius group of order 42. (b) The Sylow 11-normalizer Nu is a direct product of the cyclic group of order 2 and the F'robenius group of order 22. (c) The Sylow 23-normalizer N23 is a F'robenius group of order 23-11. Proof (a) By Proposition 3.3.3 each group G of (03-type has order IGI = 210 . 37 . 53 . 7 · 11 · 23. Let T be a Sylow 7-subgroup of G. By Lemma 3.3.6 and Sylow's Theorem we may assume that T ::; N 3 = (p, q, r, s, t) and that T = (p). Hence CN3 (S1) = 83 x T and N N 3 (T) = 83 x F21, where 8 3 = (r, s) and F 21 = (p, t). Since 7 divides IHI Table 3.4.1 and Sylow's Theorem assert that Tis conjugate to T1 = ((xy) 2 ). Using the faithful permutation representation PH of H given in Proposition 3.1.2(o) and MAGMA it has been checked that NH(T1 ) is a direct product of a cyclic group of order 2 and a Frobenius group F 42 of order 42. Hence Na (T) = 83 x F42 by Table 3.4.6. (b) Proposition 3.3.3 states that G has two conjugacy classes of involutions represented by z and u and that Ca(u) = U ~ (u) x M12, In particular, we may assume that a Sylow 11-subgroup L of G is contained in U. Hence Nu(L) is a direct product of (u) and a Frobenius group F55 of order 55. By the previous results of this section we know that 11 does not divide the orders of Ca(z) and the ones of all them-Sylow normalizers for the primes m E {3,5, 7}. Therefore Na(L) = Nu(L). (c) Now the previous results and Sylow's Theorem imply that a Sylow 23-subgroup W of G is cyclic and self-centralizing. Furthermore, Na (W) has order 23 · 11. D

Lemma 3.3.10 Let G be any finite simple group of Co 3 -type with a 2central involution z such that Ca (z) = H. Then G has 42 conjugacy classes xa . They can be ordered such that their centralizers Ca (x) have the same orders as the centralizers C® (~) of the corresponding conjugacy classes~® of the simple group © constructed in Theorem 3.2.2. They are stated in Table DVD .1.1.1 on the accompanying DVD. Proof (a) By Proposition 3.3.3 each group G of (03-type has order IGI = 210 . 37 . 53 . 7. 11 . 23.

122

Conway's simple group

(03

By Proposition 3.3.3 G has 26 conjugacy classes of elements of even order. G has one conjugacy class of elements of order 7, two conjugacy classes of elements of order 11 and two conjugacy classes of elements of order 23 by Lemma 3.3.9. From the power map information of the classification of the conjugacy classes of the groups N 1 , N 2 , N 3 , N 5 and N 6 given in Section 3.4 and the order of G it follows that G has eight 3-singular classes denoted by 3A, 3B, 3c, 9A, 9B, 15A, 15B, 21A and two classes of elements of order 5. Furthermore, Lemma 3.3.9 and the tables stated in Section 3.4 imply that the centralizers of these 41 non-trivial conjugacy classes have the same centralizer orders as the corresponding conjugacy classes of the constructed group ),GModule(sub))

we obtain the transformation matrix given by

T =

1000 0 0 0 011111 9 0 01111 0 8 0 0 4 2 5 3 12 2 03172311 0 0 0 0 1 12 0 011910 2 11 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0 000 0 0 0 0

satisfying W(x)

=

0 0 0 0 0 0 0 0 0 12 1 0 0 11 0 1 12 12 0 0 12 0 0

0 0 0 0 0 0 0 11 0 12 4 0 11 1 0 11 12 11 0 0 12 12 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 11 0 3 2 0 3 111111 1 7 4 4 3 12 2 1112 0 0 2 12 12 11 1 1 12 10 10 1 1 12 0 12 1 0 1 2 1 0 2 12 10 10 0 1 12 1 0 0 12 1 2 1 12 0 0 1 0 0 2 10 12 12 0 1 0 1 12 0 12 1 1 1 12

(522J(xi)) 7 and W(y)

=

0 0 0 0 0 0 0 1 0 1 12 0 1 12 1 2 0 1 1 0 0 1 12

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 11 2 3 12 7 3 2 11 2 0 3 11 1 12 0 1 4 11 2 0 12 1 0 0 2 11 0 0 12 0

(522J(yi)) 7

0 0 0 0 0 0 0 2 2 0 1 0 1 11 0 2 0 1 2 0 0 1 0 .

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 12 0 12 1112 11 3 12 3 1 2 9 1 7 10 10 10 11 2 1 2 10 1 2 1 0 9 0 3 1 3 9 1 0 1 0 12 1 11 12 10 3 12 3 1 2 9 1 1 0 0 12 0 1112 11 3 12 0 12 0 1 1 3 1 2 9 1 0 0 11 0 0 12 12 1 2 12

Conway's simple group Co 2

166

Let e = (W(e 1)) 7 . Then ~3 = (J:, IJ, e) ~ E. Let \!53 = (~,J:, IJ, e). Using the MAGMA command CosetAction(G, E) we obtain a faithful permutation representation of \!53 of degree 46575 with stabilizer ~3. In particular, l\!5 3 I = 218 · 36 · 53 · 7 · 11 · 23. Using the faithful permutation representation of ® 3 of degree 46575 and Kratzer's Algorithm 5.3.18 of [92] we calculated the representatives of all the conjugacy classes of \!5 3 ; see Table DVD. 1 . 2. 1. Furthermore, the character table of ® 3 has been calculated by means of the above permutation representation and MAGMA. It coincides with the one of Co 2 in [19], pp. 154-155. (c) Let 3 = (J:IJ~) 15 . Then C®,(3) contains SJ= (~,J:,IJ), which is isomorphic to H. By Table DVD .1. 2 .1 IC®, (3)1 = IHI. Hence C® 3 (3) ~ S). The character table of \!5 3 implies that \!5 3 is a simple group. This completes the proof. D Praeger and Soicher give in [110] a nice presentation for Conway's simple group Co 2 . It is used in [72] to show that \!5 3 is isomorphic to

Co2. Corollary 4.3.3 Keep the notation of Theorem 4.3.2. The finite simple group ® 3 is isomorphic to the finitely presented group G = (a, b, c, d, e, f, g) with the following set R( G) of defining relations:

a2 = b2 = c2 = d2 = e2 = f2 = g2 = l,

= (bc) 5 = (cd) 3 = (df) 3 = (fe) 6 = (ae) 4 = 1, (ec)3 = (cf) 4 = (fg) 4 = (gb) 4 = 1, (ac) 2 = (ad) 2 = (af) 2 = (ag) 2 = (bd) 2 = (be)2 = 1, (bf)2 = (cg) 2 = (dg) 2 = (eg) 2 = 1, (ab) 3

a= (cf)2, (aecd) 4

e = (bg) 2 ,

b = (ef) 3 ,

= (baefg) 3 = (cef)7 = 1.

Furthermore, G has a faithful permutation representation PG of degree 46575 with stabilizer Q = (a, b, c, d, e, g, (gfdc) 4 ). Proof The presentation of G = (a, b, c, d, e, f, g) is taken from [110], p. 106. Its subgroup, Q = (a, b, c, d, e, g, (gfdc) 4 ), is isomorphic to E 3 by Lemma 4.1.2(f) and [110], p. 106. Taking it as a stabilizer MAGMA constructs a faithful permutation representation PG of the group G = (a, b, c, d, e, f, g) of degree 46575. An isomorphism test using MAGMA

4.4

On the uniqueness of Co 2

167

and the faithful permutation representation of 18 3 given in Theorem 4.3.2(c) shows that 18 3 ~ G. D Corollary 4.3.3 will be used in Chapter 6 in H. Kim's construction of Fischer's sporadic group Fi23.

4.4 On the uniqueness of Co 2 By Lemma 4.3.1 the Goldschmidt index of the amalgam H +- D--+ E 3 is 2. Hence the author's sufficient uniqueness criterion stated as Theorem 7.5.1 in [92] cannot be applied. Therefore we do not give a uniqueness proof. However, the main result of F. Smith's article [119] asserts that Co 2 is uniquely determined up to isomorphism by the structure of the centralizer H = Cc 02 ( z) of a 2-central involution z E Co 2 • Furthermore, the largest normal subgroup 0 2 of 2-power order of H is extra-special and H = H/02 ~ Sp6 (2). In particular, the matrix subgroup 183 of Theorem 4.3.2 is isomorphic to Co 2 . Smith's proof is involved. He constructs for a simple group G of Co2type a graph Q and its automorphism group Aut(Q). But his proof does not provide an isomorphism between G and Conway's group Co 2 of [16]. Using Smith's uniqueness theorem T. Yoshida proved an even stronger uniqueness result in [141].

Theorem 4.4.1 (T. Yoshida) Let Co 2 be the simple sporadic Conway group of order 218 -36 .5 3 ·7·11-23. A finite simple group G is isomorphic to Conway's group Co2 if it has a Sylow 2-subgroup S which is isomorphic to the Sylow 2-subgroups of Co2. Proof See [141], p. 405.

D

In [141] T. Yoshida states a presentation of a Sylow 2-subgroup T of Co 2 in 18 generators. Using it, and the faithful permutation representation of the simple group 18 3 stated in Theorem 4.3.2, it can be verified that Co 2 and 18 2 have isomorphic Sylow 2-subgroups.

168

Conway's simple group Co2

4.5 Representatives of conjugacy classes 4.5.1 Conjugacy classes of H(Co2) = (x, y, h) Class

1 21 22 23 24 25 25 27 28 29 210 31 32 33 41 42 43 44 45 45 47 48 49 410 411 412 413 414 415 415 417 418 419 420 421 422 423 424 5 61 62 63 64 65 65 67 68 69 610 611 612 613 614

Representative

1 (xyh)15 /uh) 7

(y)3 /xuhu) 5 (xy)4 (xyxh 2 ) 6 (x)6

(x2 y4 )3 /xy4)2 (xh2)3 /x2 h)4 (x)4 /xhy) 4

(yh2)7 /x 2 uh) 4 (xh2y)3 (xyxh 2 ) 3

/xh2y2)3 (xhyxh 2 ) 3 (x 2uxhxu) 3

(x4y3)2 /x2y)2 (xhy) 3

/xy)2 x 2 yxhyxyh (yh3 )3

(xh3y)2 /x 3 yhxy) 2

(x)3 /xu2 )3 xhy 2

/uh4)3 X2'!12 x 2 hxu xy4 xyxh 3 y xh 5 y xh

(x 2 hyh) 3 (xyh) 5

(x3y)2 /x2 h)2 (xyxh 2 ) 2 y X4'JI x 3 hxy x2yh3 X2'!14h x 2 hvhxh x 2 h 2 'JIX'JI xuh 2 uh 2 (xhy) 2

I Centralizer! 2P 3P 1 1 2 18 · 3 4 · 5 · 7 1 1 2 18 · 3 4 · 5 · 7 1 21 270 2 17 · 3 · 7 1 22 214 . 32 . 5 1 1008 23 214.32.5 1 1008 24 216 . 32 1 1260 25 216 . 32 1 1260 25 214 . 3 1 15120 27 214 . 3 1 15120 28 215 1 22680 29 120960 2 11 · 3 1 210 26 . 34 31 143360 1 172032 1 2 5 · 3 3 · 5 32 27 . 33 33 215040 1 214.33.7 21 41 240 214 . 3 21 42 15120 213 . 3 22 43 30240 212 . 3 25 44 60480 212 . 3 22 45 60480 212 . 3 25 45 60480 212 . 3 22 47 60480 213 22 4s 90720 211 . 3 25 49 120960 120960 2 11 · 3 22 410 212 25 411 181440 212 25 412 181440 210 . 3 25 413 241920 211 22 414 362880 211 25 415 362880 483840 2 9 · 3 27 415 483840 2 9 · 3 27 417 483840 2 9 · 3 27 418 483840 2 9 · 3 27 419 210 22 420 725760 210 25 421 725760 28 29 422 2903040 28 27 423 2903040 28 27 424 2903040 12386304 5 22 · 3 · 5 5 26 . 34 31 21 143360 172032 2 5 · 3 3 · 5 32 21 27 . 33 33 21 215040 26 . 32 31 25 1290240 26 . 32 31 25 1290240 25 . 32 32 23 2580480 25 . 32 32 24 2580480 25 . 32 33 25 2580480 25 . 32 33 24 2580480 25 . 32 33 25 2580480 25 . 32 32 25 2580480 25 . 32 32 25 2580480 25 . 32 33 23 2580480 3870720 2 6 · 3 33 22 1Class1

5P 1 21 22 23 24 25 25 27 28 29 210 31 32 33 41 42 43 44 45 45 47 48 49 410 411 412 413 414 415 415 417 418 419 420 421 422 423 424 1 61 62 63 64 65 65 67 68 69 610 611 612 613 614

7P 1 21 22 23 24 25 25 27 28 29 210 31 32 33 41 42 43 44 45 45 47 48 49 410 411 412 413 414 415 415 417 418 419 420 421 422 423 424 5 61 62 63 64 65 65 67 68 69 610 611 612 613 614

169

4.5 Representatives of conjugacy classes Conjugacy classes of H(Co2) = (x,y,h) (continued) Class Representative I Classl (x)2 7741440 615 x2y4 7741440 616 xh 2 15482880 61 7 7 h 13271040 (xy3 h)3 967680 81 (x 2yh) 2 1451520 82 (xhyh) 2 1451520 83 x•y3 2903040 84 x 2 h 2 xy 2 2903040 85 x 211 5806080 85 x 3 yhxy 5806080 87 x 211 2hyh 5806080 8s xh 4 yh 5806080 89 x 2hxy 3 h 5806080 810 xy 11612160 811 xh 3 y 11612160 812 xh 211 2h 11612160 813 9 u 2h 41287680 (xyh) 3 12386304 101 xyhy 37158912 102 v 2 h 2 37158912 103 xh 2xh 2y 2 860160 121 x3y 2580480 122 xyxh 2 7741440 123 xh 2y 2 7741440 124 xy 4 h 7741440 125 xhyxh 2 7741440 12s x 2yxhxy 7741440 127 15482880 X 12s 129 x 2h 15482880 xy2 15482880 1210 xhy 15482880 1211 xh 2y 15482880 1212 vh 3 15482880 1213 (yh3)5 15482880 1214 yh• 15482880 1215 x 2y 2hy 15482880 1215 (yh2)2 13271040 141 yh 26542080 142 (yh)3 26542080 143 15 (xyh) 2 24772608 x 2yh 23224320 161 xhyh 23224320 162 18 x 2h11h 41287680 xy 3 h 30965760 24 yh2 26542080 28 xyh 24772608 30

I Centralizer!

25 25 24 23 28

·3 ·3 ·3 ·7 ·3 29 29 28 28 27 27 27 27 27 26 26 26 2 · 32 22 .3.5 22 · 5 22 · 5 25 . 33 25 . 32 25 · 3 25 · 3 25 · 3 25 · 3 25 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 23 · 7 22 · 7 22 · 7 2 ·3 ·5 25 2· 23 22 2 ·3

2' 32 ·3 ·7 ·5

2P 32 32 33 7 41 42 42 4s 4s 49 415 4s 49 415 411 414 44 9 5 5 5 61 63 65 614 61 65 614 615 64 615 614 614 64 64 615 615 7 7 7 15 82 83 9 122 141 15

3P 27 2s 210 7 81 82 83 84 85 85 87 8s 89 810 811 812 813 31 101 102 103 41 41 44 45 42 45 47 415 49 417 410 43 413 413 419 413 141 143 142 5 161 162 61 81 28 101

5P 615 615 617 7 81 82 83 84 85 85 87 8s 89 810 811 812 813 9 21 24 23 121 122 123 124 125 125 127 12s 129 1210 1211 1212 1214 1213 1215 1216 141 143 142 32 161 162 18 24 28 62

7P 615 615 617 1 81 82 83 84 85 85 87 8s 89 810 811 812 813 9 101 102 103 121 122 123 124 125 12s 127 12 8 129 1210 1211 1212 1213 1214 1215 1215 21 22 22 15 161 162 18 24 41 30

Conway's simple group Co2

170

4.5.2 Conjugacy classes of D(Co2) ulass

1 21 22 23 24 25 25 27 2s 29 210 211 212 213 214 215 215 217 218 219 220 221 222 223 224 31 32 41 42 43 44 45 45 47 4s 49 410 411 412 413 414 415 415 417 418 419 420 421 422 423 424 425 426 427 428 429 430

Representative

= (x, y)

IClassl

1 (x3y)6

I x5y)5 (x4 yxy2 )5

/x4y3)4 (xy)4 (x4yxy)4 (x)B (x2v2)2

(x2u4)3 (x4 yxy3 )3

( x2 Y3 xy3 xy )3 (xy4)2 lv)3

(x4y)3 (x 2 yxyxy) 3 (xy2xy3)3 ( x 2 uxuxu 2 x 2 11 2 ) 2 x3 '.l/X'.l/2 x2 '.l/X'.l/X2 '.l/3

( x 2 U 3 XU ) 3 (X3'.l/3X'.l/)2 x 2 uxuxu 2 xu 4

x3 '.l/4 x2 '.l/X2 '.l/X'.l/2 xuxuxuxu 4 x 3 ux 2 '.l/X'.l/X'.l/X 2 '.l/ (x3y)4

(x)4 (x3u)3

(x2 '.l/X'.l/X'.l/2 )3

I x 2 uxu 2 14 (x4y3xy)3 (x2 yxy3 xy3 )3 (x4yx2y)2 (x2yxy3)3

/xv) 2 (x4yxy)2 x 7 vxv x3yxy3x2y2 x7yxy4 x 6 uxuxu 3

X2 '.l/2 X'.l/X'.l/4 X'.l/ x 5 ux 2 ux 2 ux 2 u X3'.l/X3'.l/X2'.l/3X'.l/ x 3 uxyx 2 ux 2 y 2 xy 2 x3 y2 x2 '.l/XYX2 yxy2

/x2y)2 (x4y3 )2 x 2 vxy 3 xv (x5yx3y)2 x3yxyx3y2 x6yx2y2x2y x 2 vxuxvxvxv 2 x 2 v 2 x5y5xy4xy (x)3 (xy2 )3 (x6v)3 xyxy 4

1 1 16 16 30 60 60 240 240 240 240 240 360 480 480 480 480 720 1440 1920 2880 2880 2880 5760 5760 10240 40960 240 480 720 960 960 1440 1920 2880 2880 2880 2880 2880 2880 2880 2880 2880 2880 2880 5760 5760 5760 5760 5760 5760 5760 5760 7680 7680 7680 7680

I Uentralizerl 2P 218 . 32 . 5 1 218 . 32 . 5 1 214 . 32 . 5 1 214 . 32 . 5 1 217 . 3 1 216 . 3 1 216 . 3 1 214 . 3 1 214 . 3 1 214 . 3 1 214 . 3 1 214 . 3 1 215 1 213 . 3 1 213 . 3 1 213 . 3 1 213 . 3 1 214 1 213 1 211 . 3 1 212 1 212 1 212 1 211 1 211 1 27 . 32 31 25 . 32 32 214 . 3 21 213 . 3 24 214 21 212 . 3 24 212 . 3 24 213 24 2 11 · 3 24 212 25 212 25 212 24 212 25 212 24 212 24 212 21 212 24 212 24 212 26 212 24 211 25 211 24 211 25 211 26 211 25 211 24 211 24 211 24 2 9 · 3 27 2 9 · 3 27 2 9 · 3 27 2 9 · 3 27

3P 1 21 22 23 24 25 26 27 2s 29 210 211 212 213 214 215 215 217 218 219 220 221 222 223 224 1 1 41 42 43 44 45 45 47 48 49 410 411 412 413 414 415 415 417 41s 419 420 421 422 423 424 425 425 427 42s 429 430

5P 1 21 22 23 24 25 25 27 28 29 210 211 212 213 214 215 215 217 218 219 220 221 222 223 224 31 32 41 42 43 44 45 45 47 48 49 410 411 412 413 414 415 415 417 41s 419 420 421 422 423 424 425 425 427 428 429 430

4.5 Representatives of conjugacy classes

171

Conjugacy classes of D(Co 2 ) = (x, y) (continued) Class Representative x'y' 431 (x 3 y 2 xv 3 ? 432 X2 y' X'!/4 433 x2y2x2y5 434 x 4 yx 4 vxv 435 x 4 yxyx 2 yxy 435 x 2 yxy 2 x 2 y 2 xy 437 x 2 yxy 2 xyxu 3 433 x 4 yxyx 3 vxu 439 x 2uxyx 2 y 3 xy 2 440 x 2 vxy 4 xyxy 2 441 x2 y' x2 y3 xy3 442 x 4 yx 2 yx 3 y 2xy 443 ( x 3 yxyxy 2 ) 3 444 x2y2 445 x 3 yxyxyx 3 y 2 445 xy4 447 x 3 y"xy 443 x 8v 449 x 2 y 3 xyxy 450 x 4 vxy 2 xy 451 x1yxy2 452 x 6 yxyxy 453 x 8 yx 2 y 454 X7 yxy3 455 x 7 v 3 X1J 455 x 6 y 2 X'!/X'!/ 457 x 2 vxvxv 2 x 2 y 2 45s X3'!/X3'!JX2Y3 459 x5yx3y3xy 450 xy3 5 (x3y)' 61 x 5 y 2 xy 2 62 xyxyxyxv 3 63 x 6 y 2x 2 y 2 64 x3112xy3xy2 65 x 2 yx 2 y 2 xyxy 2 65 ( x2 yxy3 )2 67 (x)2 6s x2y4 69 x4yxy3 610 (x 3 yxyxy 2 ) 2 611 x 3 yxvxv 3 612 X3'!/2 X'!/4 613 x 2 v 3 xy 3 xy 614 y 615 x4y 616 x 2 yxyxy 61 7 x 2 11 3 xy 61s xy 2 xy" 619

iClassl I Centralizer! 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 210 11520 28 ·3 15360 29 23040 29 23040 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 28 46080 22 .5 589824 27 . 32 10240 25 . 32 40960 25 . 32 40960 25 . 32 40960 2' . 32 40960 40960 25 · 32 61440 26 • 3 25 ·3 122880 25 ·3 122880 25 .3 122880 25 ·3 122880 25 ·3 122880 25 ·3 122880 2' ·3 122880 24 ·3 245760 24 ·3 245760 24 .3 245760 24 ·3 245760 24 .3 245760

2P

3P

5P

2s 2s 25 23 25 25 24 24 24 25 24 25 2s 2s 23 2s 212 220 220 220 27 220 220 27 220 220 220 21 7 21 7 2s 5

431 432 433 434 435 435 437 433 439 440 441 442 443 444 445 445 447 443 449 450 451 452 453 454 455 455 457 453 459 450 5 21 23 22 21 22 23 24 27 2g

431 432 433 434 435 435 437 43s 439 440 441 442 443 444 445 445 447 443 449 450 451 452 453 454 455 455 457 453 459 450 1 61 62 63 64 65 65 67 6s 69 610 611 612 613 614

31 32 32 32 31 31 31 32 32 31 31 32 32 31 32 32 32 31 32

210 2s 25 25 211 213 214 215 219 215

615

616 611 61s 619

172

Conway's simple group Co2

Conjugacy classes of D(Co2)

= (x, y) (continued)

Ulass Representative IClassl I Centralizer! (x 3 11 2 x11x11) 3 15360 81 28 · 3 29 I x 2 vxv 2 ) 2 23040 82 29 /x311x112)2 23040 83 28 x 3 vx 3 v 2 46080 84 2B x 3 yx 2 yxyxy 2 46080 85 28 x2vx2v2x2v3 46080 85 28 X51/X21/2X1/2 46080 87 28 46080 x 4 vx 2 yx11x 2 v 83 28 X31/X31/3X21/ 46080 89 27 92160 X2 11 810 X41/3 27 92160 811 27 92160 x 4 11x 2 11 812 27 X31JX1/3 92160 813 27 X2 1JX1/X11 3 92160 814 27 x5yx3y 92160 815 27 x 3 11xy 2 x11 2 92160 816 27 x 2 yxvxv 4 92160 817 27 X51/2X3y 92160 813 26 xy 184320 819 26 x 4 11xy 184320 820 26 x 3 v 2 xv 3 184320 821 26 x 4 11 2 x11 3 184320 822 26 x 4 v 3 xv 2 184320 823 26 x 4 ux 2 yxy 2 184320 824 26 x 5 yx 4 11 2 xv 184320 825 x5y 589824 101 22 · 5 102 x 4 11xv 2 589824 22 · 5 X3 1JX1/X1/ 589824 103 22 · 5 121 X3 1J 122880 25 · 3 X41J3X1/ 122880 122 25 · 3 x 2 11xy 3 x11 3 122880 123 25 · 3 124 X 245760 24 · 3 X1/2 245760 125 24 · 3 x 6 v 245760 125 24 · 3 X21/X1/3 245760 127 24 · 3 x 2vxvxv 2 245760 12s 24 · 3 x3y2X1/5 245760 129 24 · 3 x 3 vxvxv 2 491520 1210 23 · 3 25 x211x112 368640 161 25 162 X3 1JX11 2 368640 24 x 3 y 2 xyxy 491520 23 · 3

2P 41 43 43 45 43 41 41 45 43 419 420 45 420 420 422 422 420 419 43 49 432 432 431 431 424 5 5 5 61 67 67 6s 63 6s 67 67 6s 611 82 83 121

3P 81 82 83 84 85 85 87 83 89 810 811 812 813 814 815 815 817 81s 819 820 821 822 823 824 825 101 102 103 41 44 45 427 428 429 47 42 430 444 161 162 81

5P 81 82 83 84 85 85 87 83 89 810 811 812 813 814 815 816 817 813 819 820 821 822 823 824 825 22 23 21 121 122 123 124 125 125 127 12s 129 1210 161 162 24

173

4.5 Representatives of conjugacy classes

4.5.3 Conjugacy classes of E(Co2) Class 1 21 22 23 24 25 25 27 2s 2g 3 41 42 43 44 45 45 47 4s 4, 410 411

412 413 414 415

415 417 418 419 420 421 422 5 61 62 63 64 65 65 61 6s 71

72

Representative 1 (xye) 5 (x)6 (xe) 10 (x 2 ey) 7 (xy 2 e)7 (e)2 (y)3 (xe 3 )5 (x2ye3y)2 (x)4 (x3 y)3 (y2 e2 )3 (xe' ) 4 (ye)3 (xy' )3 (xy)2 (x)3 x 2 yx 2 eyxe

(x 3 yey 2 ) 2 xyxe 3 xe xy 3 exe 2 (xe) 5 e (xy 2 e 2 ) 2 x 2 e 2y 3 e xsy (x' y 2 e) 3 x'y2 x2 y2 xe3 x 2 ye 3 y x 2 exe11e xye'

)1°

(xe ) 4 (x")J)" x')J4 X)Jxe' y (x)2 x4y x 2 yxe x3 e3 y

x 2e (x 2 e) 3

1czass1

= (x, y, e)

I Centralizer!

1 2 18 · 3 2 · 5 · 7 · 11 218 . 32 . 5 77 330 2 17 · 3 · 7 215 . 32 . 5 616 2640 2 14 · 3 · 7 2640 2 14 · 3 · 7 18480 2 14 · 3 213 . 3 36960 44352 2 12 · 5 214 55440 27 . 32 788480 18480 2 14 · 3 213 . 3 36960 55440 73920 73920 110880 147840 221760 443520 443520 443520 709632 887040 887040 887040 887040 1182720 1774080 1774080 3548160 3548160 3548160 22708224 788480 3153920 3153920 4730880 9461760 9461760 9461760 18923520 32440320 32440320

214

2 12 · 3 212 . 3 213 211 . 3 212 211 211 211 28

• 5 210 2 L0 210 210

28 · 3 29 2' 28 28 28 23 27 . 25 . 25 . 26 25

·5 32 32 32

.3 .3 2' .3 25 .3 24 · 3 22 .7 22 · 7

2P 1 1 1 1 1 1 1 l 1 1 3

3P 1

5P 1

7P 1

llP 1

21 22 23 24 25 2,

21 22 23 24 25 25 27 2s 2g

21 22 23 24 25 25 27 2s 2g

22 23 24 25 25 27 2s 2g

1

3

3

3

21

41 4,

41 42 43 44 45

41 4,

41 42 43 44 45 45 47 4s 49 410 411 412 413 414

22 21 22 22 22 22 21 22 22 22 23 25 25 25 25 25 25 2o 2g 25 2g 5 3 3 3 3 3 3 3 3 71 72

21 2s 2g

43 44 45 45 47 43 4g 410 411 412 413 414

45 47 4s 4g 410 411 412 413 414

415 415 415 415 41 7 417 4[8 41s 419 419

420 421 422 5

420 421 422 1

21 23 21 22 24 2s 25 21 72 71

61 62 63 64 65 65 61 63 72 71

43 44 45 45 47 4s 49 410 411 412 413 414 415 415 417 413 419 420

21

415 415 41 7 41 8 419

422 5

420 421 422 5

61 62 63 64 65 6s 67 6s 1 1

61 62 63 64 65 65 61 6s 71 72

421

174

Conway's simple group Co2

Conjugacy classes of E(Co2)

= (x, y, e) (continued)

Class Representative 1Classl (x' exe 2 ) 3 1182720 81 (xe 2 )2 1774080 82 x"ye 1774080 83 x2y 3548160 84 x3 e2 3548160 85 x 2 yey 2 3548160 85 x 2 eyxe 3548160 87 xy4e2 3548160 8s x"3y2e2y 3548160 89 xy 7096320 810 x 2 e 2 y 14192640 811 xy 2 e 2 14192640 812 yeye 2 14192640 813 x 2 e 2 y 2 14192640 814 x 3 yey 2 14192640 815 (xe) 2 22708224 101 xye 45416448 102 xe 3 45416448 103 11 xy'xe 82575360 ye 9461760 121 xy2 9461760 122 x3y 9461760 123 18923520 124 X y2 e2 18923520 125 125 x 3 y 2 e 37847040 141 x 2 ey 32440320 (x'ey) 3 32440320 142 xy 2 e 32440320 143 (xy 2 e) 3 32440320 144 xeye 32440320 145 (xeye) 3 32440320 145 xe 2 28385280 161 xyxey 28385280 162 20 xe 45416448 24 x 2 exe 2 37847040

I Centmlizerl

28 · 3 29 29 28 28 28 28 28 28 27 26 26 26 26 26 23 · 5 22 · 5 22 · 5 25 25 25 24 24 23 22 22 22 22 22 22

11 .3 .3 .3 .3 .3 .3 .7 .7 .7 .7 .7 .7 25 25

2P 41 43 43 45 45 41 43 43 41 45 413 414 413 414 4g 5 5 5 11 64 64 61 64 64 65 71

72 72 71 72 71 82 83

2 2 · 5 101 2 3 · 3 123

3P 81 82 83 84 85 85 87 8s 89 810 811 812 813

5P 81 82 83 84 85 85 87 8s 89 810 811 812 813 814 814 815 815 101 23 102 21 103 2s 11 11 44 121 45 122 41 123 47 124 42 125 41 7 125 142 142 141 141 144 144 143 143 145 145 145 145 161 161 162 162 20 412 24 81

7P 81 82 83 84 85 85 87 8s 89 810 811 812 813 814 815 101 102 103 11 121 122 123 124 125 125 24 24 25 25 22 22 161 162 20 24

llP 81 82 83 84 85 85 87 8s 89 810 811

812 813 814 815 101 102 103 1 121 122 123 124 125 125 141 142 143 144 145 145 161 162 20 24

4.6 Character tables of local subgroups of Co2

175

4.6 Character tables of local subgroups of Co2 4.6.1 Character table of E3 2 3 5 7 11 2P 3P 5P 7P llP X.l X.2 X.3 X.4 X.5 X.6

X.7 X.8 X.9 x.10 x.11 x.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19 X.20 x.21 X .22 X .23 X.24 X.25 x.~6 X.27 X .28 X .29 X .30 X.31 X.32 X.33 X.34 X .35 X .36 X.37 X.38 X.39 X.40 X.41 X.42 X.43 X.44 X.45 X.46 X.47 X.48 X.49 X .50 X.51 X.52 X.53 X .54 X .55 X .56 X.57 X.58 X .59 X.60 X.61 X .62 X .63 X.64 X.65 X.66 X.67 X .68 X .69 X.70 X.71 X .72 X.73 X.74 X.75 X.76 X.77 X.78 X.79

18 17 18 2 2 1 1 1 1 1 la 2a 2b la la la la 2a 2b la 2a 2b la 2a 2b la 2a 2b 1 1 1 1 1 1 21 21 21 21 21 21 22 -10 6 -10 22 6 45 45 45 45 45 45 45 45 45 45 45 45 55 55 55 55 55 55 99 99 99 99 99 99 154 154 154 154 154 154 210 210 210 210 210 210 231 7 39 231 7 39 231 231 231 231 7 39 231 7 39 231 231 231 385 385 385 385 385 385 440 -200 120 440 -200 120 560 560 560 -30 -14 770 210 770 -350 770 -30 -14 -30 -14 770 -30 -14 770 210 770 -350 924 28 156 924 28 156 924 28 156 924 28 156 270 990 -450 270 990 -450 270 990 -450 270 990 -450 1155 195 35 35 1155 195 35 1155 195 1155 195 35 1386 234 42 234 42 1386 384 1408 -640 1408 -640 384 1540 -60 -28 1540 -700 420 2772 -1260 756 3080 -120 -56 3080 -120 -56 3080 -120 -56 3080 -120 -56 3465 105 585 105 3465 585 3465 105 585 3465 585 105 3465 585 105 3465 585 105 105 3465 585 105 3465 585 4620 -180 -84 4620 -180 -84 4620 -180 -84 4620 -180 -84 210 6930 1170 210 6930 1170 6930 -270 -126 6930 -270 -126 6930 -270 -126 6930 -270 -126 9240 -360 -168 9240 -360 -168 13860 -540 -252

= E(Co2) = (x, y, e)

15 2 1

14 1

14 1

14 1

13 1

12

14

2c la 2c 2c 2c 2c 1 1 21 21 -2 -2 45 45 45 45 55 55 99 99 154 154 210 210 -9 -9 231 -9 -9 231 385 385 -40 -40 560 10 -70 10 10 10 -70 -36 -36 -36 -36 -90 -90 -90 -90 -45 -45 -45 -45 -54 -54 -128 -128 20 -140 -252 40 40 40 40 -135 -135 -135 -135 -135 -135 -135 -135 60 60 60 60 -270 -270 90 90 90 90 120 120 180

2d la 2d 2d 2d 2d 1 -1 -7 7 8 -8 -3 -3 3 3 13 -13 15 -15 14 -14 -14 14 -21 -35 -7 35 21 7 21 -21 -48 48

2e la 2e 2e 2e 2e 1 -1 -·7 7 -8 8 -3 -3 3 3 13 -13 15 -15 14 -14 -14 14 -21 -35 -7 35 21 7 21 -21 48 -48

2/ la

2g la 2g 2g 2g 2g 1 -1 -7 7

2h la 2h 2h 2h 2h 1 -1 1 -1

84 56 28 -84 -28 -56 -84 -28 84 28 -24 -24 24 24 63 7 -7 -63 42 -42 -64 64

-28 -56 -84 28 84 56 -84 -28 84 28 24 24

2i la 2i 2i 2i 2i 1 1 5 5 -2 -2 -3 -3 -3 -3 7 7 3 3 10 10 2 2 -9 7 7 7 -9 7 1 1 -8 -8 -16 -6 -6 -6 -6 -6 -6 28 -4 28 -4 6 6 6 6 19 3 3 19 26 26

56 -112 -56 112 105 -105 63 63 -105 -63 105 -63 -112 112 56 -56 -42 42 84 -252 -84 252 56 -56

H H1

1 5 5 6 6 -3 -3 -3 -3 7 7 3 3 10 10 2 2 7 23 7 23

-3 5 -3 5 3 -5 3 -5 13 5 -13 -5 15 -1 -15 1 14 6 -14 -6 -14 10 14 -10 11 11 -3 -11 -7 9 3 11

7 -11 -11

7 7 -9 1 21 5 1 -21 -5 24 24 -16 26 34 -4 18 4 -20 34 4 -20 34 20 34 -4 18 28 -20 -4 -4 36 4 28 20 4 -4 -36 -4 -18 -18

-24 -18 -24 -18

63 35 19 7 -7 19 -63 35 42 -6 -42 -6 64 -64 . -28 36 : -12 40 56 112 40 -56 40 -112 40 105 9 -105 9 63 -39 63 9 -105 57 -63 -39 105 57 -63 9 -112 44 44 112 56 -52 -56 -52 -42 -30 42 -30 -252 18 84 18 18 252 -84 18 56 -8 -56 -8 . -60

3j 15 15 39 -39 -15 -31 -15 42 -14 -42 14

-8 8

9

-9 -33 -33 -9 33 9 33 16 -16 -8 8 -42 42 12 12 -12 -12 -8 8

26 . -12 4 40 8 8 -40 8 8 25 25 -25 25 -25 -23 15 -39 15 9 25 -23 -15 9 -15 -39 -4 -4 -40 28 40 28 30 2 -30 2 20 -22 20 -22 -20 -22 -20 -22 24 40 -40 24 . -12

14 1

3a 3a la 3a 3a 3a 1 1 3 3 4 4

6 6 -3 6 6 -3 -2 -2 8 8 2 5 -4 5 5 5 -4 6 6 6 6 . . . . -6 -6 -6 -6

4a 2a 4a 4a 4a 4a 1 1 5 5 6 6 -3 -3 -3 -3 7 7 3 3 10 10 2 2 23 7 7 7 23 7 1 1 24 24 -16 -14 18 -14 -14 -14 18 -4 28 -4 28 -18 -18 -18 -18 19 35 35 19 -6 -6

10 68 -8 36 . -12 -7 40 2 40 -7 40 2 40 . -39 . -39 9 57 9 9 9 57

3

3 3 3 . . . .

-52 -52

44 44 -30 -30 -30 -30 -30 . -30 -3 -8 -3 -8 36

13 1

14

12 1

12 1

13

11 1

12

4b 4e 4c 4d 2b 2a 2b 2b 4b 4e 4c 4d 4b 4c 4d 4e 4b 4c 4d 4e 4b 4c 4d 4e 1 1 1 1 1 1 -1 -1 -7 -7 5 5 5 5 7 7 -6 -2 -4 4 -6 -2 4 -4 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 3 3 -3 -3 3 3 7 7 13 13 7 -13 -13 7 3 3 15 15 3 -15 -15 3 10 10 14 14 10 -14 -14 10 2 2 -14 -14 2 2 14 14 15 7 -5 -5 15 -9 -3 -3 7 7 -7 -7 15 -9 3 3 15 7 5 5 7 7 7 7 1 1 21 21 1 -21 -21 1 -24 -8 24 -24 -24 -8 -24 24 -16 -16 . -10 10 -8 -18 -6 -28 28 -10 10 8 -10 10 8 -10 10 -8 -18 -6 28 -28 12 -4 -4 -4 28 -12 -12 12 12 -4 4 4 12 28 12 12 12 -12 18 6 12 -12 18 6 6 -12 18 12 6 -12 18 12 27 3 -1 -1 27 19 -9 -9 27 19 9 9 27 3 1 1 -6 26 -6 -6 -6 26 6 6 32 -32 : -32 32 -20 -12 -36 -12 12 4 -40 8 -8 -8 -40 8 -8 8 -40 8 8 8 -40 8 8 -8 -15 -23 9 9 -15 -23 -9 -9 -15 15 25 15 15 33 9 15 33 -39 -9 -9 -15 25 -15 -15 33 -39 9 9 33 9 -15 -15 4 16 16 28 28 -16 -16 4 4 -4 -8 -8 4 -4 8 8 -30 2 6 6 -30 2 -6 -6 6 -6 24 6 -6 24 6 -6 -24 . -24 6 -6 8 24 -8 -8 8 24 8 8 12 -44

4f 2b

4g 2b 4g 4g 4g 4g 1 -1 -7 7

4h 2a 4h 4h 4h 4h 1 -1 1 -1

11

11

11

-1 -1 1 1 9 1 -6 18 6 -18

4i 4j 4k 2b 2b 2b 4i 4k 4i 4k 4i 4k 4i 4j 4k 1 1 1 1 1 -1 5 1 5 5 5 -1 2 -2 2 -2 -3 -3 5 -3 -3 5 -3 -3 -5 -3 -3 -5 7 7 5 7 7 -5 3 3 -1 3 3 1 10 10 6 10 10 -6 2 2 10 2 2 -10 -1 -1 3 -1 -1 -3 7 7 9 -1 -1 3 -1 -1 -3 7 7 -9 1 1 5 1 1 -5 8 -8 8 -8 -16 -16 -2 2 -4 6 -6 -2 2 4 -2 2 4 -2 2 -4 6 -6 -4 -4 -4 -4 -4 4 -4 -4 4 -4 -4 -4 -6 6 -6 6 -6 6 -6 6 -5 -5 i -5 -5 7 -5 -5 -7 -5 -5 -7 -6 -6 2 -6 -6 -2

-4 12 -4 -8 8 -8 -8 -8 -8 -8 1 -15 -23 1 15 23 1 -9 23 -15 -9 -1 -15 15 -1 9 -23 1 -15 -15 -15 9 -12 -16 -12 16 -12 8 -8 -12 -8 8 2 6 -2 2 -6 2 14 -12 4 14 -12 4 14 12 -4 14 12 -4 -24 8 8 -24 -8 -8 28

-4 4 12 -12 -4 4 -8 8 -8 -8 8 -8 8 -8 8 1 1 i 1 1 -1 1 1 -1 1 1 7 1 1 7 1 1 1 1 1 -7 1 1 -7 4 -4 4 -4 4 -4 8 4 -4 -8 2 2 14 2 2 -14 -2 2 -4 -2 2 -4 -2 2 4 -2 2 4 8 -8 -8 8 -8 8 -4 4

H

u

1 1 5 5 2 2 -3 -3 -3 -3 7 7 3 3 10 10 2 2 -1 -1 7 -1 -1 7 1 1 8 8 -16 -2 6 -2 -2 -2 6 12 12 12 12 -6 -6 -6 -6 11 11 11 11 26 26

-3 5 -3 5 3 -5 3 -5 13 5 -13 -5 15 -1 -15 1 14 6 -14 -6 -14 10 14 -10 3 -5 5 5 -7 9 -5 -5 -3 5 7 -9 21 5 -21 -5 : 4 -4 -4 4

-4 -4 4

12

-4 4 4 -4

4 -12 -4

-9 -i

:1.4}

Conway's simple group Co2

176

Character table of E(Co 2 ) (continued} 10 10 10 10

2 3 5 7

11

2P

3P 5P

7P

llP X.1 X.2 X.3 X.4 X.5 X.6

X.7

4l 2c 4l 4l

4m 2/ 4m 4m 4l 4m 4l 4m 1 1

-1

1 -1 5

4n 2/ 4n 4n

4o 2/ 4o 4o

4p 2/ 4p 4p

4q 2/ 4q 4q

4r 2/ 4r 4r

4s 2/ 4s 4s 4n 4o 4p 4q 4r 4s 4n 4o 4p 4q 4r 4s 1 1 1 1 1 1

4t 2i 4t 4t 4t 4t 1

4u 4v 5a 6a 6b 2f 2i 5a 3a 3a 4u 4v 5a 2a 2c 4u 4v la 6a 6b 4u 4v 5a 6a 6b 4u 4v 5a 6a 6b 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 1 1 -3 -3 1 1 -3 1 1 1 1 3 1 1 3 3 -1 1 3 1 1 -1 1 3 3 2 -4 -2 2 2 -4 4 . -2 . -2 2 . -2 . -2 2 . 2 2 4 -4 2 -4 -2 1 1 1 1 -3 1 1 1 1 -3 1 1 1 1 -3"' 1 1 1 1 -3 1 1 -1 -1 3 1 -1 1 1 3 1 1 -1 -1 3 1 -1 1 1 3 3 3 1 1 1 3 1 -1 -1 1 3 3 -1 -1 -1 3 -1 -1 -1 -1 . 3 3 -1 -1 3 3 -1 -1 -1 3 -1 3 3 1 1 -3 3 1 -1 -1 -3 -1 -2 -2 2 2 2 -2 2 2 2 2 -1 -2 -2 -2 -2 -2 -2 -2 2 2 -2 -1 -2 -2 -2 -2 2 -2 -2 -2 -2 2 . -2 -2 2 2 -2 -2 2 -2 -2 -2 -1 -1 -5 -5 3 -1 3 3 -1 -1 1 6 3 3 -7 -7 -3 3 1 -1 3 1 1 6 . 1 -3 -3 -1 -1 1 1 1 -1 1 -1 -1 1 3 3 7 7 3 3 -1 -1 3 -1 ~ -1 -1 5 5 -3 -1 -3 3 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -3 -3 1 1 -3 -3 -3 1 -3 1 1 -3 . -2 -2 1133313113 . 8 -8 -8 8 . -8 -4

X.8 5 X.9 -5 X.10 -5 X.11 5 X.12 -5 X.13 -1 X.14 1 X .15 6 X.16 -6 X.17 10 X.18 -10 X.19 -1 X.20 1 X .21 9 X.22 -1 X.23 1 X .24 -9 X.25 5 X. 26 -5 X.27 X.28 X.29 X.30 10 -6 -4 4 4 -2 2 2 6 6 4 -4 . -6 X.31 2 -2 X.32 -6 10 -4 4 -4 -2 2 2 10 -6 4 -4 -4 -2 X.33 2 2 X.34 -6 10 4 -4 4 -2 2 2 4 . -6 . 6 6 -4 2 -2 4 4 4 -4 -4 . 4 -4 X. 37 -4 -4 -4 -4 -4 . -4 -4 X.38 -4 4444.44 X .39 4 -4 -4 4 4 . -4 4 . 2 2 -4 4 . -2 . -2 X.40 X.41 2 2 -4 4 . -2 . -2 X .42 2 2 4 -4 . -2 -2

r

H X.45

X.46 X.47 X .48 X .49 X.50 X.51 X.52 X .53 X.54 X.55 X .56 X.57

HS X.60

X.61 X .62 X .63 X.64 X.65 X.66 X.67 X.68 X.69

-5

~

§ _f

-6

-~ -~ _§ _§

6 -2 -2

7 -§

2

2

_f3 3

=~ -~ =~ -~ =~ 6 -2

-4 -4 -4 -4

. -3 . -3 . -3

-1

i6

. -1

6

-i .-1

-i -1 1

-5 -5 -5

1 5 1 -3 5 -3 1 -4 -4 4 4 -2 -2 10 -6 10 -6

X.78 X.79

-4 -4

-5

5 5 5

.

7 7 3 1 -5 -5 3 5 -1 -1 3 1 -5 -5 3 -3 5 5 -3 5 5 5 -3 -3 1 1 -3 1 4 4 -4 4 . 4 -4 . . -4 -4 2 2 -6 -2 -2 -2 6 -2 4 -4 . -2 -4 4 . -2 -4 4 . -2 4 -4 . -2 . -8 8

1 1 1 2

6

i

-1 3 -9 3 -3 -3

i-:i

i

i -i

i

1 -3 -1 -3 1 -1 1 -3 -1 -3 1 -1 -3 1 1 -3 1 1 9 1 -3 1 -4 4 4 . . . -4 2 2 2 2 -2 2 2 -2 . -2 -2 4 . -2 -2 4 . -2 -2 -4 . -2 -2 -4

-6 4 9 -5 -6 4

i

i

i

i -i -i

1 1 1 -1 1 1 -1 -1 -1 -2 . 2 2 -2 . 1 -1 -1 2 -2 1 1 -2 -2 :

:i

1 -1 -1 . -2 . 1 -3 1 1 1 1 1 3 -1

. 2 -3 1 1 -3 1 -3 -3 1 -1 2 2 -2 . -2 2

-~

2

_§

2

-2

2

.-2

. -2 -2

.

:

. 2

1 1 1 -1 1 i -1 1 A -3 A -3 A 3

r

2

. 2 . -2 . 3 -3 1

. -2

2

. -2 2

3 i -i

i -i

. -3

3 -i -i -i

i

3 1 -1 1 -1 3 -1 -1 -1 1 3 1 -1 1 -1

3 -3 -i 3 -3

1

§ _§ -~ 2 1 1 1

i -i 1

-2 -2 3 -1 -1

1 1 1 -1 1 1 -1 2 1 -3 1 -3 1 3

§

-2 -2 -2 -2 -2 2 -2 2 2 2 -2 -2 3 -1 -1 3 3 -1 -1 1 1 1 -1 1 -1 1 1 1 -1 1

:

-i -1 -i

:

-4 -2

-r -r =1 -1 -r i -~

2 2 -2

. 6 -6 2 -2 4 -2 2 2 -2 . 4 -2 2 -2 -4 -2 2 -2 2 . . 4 4 .

-l =~ -

A

A

A A A

A A A

.-4~

i -i

3 3 -3 -3 -6 6

2 2 2 -1 3 3 -1 -2 -2

. -8

-4 -4

. . . .

.

-2 -2 -2 -1 3 3 -1 -2 -2

i

1 -1

. . . . . .

3

3 -3 -3 -3 -4 4 -4 . 4 . -6 6 . . . .

.

8

. -8

5

1 -3 1 1 1 -3 4 4 -4 -4 -2 -2 -2 -2 -2 -2

-2

4 -4

. . . .

4 4

-2 -2 -2 -2 -2

-2 2 2 -1 -1 1 1 2 -2

2 -4 2 -2 -2 . -1 -1 3 -1 -1 3 -1 -1 1 -3 -1 1 1 1 3 1 2 -2 -2 2 -2 2 -2 -2

4

4 4

5 1 -3 1 1 1 -3 -4 -4 4 4 -2 -2 2 2 2 2

: -2

.-4

. 4 . -4

: -4 8 : -3 5 5 3 3

: -3

1 1 1 1 1 -1 -1 -1 1 -3 -3 -3 ~ 3 -2 2 -2 1 1 1 1 1 1 1 1 1 -1 -1 -1

4 -4

. -2 2 -2 -2 3 1 1 1 1 -1 . -2 -2 -2 2

:i -3 -3

1

. -1 -1 -1 -1 -1 -1 -1 -1 . -3 1 1 -3 -3 -3 1 -3 -i -i 3 1 1 3 1 3 -1 -1 . -4 4

.

2 -2

.

l

2

-2

3 -3 -3 3 -3 -3 3 -3 -3

Bb Be Bd Be Bf 8g Sh

!3 i3 -i -i -i -l r -1 -1 -1 -1 -1 3 3 -1 -1 -1 -1 3 ~ -i -i -i ! ! ½ -½ -i . -2 -2 -2 -2 -2 -2 2 -2

-1 -1

~ -2

2 -2 -2 -2 2 2 -2 2 2 2

=i8 -§4 -~ -4 g -5

i 1

A A

7b Ba

1 -1 -1 -1 -1 -1 1 1 3

1 1 3 3 1 3 3 -1 . -2 . . -2 -2 -3 -3 -1 2 -3 -3 -2 -2 . -2 4

. -6 . -6

. -~

_§

1 1

,1

i

1 -1

: -6

§ i

8 -8 -8 i

§

.. -~4 2f -2 -~

. -4 4 4 -4

ijj_:i

5

~

2

1 1 1 1 1 1 -1 -1 1 -1 3 -1 -1 -1 -1 3 -1 1 . -2 2

. : =§

-6 2 -2 -2 -2 1 . , . -2 -4 -2

: -4 -4

-4 -4

X.71 -10 X. 72 10 X.73 X.74 X.75 X.76

X.77

3 -§

-5 -1 -1 3 3 -1 -5 -1 5 -1 -1 -3 -3 -3 -1 5 -1 5

5 1 -3 5 -3 1 -4 -4 4 4 -2 -2 -6 10 -6 10

X.70

=t3

1 1 3

:=~ =i -i

§jg

§

6c 6d 6e 6/ 6g 6h 1a

t -1

-1 -1 3 3 1 -3 -3

-1 -1 3 3 1 -3 -3

. . -2 -2 2 2 -2 2 2 -2 2 -2 -2 2

j

-1 3 -1 -5 -3 5 1

i

1 -3 1 -3 -3 -3 1 . . . . . 2 2 -2 2 2 2 2 2

: -4

i

-1 -1 -1 -1 1 1 1 4 -4 -4 4 2 -2 -4 -4 4 4

4.6 Character tables of local subgroups of Co2

177

Character table of E(Co 2 ) (continued) 2

3

5

7 11

i

-1

-1 -1 -1 -1 -1 -1

-1 -1 J 1 1 -1 1 1 -1 -1 -1 -J -1 -1 -1 2 2

-A A

i

-i -i -i -1 -1 -1

-1 -1 -1

1 1 -1

-2

-i -i

=1

i

-i

1 1 -1 -1 -1 -1 1 1 -1 - l -1 -1 -1 -1 -1 1 1 1 1 1 1

-1 -1 1 -1 -1 1 -1 -1

1

2 -2

i

i

-1 -1

-i -i

-2 -2

-i -i

1

-~

1

J

-1 -1 1

-1 -1

-~

1

1

2 2

-1

i

-1

i

-3 -3 -1 -1

i

1

-1 -1 1 -1

§ 2 -2

2

i -i

-l

-2

1 -1 -1

1 -1

1

i

I

i I

-1

-i

-2

-1

.

1 1 -1

A

A -A

-A

i

-1

i

-1

1 -1 -1

1

1

i

-1

1

-1

-1

-i 1

i -i -i

-1

-i -i 2

-i

1 -1

2

i -i -i

-1 -1 -1 1 1 -1 -1 -1 -1

-i

1

-1 1

-1 1

-2

2 -2

1 1

-1

-i -i

i

--l

1 -i

=1 -1l i

-1

-i -i

1

2 -2 -

i -.1

i -i

-2

-1 -1

i

-l

i

i

1 -3 1 1 -1 -1 -3 1 -1 -1 1 1

1 1 1 1 -1 -1 1 -3 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -3 1 1 1 1 1

-l

1 1

-i -i -i

2 -2 -2

i

-1

-1

-1

i -i

-2

2 -22 -2 i

-1 -1

A

1 -1 1 -1 -1 -1

2 -~ 2 -2 -2 -~ - 2

-2

1 1

1 -1

-1

A

-2

-4

4

-1

1 -1

A

A

A

;

A

-i

-1

A

-1

-1

A

1 1 -1 1 1 -1 1 -1 -1 -1

A

A A

-1

. -A

-i -i -i

A

1 -1

. -A

4

i

1

-1

-1

-2 -2

-3 -3

J -1

;

3

-2 -2

-1

. 1 -1

i

3 -1 1 -3

2 -2 -2 2

A

-1

i -i

-2 -2

-1

-J

-2 -2 -1 1 -1

1

-2 -2

2 -2

2

A

-1 J

-1

1 1 1 -1 -1

A A

1 -1

-i

2 1

-i -3

-i

2 2 -1 J -l 1

1

i-11

-1

-1

1 1 1 1 1 -] -1 -1 -1

1

2 -2

-2

i -i -i -i -i

-1

-A -A -A -A -A -A -A

1 1 1

1 -1 1

i

-1 1 -1

1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1

-i l -1

1

2

-2 2 2 -2 -2

2

i

i

-1 -1

where A= ½(-1

+ iv?).

-i -i 1 -1

i

-1

-i

1

Conway's simple group Co2

178

4.6.2 Character table of H(Co2)

2P 3P SP 7P

18 4 1 I la la la la la I 7 15 16 21 21 27 35 35 56 70 84 105 105 105 I 12 120 120 120 135 168 189 189 189 210 210 216 240 280 280 3l5 336 336 336 378 405 405 405 420 432 512 560 560 720 720 810 840 840 896 945 945 945 945 945 1080 1120 1344 1680 1680 1680 1680 1680 1680 1890 1890 1920 2520 2520 2688 2835 2835 2835 2835 2835 2835 2835 2835 3024 3024 3024 3240 3240

18 I7 14 14 4 I 2 2 I I I I i 2a 2b 2c 2d la la la la 2a 2b 2d 2c 2a 2b 2c 2d 2a 2b 2c 2d I I 1 I 7 7 -5 -5 15 15 -5 -5 -16 -4 4 21 2i -11 -11 9 21 21 9 27 27 15 15 35 35 -5 -5 35 35 15 15 56 56 -24 -24 70 70 -10 -10 84 84 4 4 105 105 25 25 105 105 5 5 105 105 -35 -35 -112 -20 20 120 -8 120 120 40 40 120 -8 135 7 15 168 168 40 189 189 21 189 189 -39 189 189 -51 210 210 10 210 210 50 216 -24 216 -240 -20 280 2sci 40 280 -40 280 3I5 315 -45 -336 36 -336 -44 -16 336 -30 378 405 45 405 45 405 45 420 20 -432 60 512 512 -560 60 -60 -560 -20 20 720 720 810 90 90 840 840 -896 -96 96 945 49 -75 -75 945 49 -75 -75 945 49 -15 -15 945 49 -15 -15 945 49 105 105 1080 56 120 120 -1120 -40 40 -1344 16 -16 1680 -112 -1680 1oci -100 1680 1680 -1680 -14ci 14ci -1680 20 -20 1890 98 -30 -30 98 -150 -150 1890 -1920 160 -160 2520 2520

-2688 2835 2835 2835 2835 2835 2835 2835 2835 -3024 -3024 -3024 3240 -216 3240 -216

16ci -16ci 135 135 135 135 -45 -45 -225 -225 I 35 135 -225 -225 135 135 -45 -45 -204 204 -156 156 84 -84

16 2

16 2

2e la 2e 2e 2c I -1 7 8 5 -3 3 3 11 -8 -10 20 -7 I7 I -8 24 -8 24 39 8 -3 21 -3 -14 2 24 56 24 -8 -21 -24 40 16 -6 -27 -27 -27 4 24

2f Ia

H 2f 1 -1 7 -8 5 -3 3 3 11 -8 -10 20 -7 17 1 8 24

14 1

= (x,y,h) 14 I

15

11 I

2g 2h 2i la la Ia 2g 2h 2i 2g 2h 2i 2g 2h 2i I I 1 3 3 -1 7 3 3 4 -4 5 5 5 I I -3 7 3 7 -5 -5 3 7 11 7 -8 8 8 6 -10 6 4 4 20 9 9 -7 -3 -3 I7 1 5 5 12 -12

2j la

-8

24 39 8 -3 21 -3 -14 2 24 -56 24 -8 -21 24 -40 16 -6 -27 -27 -27 4 -24

H 2.i 1 -1 -1

-3

-3 3 3 3

-2

4 1 -7 1

-8 8 7 8 -3 -3 -3 2 -6

16 -6 -27 5 5 4

-6 -3 -3

-3

4

28 -28

-20 4;

20

4;

32 -32

-3 -3 45 45 33 33 -15 -15 9 9 24 24 24 -24. 16 -16

36 -36

2ci -2ci

-12 12 18 18 42 42 32 -32 3; 15 63 -45 -9 -33 39 15 3 52 4 -44

16 16 26 8 8

-48 16 16 50 2 -40 -40 1i -5 51 -5 -5 -21 -21 67

-24 -24

3b 3b la 3b 3b

1 -2 -3 -2 3 3

i

3a

-i

-1 2 7 3 6 6 -3 4 3 -6 3

18 8 -8

6 6 9 5 5

11

-5 -6 15 4

15

3c 3c la 3c 3c 1 I 3 4

2

2 2 1 3 3 3 -3 4 6 6 9 -3

-6 3 6 -8 10 -9 -6 -6 -6

-3 8 2 2 -9 -9

9 -7

-7 33 I7 I7 24

3a 3a la 3a

-6 -6 -4

-7 -7 17 I 8 . -14 -6 : -12 -12 15 15 6 -12 12 9 9 -12

-5

10

,;

14 3

14 1

13 I

12 I

12 I

4a 2a 4a 4a 4a I 7 15

4b 2a 4b 4b 4b I -1 7

4c 2b 4c 4c 4c 1 -5 -5

4d 2f 4d 4d 4d I 3 -1 4 I 5 3 7 -1

4e 2b 4e 4e 4e I -1 -1 -4 -3 -3 3 3 3

2i 5 -Iig 21 -3 27 15 3 35 3 -5 35 11 15 -8 -24 56 70 -10 -10 84 20 4 105 -7 25 105 I7 5 105 1 -35

8

120

40

-9 168 189 189 189 210 210 216

-i 40 21 -39 -51 10 50 -24

8

-2 I

24 40 -8 -40 -21 -45

6 6 6 -9

16 -16

-6 -30 -27 45 21 -3 21 -3 4 20 -4 8 8

sd

6 6 1i 8 9 9 9 9 9 -9 4 -5 -6 12 12 112 12 -6 -6 15 -12 12 -9 -126 -9 -126 15 168 168

12 1

12 I

4/

4g 2b 4g 4g 4g I

H

4f 4/ I 3 -1 -4 1 5 3 7 -1

2 -2

4 -3 -3 5 12 12

-4 3

9 -3 -3 6 -2

-2 4 I -7 I -4 8

4 1

-7

I 4 8

-8 7 8 -3 -3 -3 2

-6

-4 3 9 -3 -3 6 -2

20 4

8 3 -12 -12

6 -3 -3 -3 -4 -12

-6 -3 -3 -3 4 12

-3

-3 -4 12

-4

4

48

9 -3 9 -3

8

12

-8 18 6 -8 36 8 -12

18 -8 8

-8

-7 I7 I 8 8

9 -3 9 -3

-8

16 -16 -16 -8 -8

-4

16

24 -24 32 3 3 3 3 -21 3 - 21 -21 -12 -12 -12 24 -24

2

-4 12

-4

-9 -9

-9 3 3

24 24

-8

4 -16 16 4 -28 10 -14

. -189 -69 . -189 -21 . -189 99 -189 -21 -189 -21 -189 27 -189 27 -189 51

216 216

-7

-7 -7 17 I 8

-12 12 24 -16 24 -8 16 -8 20 -4 -20 -12 28 12 6 10 6 6 -14 10 6

40 40

6 -12

,;

-28

5 -3 -7 -3 -7

5 I I -7 -8

7 8 -3 -3 -3 2 -6

8 3 -5 12 -20 12 -4

6 -3

28 -8 -8 -6 6 36 . -12

-8

-4

-4

-5

-1 -1 4 -3 -3 3 3 3

-9 3 3 3 15 -9 -9 3 15 3 -9 15 3 -9 . -12 . -12 36 36 : -12

12

-9

3 3 15 -9 12 12 -36 36 -12

4-6

Character tables of local subgroups of Co2

179

Character table of H(Co2) (continued)

2P 3P 5P 7P X.l X.2 X.3 X.4 X.5 X.6 X.7 X.8 X.9 X.10 X.11 X.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19 X.20 X.21 X .22 X .23 X.24 X .25 X .26 X.27 X .28 X.29 X.30 X.31 X.32 X.33 X.34 X.35 X.36 X.37 X.38 X.39 X.40 X.41 X.42 X.43 X.44 X.45 X.46 X.47 X.48 X.49 X.50 X.51 X.52 X.53 X.54 X.55 X.56 X.57 X.58 X.59 X.60 X.61 X.62 X.63 X.64 X.65 X.66 X.67 X .68 X.69 X.70 X.71 X.72 X.73 X.74 X.75 X.76 X.77 X.78 X.79 X .80 X.81 X .82

13

11 1

11 1

12

12

10 1

11

11

4h 2b 4h 4h 4h

4i 2e 4i 4i 4i 1 3 -1

4j 2b 4j

4k 2e 4k 4k 4k 1 -1 3 4 1 1 -1 -1 3

41 2e 4! 41 4! 1 -1 3 -4 1 1 -1 -1 3

4m 2e 4m 4m 4m 1 3 -1

4n 2b 4n 4n 4n I 3 3

4o 2f 4o 4o 4o 1 -1 3

i

2 4 -3 I 1 -4 4

2 4 -3 1 1 4 4

2 4 -3 -3 5

5 1 7 -5 7 8 6 4 9 -3 5

4

4 11

-4 3

5 1 7 -5 7 8 6 4 9 -3 5

-1 2 4 -3 -3 5

!} 1 -1 7 5 -3 3 3 11 -8 -10 20 -7 17 1

-4

-i

8 -11 1 13 10 2 8

-8 12 3 -i 8 9 -3

-3 21 -3 -3 6 -14 -2 2 24

8

-8 3

-16 2 -3 13 13 -12

24 -8

-5

-21

6 -3 -3 -3 -4

16 -6 -27 -3 -3 4

11

i I -3 -2 -2

5 3 7 -1

2 4 -3 1 1

-4

-i

8

9 -11

-3 -3 -3 -2 6 -2" -2

12 -12

-2 5 5 5 -4 -4

i

1 -1 -1 3

1 13 10 2 8

4 -5 i I -3 -2 -2

8

:i -4 -4

-5

-2 5 5 5 -4 4

6 -3 -3 -3 -4

-8 3

-16 2 -2 -3 5 -3 -11 -3 -11 -12 -4

12 -12

10

6 -12 36

-6

-4 8 8 6 -4 -4

1:i -3 -15 I 25 24

-:i 9 -3 9 -3

i

9

-8 -8

1 -11 -11 -7 13 13 -7 -7 -7 -7 -11 -11 -8

-:i 9 -3 9 -3

8 -8

12 -4 -i -17 3 39 15 23 -1 -13

9

16 -16 8 8 -12 12 8 8 8 8 4 -4 4 -4 6 -14 6 6 2 -2 -2 6

-8 -8 24 -14 10

4 8 8 8 8 6 6 -4 -12 -4 -12

-9

8

-6 -:i -3 1 1 -7 -8

8 6 -4 -4

-7

5 -3 9 5

10 10

4p 29 4p 4p 4p I I -3 -2 -3 -1 I -1 5 4 2

4q 29 4q 4q 4q 1 -3 1 -2 -3 3 5 -1 1 -4 2

4r 29 4r 4r 4r 1 I -3 2 -3 -1 I -1 5 4 2

-:i 3 -1 -2

-:i -1 -5 6

-:i 3 -1 2

-4

4

-4

i -5 I -2 2 -4 6 4

2 -3 -3 -3

-8

2 -6

:i :i :i -9 3-17-17 3 3 3 -21 3 3 3 -9 3 -5 -5 -9 3 3 -1 -1 3 3 3 7 7 3 15 11 15 3 11 -9 -21 -9 -9 -9 -12 12 4 -4 4 -4 -12 -4 -4 -12 -4 -4 -12 36

6

-2

,j

-4 -4

-i -1

-1:i -1 3 -13 -9 11 -1 15 -9 -9 -1 -5 3 7 -4 -4

9

-4 -4

:i 6 -6

2 -3 -3 -3

2 -3 -3 -3

2 -6 -2 -2 -3 -3 5 5 -3 5 -3 I -3 5 -3 I 4 -4 -4 10

-2 2

8

-4

-3 3 -3 9 9 -3 3 -2 10 -2

-2 2 2 -2 -2 4 4 4 -i -:i i i I -3 -3 -1 -7 I 1 -1 -1 -5 I 1 -1 -1 I -6 2 -2 2

:i -2 -6

-4

4

4x 5o 2i 2g 29 5o 4v 4w 4x 5o 4v 4w 4x la 4v 4w 4x 5a I I I 1 -1 2 1 1 3 I 1 2 -2 1 1 i I -1 -1 -1 I I -1 -1 -1 1 3 1

:i -6 6

3 -9 -3 -3 -3

2 -6 -6 -6 8

-:i i -3 I 3 -1 3 -1 3 3

4v 4w

i -5 I -2 2 -4 -6 4

6

-8

4u 2e 4u 4u 4u I -1 -1 -1 3

i -1 I -2 2 4 -2 -4

-9 3 -3 -3 -3

-8

4t 2b 4t 4t 4t I

:i - i 3 -i :i 8 i -3 i i i i -1 -3 I -1 -1 I I -3 -3 -3 I I -2 2 -2 -2 -2 -2 2 -6 -2 -2 -2 -2 4 2 2 -2 -4

-2 -10 -10 2

4s 29 4s 4s 4s 1 -3 I 2 -3 3 5 -1 1 -4 2

10 2 -6 -6 -8

10 -2

:i - i - i -2 2 2 -2

2 -2

i I I 2 -2 1 1 1

2 -2 -2 2

-8

3 i i -:i - i - i -9 1 -3 I 3 3 -3 1 5 1 I 1 -3 -7 1 -3 -3 -3 -3 9 -3 1 -3 -3 8 -4 4

-6

-6

-8 -8

-:i 9 3 9 -3 -3 9 3 2 -2 2

-5 -5 -5 -5 3 -5 3 3

-5 -9 -5 3 7 -1 3 -1

-i 3 -1 -1 -5 3 -1 3

-:i I 3 I I -3 -3 -1 2 -2 2

-:i

1 3 1 1 -3 -3 -1 -2

6 6 9 5 5 11 -5 -6 15 -4 15

2 2 2 I 3 3 3 -3 -4 6

6

6d 3o 2e 6d 6d I 2 -1 3

6e 3o 2/ 6e 6e I 2 I -2 -1 3

6f 69 3b 3b 2c 2d 6g 69 I 1 -2 -2 -2 -2 1 -1 -2 -2

~f

:i 1

:i

-1 -1 3 3 -2 -2 -3 -3

-1 -1 2 2 I 4 3 -2 3

-1 -1 -1 -1 -2 -2 2 4 4 2 2 2 1 I 1 -4 -2 2 3 -2 3

9 -3 2 -2 -3 9 9 3 -3 9 -6 -15 :i -2 -2 I -1 -1 -1 3 15 -9 -3 -6 -12 2 -2 -2 -8 -5 -2 I 10 10 I -2 -2 2 -9 3 3 6 -6 6 -6 -2 6 -6 2 -2 -6 -2 -2 6 2 -9 3 -:i

-5 -5

-2 -3 3 -3 1 -1 -3 2 1 2

i -4 -4 3 -3

-4 -8 -2 2 -8 6 -6 3 3 3 3

:i -:i I -1

6 6 6 6 6 6 -8 -4 4 -:i 9 9 9 9 9 -9 5 -4 -2 2 -i 6 -12 -2 2 -2 12 -12 4 -4 4 -4 -6 3 3 -6 3 3 -15 2 -2 12 i -i -12 4 -4 2 -2 -9 -9 -15 4 i -i -4 -3 -3 -3 -3 -6 12 4 -4 -2

-6 -6 4 -Ii

-i

2 -2

-8 -2 -10 -2 2 -2 6 -2 2 -6 -6 -6 6 -2 -2 -2 -6 -6 2 -2 -2 2 2 -8 8 -4 -4

-i

6c 3c 2a 6c 6c 1 1 3 -4

6

-2 -2 -9 -9

-8 6 -2 4 4

8 -8

-i -1 2 7 3 6 6 -3 -4 3 -6 3

6b 3b 2a 6b 6b

-9 8 -16

2 -2

-9 3 -3 -3 -3

-:i 9 9 -3 3 3 9 -3 -3 9 -3 9 9 -3 3 3 -2 2 2 -10 -2 2

-2 -1 -1 -1

6a 3o 2a 6a 6a I -2 -3 2 3 3

-2

-i

2 -1

-2 -1

14 6 -12 12 15 15 -6 12 -12 9 9 12

-9

-9 -9

-:i :i 3 -3 -3 3

180

Conway's simple group Co2

Character table of H(Co 2 ) {continued)

2P

6h 6i 3c 3c 2 6h 6i 1 1 -1 1 1 1

gt lK 61

7P X.l X .2 X 3

§:t

6j 6k 6l 6m 6n 60 6p 6q 7a 8a Bb Be 3c 3b 3b 3c 3c 3b 3b 3c 7a 4a 4b 4b 2 6;,; 6j 6k 6l 6m 6n 60 6p 6q la Ba Bb Be 1 1 1 1 1 1 1 1 1 1 1 1 -1 2 2 1 1 -1 . -1 -1 3 1 2 2 1 3 -1 1 -1 3 -1 2 -3 i 1

~~

§ -~ -§

ik /i

l~

~~

i; it

=! -! =§ . ~ -~

§x:_¥3

. -2·

. _33' _33' -2·

2: -11 -rl

X .9

2

2 -1 -1

2

.

~~ ~~

1

-i

1

~g

8d Be Bf Bg Bh 4i 4o 4h

4h 4h

~~ ~~

~: ~1 ~i

X .15

1

3

3 -1

1

5 -1

-i

1

1

1

§:i~ -: : : -

2

§:is-2-2-2 §Jr X .22 X. 23 X .24 X. 2 5 X.26

§ j~ X .29

X. 30 X.31

-:

-2

.

r _1

1

. -2

: -2

2

.

2 -2 -2 -2 3 3

us

i-i

}J! X .43

4

.j . -3

: -4

. -4

-i _.j

3 -3

§1¥

i -i i -i

1 -1

§JL.i . 4 X.50-3 3-3 X.51-3 3-3 X.52 3-3 3 X.53 3-3 3 X.54 3 3 3

=~ =~ -~ §J~ -~ }Jg

2

-i

:i

3 -3 -3

-

3

1

i -i -~

-l

2 -2

2 -2

-4

-½

-i

.

1-1-1 1 -1 -1

. -~ -~

-3 -3 -3 1 1 2 -2 6 -2 -2 -6-2-2 2 2 4 -4

3 -3

1

1 1 . -2 .-2

: -6 -2

1

1

. -2 -2 . -2 2

6 2 2

1

~ ~ -2

2

1

1 -1

-i -i -i -1 1

.

2 -2

i -i

-6 -1

=1 -1

i

1

i

1

-1 -1 -1

3 i -i

-6 3

-i -i

: -2 -2 . 2

: -i

. -1 . -1

. -2

: -2

i -i -i -i

i

10 -9

-l

-6 -:i

-2

2

2 2 . i -1 -i

i -i

3 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 -3 -1 -1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 -1

i -i

: -2

-2

2

2 -2 2

2

6 -2

3 3 3 3 3 3 3

i

3 7 -1 3 -5 -1 -5

:i

-1 -1 3 -1 -1 -5 3

-5

2

.

2

: -2

. -2 -2

2 -2

2

i -:i -i

i

i -i

i

i

-3 -1 -3 1 1 -3 -1

1 -1 1. -3 -3 1 -1 . .

1 -1 -1 1 1 -1 1 2 -2 -2 2 2

-1 -1 1 -1 -1 1 1 -2 -2 2 -2 2

2

. -2

. -2

. -2 -2

:i

-5

: -2 -2

2 . -2 -2 . -2

2 -2 -2

. -1

-i

-1 -1 1 -1

3

5 i -i

~

6

-3

-2

-2

.

-1

-8

2

.

-i

~

-i -i -i

:-2

-i

2 -8

1

1 -1 -1 1 1 1 . -2

. -2 2 -2 . -2 -2 2

: -i

ig~

.-2

2 2 2 -2

.....

i

-2

2-2 1

1 1

1 1 1 -3 1 -3

: -2 -2

1 1 -3 1 -7 1 1 5 -3 1 1 . -7

i -i

2

X.59 -2 4 -4 -2 X .60 X.61 . . . . X.62 2 2-2-1 -2 . -1 X.63 -2 -2 2 -2 2 . X.64 -3 3 -3 3 -1 X.65 3-3 3. -3 -1 4 . -1 X.66-4-4 4-1 X.67 X .68 . X.69 -2 2 -2 2 -2 2 -2 X.70 X.71 X.72 X.73 X.74 X.75 X.76 X.77 . :i -:i i -i X.78 . -3 3 1 -1 X.79 3 -3 1 -1 X.80 X.81 X.82

~~

1 -1 -1

1-1-1-1 1 -1 -1 -1

. -2 -6 -2 -2 -2 -2 . . 2 . -2 2 -2 -2 -2

-l

1 1!~ I Ob 10c 12a 1 1 1 -2 -3

_z · -1

:i -i -i -i -i -i -i -i -i

. -2

: -i

~~

: -2 -2

-4

. -1 -3 5 -3 -3 -3 . -1 -3 1 1 1 1 -1 -3 1 1 1 1 4 -4 -4

.-1 .-1 .-1 1 -2

.

:i -5

. -1

. -2 2

5 -5 -1

i

2

2 -2 2 2

. -1 -1

i -i

1

1 -1

-2 : --i -i i -i -i -i

l . . . 4-4 2 - 2 . 2 -1 -1 -1 -1 -1 1 1 -1 2 -l 1 : -i i 9 i i -i -i i 1 . -3 1 -3 -1 -5 -1 1 -1

=~

-4 -4 -2

1

1

2

2 2 . -3 -3 . 3 3 . -3 -3 1 . 1 1 1 1 1 -1 2-1-1 .-1-1 -2 -~ -2 : -1 -1 : 2 . -2 . -3 -3 -2 -2 -1 -1 . 1 -1 1 -2 -2 -1 1 -2 -2 -1

.j X.34 -2 X.35 X .36 X.37

X.44 X.45

:

1 -3 -1 -1

1-2-2-1-1

-r r -r

§ :H

X.40

1

ii ii ~t ~i ~:

I Ob 10c 12a 5a 5a 6a

-3 -3 _f =f i =f _f -i i -i -f _f =f -i3 -l i7 -1~ -1 -1 -11 -l1 -1 -1 -11 -l1 -i1 -1 -1 -11 -1. ~ - l - l -1

=f -l -l -i i -1 -1 i -2 2 2 -i ~ i:H =i i =l -~2 -12-1i ~3 - 2.- .2 -1l ..-71 _§1-3-1 -1 -3 -33 X.14-1-1-1

§:l~ =I

~~

8d Be Bf By Bh 1 1 1 1 1 -3 1 -1 1 1 1 -3 1 -1 1

Bi Bj Bk Bl Bm 9a 10a 4i 4o 4k 4n 4d 9a 5 a 1 2°~ Bi Bj Bk Bl Bm 9a 1 Oa 1 1 1 1 1 1 1 -1 1 1 1 -1 1 2 1 -1 -1 1 1

-:i

-3

i -i

-3 1 -1 -1 1 -1 -1 -1 -1 -1 -3 -1 1 1 1 -3 1 -1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 3 1 1 1 -1 -2 2 2 2 2 -2 . 2 -2 2 2 2 -2

1 -1 -1 1 1 -1 1

-2 -2

-i

i

1 -1 1 -1

4- 6

Character tables of local subgroups of Co 2

181

Character table of H(Co 2 ) {continued)

-i -i

1

3

i

-i

2 -2

-1 i -1 1 -1 -1

-2

2

-J

-1

=i

-i

1

2

1 -1

-1

i

-2 2 -1

-i -2

1 -1

=T

1

i

i

-]

1

i

3 -1

-1

-1

-i -2 i 1 -1 -i

-i

-2

1

-i

i

1 1

-r

-i

-2 i

i

1

1

-1 -1 l 1

i -2 -2 -i -]

i

-1

i

-i i -l

i

-1

-1

-i

-i

1

-i -i -i

1

-i -i -i

=1

-i -~

-i

t -i

-1

-1

1

-1

-2

-1

-2

;

-i

2

J

-1 -1

. -2

-i -i -i

1

i

2

2

-2

1 1

-2

-2

=~

2 -2 -2 -2

-i -3

-i

B

-l -l

-1

-i

-i

CJ

-i -1

-i

-1 1 -1

1

l

A A 1

1 1

-2

i

-1 -1 -1

-i -i -i

-i

i

i

-i

-1 -1

-1 -1

1 1

. -i

-1 l J

i

-i

-1

-1

1 1

1

-1

-1 -1

-1

-2

2

i -i

i -i

-1 -1 1 1

-i

-i

-i

i

1 1 1

-1 1 1 1 -1 -1 1 1

1 1

-2

-1 1

-i

i -i

-3

-i -2

i -i

j

1

A A

-i

-1 1

i

-i -i

-i -i

i -i -i

B

i -i

-i

1 1 1

-2

1 -1 -1

2

1 -2 -2 -2 2

-i

-i

B

JJ

-1 -1 -1 1

1 -2 -1

-i

-1

-i

2 2

-i

i

-1

2

-i

-1

i -i

-1

J L

1 1 -1

-1 -1

-i -i -i

-i -i -i -i -i

-2

-1 -1 -1

-1 -1 -1

1 -1

1 1

2 -2

i

i

1

-i -2

-i

-2 -2

1 -1 -1

-1 -1

-i

-i 1

-i

-i

-i -i

1

-2

2

1 l

-i -i -i

1 1

=i

-1

-i -t1 -1i 1 -1 -i

i -i

2

-2

-2

i

1

-i -i

=! -l

-1 1 . 1 1 -J 1

-J

i -i

3

-i

-1

-2

2

-2

2

-J

-i -i

1 1 1 -1 -1

2

=i -1 =i

1

-1

-3

1

-i -2 -i

-i -2

i -i -~

-i

~ -2 -i

-1

2 -2

-i

-i i

-J

-i

1

-i

3

-1

-1

-i -i

-i -J

i -2

-1

-i

-1 i 1 -1

l -1

i

-1

1

-i

-i -i 1 -1

-1 -1

-i -1

-1 -1

1 1

1 -1 1 -1 -1 1 -1 -1

-i -1

Conway's simple group Co2

182

Character table of H(Co2) (continued) 18 4

2P 3P 5P 7P X.83 X.84 X .85 X.86 X.87 X.88 X.89 X.90 X.91 X.92 X.93 X.94 X.95 X.96 X.97 X.98 X.99 X.100

1 1 la la la la la 3360 3360 3456 3780 3780 4480 4480 5040 5040 5376 5670 6048 6480 6720 6720 7560 7680 8192

18 17 14 14 16 4 1 2 2 2 1 1 1 1 2a 2b 2c 2d 2e la la la la la 2a 2b 2c 2d 2e 2b 2e 2a 2c 2d 2a 2b 2c 2d 2e -3360 40 -40 -112 -3360 200 -200 16 -3456 -96 192 96 132 3780 196 -60 -60 3780 196 -60 -60 -156 -4480 . -160 160 -64 -4480 160 -160 192 5040 -336 48 -5040 . -180 180 -168 -5376 -64· 64 128 270 -90 5670 294 270 -6048 . -120 120 -48 180 -180 -216 -6480 6720 -448 . -192 -6720 32 80 -80 392 -120 -120 -24 7560 7680 -512 -8192

16 2

14 1

2/ la 2f

2g la 2g 2g 2g 40 8 32 -60 36 -32 32

H 112

-16 -192 132 -156 64 -192 48 168 -128 -90 48 216 -192 -32 -24

12

-64 -18 8 -12 -48 -24

14 1

1511 .1

14

14

13

12

4a 2a 4a 4a 4a

4b 2a 4b 4b 4b

4c 2b 4c 4c 4c

4d 2f 4d 4d 4d 24 -8

3

2i 2j 3a 3b 2h 3c la la 3a 3b 3c la 2i 2j 2h la la la 3a 3b 3c 2h 2h 3a 3b 3c 12 -15 -40 12 -8 -6 15 -9 -32 -60 4 4 9 36 -28 4 9 . -20 10 4 32 -32 16 -5 ~8 -16 18 -12 18 64 12 -18 -26 6 . -8 -9 12 64 12 12 48 -24 -24 -9 : -12 . -24 . -16 -16 -16

~! H

1

1

-252 -60 -252 36 336 -378

16

1

12

1

1

11 11

12

12

4i 4j 4h 2P 2b 2e 2b 3P "4h 5P 4h 7P 4h 4i 4j X.83 X.84 X.85 . . X.86 4 12 X .87 -28 -12 X.88 X.89 : -24 X.90 X.91

4k 2e 4k 4k 4k

4l 4m 2e 2e 4l 4m 4l 4m 4l 4m 8 8

1

1

!! !1

:H~ X.94

14

-6 6

X .95 X.96 . X.97 X .98 -24 X .99 X.100

8

-8 -8 -4 4

12 -12

24

H H

4 4 32

. -24 : -24 . -20 -12 20

12

-6 24 : -12

6 -6 6 24 -24 -24 12 -12 12

-16 -16 8

-504 -24 512

16

10 10

11

4q 2g 4q 4q 4q 4 -4 - 8

. : -8 8 . -8 . . . -6 -6

2 -6 -2

-8 8 20 -20

: -16

1

4n 4o 4p 2b 2/ 2g 4n 4o 4p 4n 4o 4p 4n 4o 4p . 4 . -4 8 -4 12 4 -4 4 -12 4 4

-8 -8 2

10 11

4r 2g 4r 4r 4r -4 4 - 8

4s 2g

4s 4s 4s -4

4t 2b 4t 4t 4t

4u 2e 4u 4u 4u

4v 4w 4x 2i 2g 2g 4v 4w 4x 4v 4w 4x 4v 4w 4x . -4 4 . -4 4

4 -4 4 4 4 -4

. -4 -4 4 4 6 6 -6 -6

-10

6a

6b

3a 2a

3b 2a

Ia 6a 6b 5a 6a 6b 15 . -12 6 -15 9

. 20 -10 . -16 5

8 -8

2 -6 -6 -6 -6

5a 5a 5a

: -2 2 2

-U 1 -12 :

4 -4 -2 2 -2

6

: -24 16

2

6c 3c 2a 6c 6c -12 .

9

6d 6e 3a 3a 2e 2f 6d 6e 6d 6e -4 4 -2 2

.

9

.

8

6 6

-4 -4

.

4

. 6 -6 . -4 4

6f 6g 3b 3b 2c 2d

6f 6g

6f 6g 1 -1 -1 1 -3 3

2 -2

1 -1

2 -2

s -3 -6

16

-6 .

1

4e 4/ 4g 2b 2b 4e 4g 4e 4g 4e 4g -8 -24 8 24 8 -24

Character table of H(Co 2 ) (continued) 13

12

4 12 4 12 4 -12 4 -12 . -32

6 -18

448 -64

12

12 -6 -6

. -12 -9 16 -1g

.

2 -2 -4

16 8

4. 6 Character tables of local subgroups of Co2

183

Character table of H(Co2) (continued)

2P

~~

7P

6h 6i 6j 6k 3c 3c 3c 3b

X .84

-:i

2 -2 -1 4 -4 1 -:i -:i 3

HI -~. =~ =~ X.89 -4 X.90 X.91 . X.92 -4 X.93 X.94 X.95

§J~ X .98 X .99

2 -2 -~

3 -3

~z 6g gj

Ji

Gk ~16i ~{ it /z 6j 6k

6h

X .83

§Jg

6l 6m 6n 60 6p 6q 7a 3b 3c 3c 3b 3b 3c 7a 7a a~ ~~ 7a la 6l 6m 6n 60 6p 1 -2 . 1 -1 -1 -4 . -1 1 -3 -:i i -1 1 1: -2. . 1 1 2

4

2 -2 -4 : -:i

-2

3

8

Sc 4b Be Be Be

8d 4h 8d Bd 8d

8

7

7

Se Bf Bg Bh 4h 4i 4o 4h Be Bf Bg Bh Be Bg Bh Be Bg Sh

H

7

6

Bi Bj Bk Bl Sm 4i 4o 4k 4n 4d Bi Bl Sm Bi ~q~ Bl 8m Bi Bk Bl Sm

d

9a 10a I Ob 10c 12a 9a 5 a 5a 5a 6a 3a 1 Oa 10b 10c 4a 9a 2 a 2d 2c 12a 9a 1 Oa !Ob 10c 12a

-i

-4 -4 -4 -4

4

-i

4

. -1

2 -2 : -i

2 -2 -2 -4

3

Ba Bb 4a 4b Ba Bb Ba Bb Ba Sb

9

6 -6

: -2 : -i

3

: -2 2 2 2 2 -2

-6

2 -2

-i -i

: -2 : -2

-2

-8 : -i

X.100

-2

Character table of H(Co2) (continued)

2P

12b 12c 12d 12e 12/ 12g 12h 12i 12j 12k 12l 12m 12n 120 12p 14a 14b 14c 15a 16 a I6b lBa 24a 28a 30a 6c 6e 6n 6a 6e 6n 60 6d 60 6n 6n 6d 6d 60 60 7a 7a 7a 15a Bb Be 9a 12b 14a 1 5a

1ii i~ 11b 1t~ 1ij 1i~ 1if 1iZ 1iK 1il 1il 1ii 1~l 142~ 1i: 1i~ 1i; i1~ il~ Hi \b ?~~ l~i 1~~ 2l~ 228s: 7P I2b 12c 12d I2e 12J 12g 12h 12i 12j I2k 12l 12m 12n 120 12p 2a 2b 2b 15a 16a 16b 18a 24a 4a 30a 2 2 §:U -2 -l -l -i -i -i §Jg

-3

i

X.87 -3 X.88 X.89 X.90 X.91 X.92 X.93 X.94 X.95 X.96

1 -2

ui

:i

where A

2

-1

1

-i

-i

-i -1

1

-i

-2

-2

X .99 -4 X .100

-1

-2 -2 -1

-i -i

-i -i -i

i -i -i

-2

= i\1'7,

-i

-1

-1

B=-l-iv3.

-i

5

Fischer's simple group Fi22

In 1970 B. Fischer found three sporadic simple groups by characterizing the finite groups G which can be generated by a conjugacy class D = z 0 of 3-tmnspositions. That means that the product of two involutions of D has order 1, 2 or 3; see [30]. He proved that, besides the symmetric groups Sn, the symplectic groups Spn (2), the projective unitary groups Un (2) over the field with four elements and certain orthogonal groups, his two simple groups Fi22, Fi 23 , and the automorphism group Fi24 of his third sporadic group Fi; 4 comprise all 3-transposition groups [29]. For such a group G = (D) Fischer constructed a graph g and its action on Q. As its vertices he took the 3-transpositions x of D. Two vertices x, y E g are connected if x and y commute in G. In that case they are joined by an edge ( x, y) in g. Fischer showed in his lecture notes [30] that each of the groups G considered in his theorem has a natural representation as an automorphism group of its graph Q. Furthermore, Theorem 17.2.3 of his lecture notes [30] contains the first existence proof for his sporadic group Fi22. It is the purpose of this chapter to provide an existence proof for the smallest sporadic Fischer group Fi 22 by means of Algorithm 1.3.8. This proof is due to H. Kim and the author [72]. The methods presented here are purely algebraic and do not require any geometric insight. In particular, they are independent of [30] and [4]. In Section 5.1 Algorithm 1.3.8 is applied to the split extension E 2 of the Mathieu group M22 by its simple module Vi defined in Lemma 4.1.1. Proposition 5.1.1 describes a construction of a finitely presented group H 2 with center Z(H2 ) = (z, v) of order 4 and exponent 2. It has a Sylow 2-subgroup S2 containing a unique elementary abelian normal subgroup A 2 of order 210 such that D 2 = NH 2 (A 2 ) "" D = CE, (z). Hence all conditions of Step 5 of Algorithm 1.3.8 are satisfied. 184

5.1 Construction of the 2-central involution centralizers

185

In Section 5.2 we take the constructed presentation of H 2 as the input of Algorithm 7.4.8 of [92]. It returns a finite simple subgroup of 1!5 2 GL 78 (13) having order 217 · 39 · 52 • 7 · 11 · 13 and a 2-central involution 3 such that C® 2 (3) ~ H2; see Theorem 5.2.1. In [72] H. Kim and the author verified that the character table of 1!5 2 is equivalent to the one of Fischer's simple sporadic group Fi 22 . For applications in Chapters 6 and 9 a nice presentation of 1!5 2 is determined in Corollary 5.2.2, which, by C. Praeger's and L. Soicher's book [110], is also satisfied by Fischer's original simple group Fi22. Section 5.3 contains a sketch of a uniqueness proof for Fi 22 . This result is mainly due to D. Parrott [107] and D. Hunt [55]. In particular, it can be shown that any simple group G having a 2-central involution z with centralizer Cc(z) ~ H2 is isomorphic to the simple matrix subgroup 1!5 2 of GL78(13). In Section 5.4 it is shown that Algorithm 1.3.8 constructs the automorphism group Aut(Fi22) of the Fischer group Fi22 when it is applied to the extension group E4 constructed in Lemma 4.1.1; see Corollary 5.4.1. However, it is also mentioned in Section 5.4 that it fails to construct a centralizer Hi of a group when it is applied to the extensions Ei, where i = 1 or 5. The systems of representatives of conjugacy classes of the local subgroups E 2 , H 2 and D 2 are stated in Section 5.5. The final section contains the character tables of these groups. The four generating matrices of the matrix group 1!5 2 ~ Fi 22 and a system of representatives of its conjugacy classes are documented on the accompanying DVD.

5.1 Construction of the 2-central involution centralizers

In this section we apply Algorithm 1.3.8 to the extension group E 2 of Lemma 4.1.2 in order to construct a group H 2 which is isomorphic to the centralizer of a 2-central involution of the simple group Fi 22 . Proposition 5.1.1 Let E 2 = (a, b, c, d, t, g, h, i, v1 ) be the split extension of M22 by its simple module ½ of dimension 10 over F = GF(2) defined in Lemma 4.1.1. Then the following statements hold.

(a) z = (iv 1 ) 2 is a 2-central involution of E 2 with centralizer D = CE 2 (z). (b) D is a finitely presented group D = (Yi I1 :::; i ::::; 11) having the following set R(D) of defining relations:

186

Fischer's simple group Fi22 y;

=

Y;

(ysyg) 2

= y; = Yi = Y: = Y~ = yJ = Yi1 = 1, = (YsYu) 2 = (ygyu) 2 = (Y1oy3y5y4) 3 =

1,

Yu Y3 Ys Y4 Y10Ys Ys Y4 Y10Ys 1 Y101Y"i: 1 Y~;1 Y3 1Yu/ Y"i: 1 y~;1 Y3 1

=

1,

Yu Ys Ys Y4 Y10 Y3 Ys Y4 Y10 Yg 1 Y101Y"i: 1 Ys 1 Y3 1 Y101 Y"i: 1 Ys 1 Y3 1Y5 = 1, Yu Y101Y"i: 1 Ys 1 Y3 1 Y101Y"i: 1 Ys 1 Y3 1 Yg 1 Ys 1 Y3 Y5 Y4 YlO Y3 Y5 Y4 YlO = 1, Y10Ys 1 Y101Y"i: 1 Ys 1Y3 1Y5Y9Y3Y5Y4 = YuY3Y5Y4YlOYg 1 Y101 Y"i: 1 Ys 1Y3 1 YsY9 = 1, YuY101 Y"i: 1 Ys 1 Y3 1 Ys 1 YsYsY4Y10

=

Y9Y4 1 Ys 1 Y4

=

Y9Y4 1 Yg 1 Y4Y5

=

1,

Y11Y4 1 Y1/Y4Y5 = YuY3Y5Y4Y10Y4 1YsYsY4Y10Y4 1Y9 = 1,

Y10Y"i: 1Y10 1Y"i: 1Ys 1 Y3 1Y"i: 1YsY5Y4YlOY4YsYsY4

=

1,

Y11YsY5Y4YlDY3Y5Y4Y10Y"i: 2Y101 Y"i: 1Ys 1 Y3 1 Y4Y9 = 1, Y11 Y10 Yfi 1 YlQ1 Y:LJ.1 Y3 1 Y5 yg Y3 Y5 Y4 Y10 Y3 Y5 Y4 Y10 Yi 1 YlQ1Yi 1 Y5 1 = 1, Y11Y10Yg 1Y10 1 Y1/Y3 1 YsY3Y5Y4Y10Y4 1 Ys

=

1,

Y11 Y101 Y1/Y3 1Y4 Y101Y"i: 1Ys 1Y3 1 Y1/ Yg 1 Ys 1Ys Yu Y10

=

1,

Yu Y"i: 1 Ys 1 Y3 1 Y101 Y"i: 1 Ys 1 Y3 1 Y101 Yu1 Y3 1 Ys yg Y11 Ys Ys Y4 Y10 Y3 Y5 Y4 Y10 Y3 -1

Y11 Y4

-1

Ys

-1

Ys

-1

-1

Y4 Y10 Y4

-1

-1

Ys

-1

-1

Ys Y11 Yg 1 'Y5Y3Y5Y4Y10YsY5Y4Y10Y4 Ys = 1,

-1

-1

= 1,

-1

Y10 Y11 Ys

Y11Y10YsY11Y10YsY4 1 Ys = Y11Y10Y3Y5Y9Y101 Y5Y3Y5Y4Y10Y 4 1 Y101 Y4 1 Y5 1 = 1, Yll Y3 Y5Y4YlOY3 Y5 Y4 Y10Y4 1 Y3 Y5 Y101Y"i: 1 Ys 1 Y3 1 Y101Y"i: 1 Ys 1 Y3 1 Ys 1 Y3 1Y5 Yll Y"i: 1 Y101Y"i: 1 Ys 1 Y3 1 Ys 1 Y3 1 Y101 Yu1 Y3 1 Yu1 Yg 1 Ys 1 Y3 y; Yll YlO Ys 1Y3 1 Y101 Yu 1 Y3 1Y4 Yg 1 Y3 Y5 yg Y3 Y11 Ys 1 Y3 1 Y101 Yu 1Ys Y4 Y10Y4 Yu1 Ys Ys yg

Y9Y2 1 Ys 1 Y2

=

=

=

yg Y4 1Y2 Y"i: 1 Y2 1

Y10Y2 1 Ys 1 Y"i: 1 Y101 Y2

Y9Y2Y1Y2Y1Y2 1 Y1 1 Y2 1 Y1 1 Ys

=

=

= 1,

1,

= 1, (Ys Ys Ys )2

Yll Ys Ys Y4 Y10 Y"i: 1 Y3 Ys Y4 Y10 Ys Ys Y4 Y3 Y5 Y"i: 1 Ys 1 Y3 1Y5

Y9 Ys Ys Y"i: 1 Y2 Y3 1 Y2 1Ys

=

= 1, = 1,

yg Y2 Ys 1 Y2 1

=

= 1, = 1,

Y11Y2 1 Yu 1 Yg 1 Ys 1 Y2

Y9Y4 1 Y1Y2YIY3 1Y1 1 Y2 1 Y1 1YsYs

= 1,

Y9Y"i: 1 Y1Y2Y1Y4 1 Y1 1 Y2 1 Y1 1 Ys = Y9Y1Y2Y1Ys 1 Y1 1 Y2 1 Y1 1 = 1, Y9Y1 1 Y2 1 Y1 1 Ys 1 Y1Y2Y1

=

Y10Y1 1 Y2 1 Y1 1 Y"i: 1 Yg 1 Y10 1 Yg 1 Y1Y2Y1

Y11Y1 1 Y2 1 Y1 1 Yu 1 Yg 1 Ys 1 Y1Y2Y1

=

-1

(Y"i: 1 ,Yi)

-1

Y11Y10 Y9Y2Y1Y2Y1Y6YsY11Y10Y1Y1 ·Yu1 Ys 1Y1 1 Y101 Yu1 Y3 1 Y6 1

Y9Y1Y2YiY2Y1Ys

-1

Y2

=

=

1,

Y9Y1Ys 1 Y1 1

-1

Yi

=

Y9Y4Y1Y2YiY2 1 Y1 1

= 1, = Y9Y1 1 Yg 1 Ys 1 Y1 = 1, Y10Y1 1 Y4Y10Y1 = Y11Y1 1Yu1 Ys 1 Y1 = Y9Y1Y4 1 Ys 1 Y1Ys = Y4Y1 2 = 1, Y11 Y2 Yi YB Ys Y11 Y10Y1 Y1 1 Y2 1 Y"i: 1 Y101 Y3 1 Ys 1Y1 1 Y101 Yu 1 Y3 1 y;;1 Ys yg Yll Yio = 1, Y11Y4 1 Y1Y3 1 Y1 1 Y9Y11Y3Y9

=

-1

Y2

-1

Y4Yg

-1

Y10Y4

= 1,

Y11 Y10 Y7Y4 Y101 Y4 Ys 1 Y1 1 Y101 Yu 1 Y3 1 Y5 1Ys yg Y11 Yio YB Ys

= 1,

Yu Y"i: 1 Y101Ys Y2 YB Ys Yu Y10Y1Y2 1 Y3 1 Y10 Y3 1 Y1 1 Y101 Yu1 Y3 1 Y5 1 yg

=

1,

Yll Y3 Y4 yg YlO Y"i: 1 Y6 Y3 Yll Y10 Y7 Y"i: 1 Y101 Yg 1 Y3 1 Yu 1Yg 1 Ys 1 Y1 1 Y101 Yu1 Y3 1 Y6 1 Y11 Y"i: 1 Y10 yg Y1 1 Y101 Yu1 Y3 1 Y5 1 Y4 1 Y101 Y4 1 Yu1 Yg 1YB Ys Yu Y10 Y7

= 1,

= 1,

YllY1 1Y10 1Yu 1Y3 1Y5 1Yu 1Yg 1Y6Y3YllYlOY7Y9 = 1, YuY10Y1Y10 1YBYsY11Y10Y1Y5Y9Y10 1 Y"i: 1Y10Y6Y3 = 1, Yll Ys Y2Y1 Y2 Yi Y6 Ys Y11 Y10Y1Ys Y10Y6 Y3 Y11 Y10Y1Y4 Y1 1 Y101 Yu1 Y3 1 Y5 1Y5 Ys Y4

= 1,

5.1 Construction of the 2-central involution centralizers

187

Yll Y10 Y7 y:; 1Y:; 1Y1 1 y;;/ Yu 1 Y3 1 Y6 1 Y5 Y9 Y10 Y3 Yl o Y1 Y2 Yi Y6 Y3 Yll Y10 Y7 Y9 Y6 Y3 Yll Y3 Y11 Y10 Y6 Y3 Y11 YJO Y7 Ys Y6 Y3 Y11 Y10 Y7 Yg 1 Y:; 1Y1 1 y;;/ Yu 1 Y3 1 Y6 1 Yo y7y; 1 y:; 1 y;: 1 y:; 1Y4Y5 1 YoY:; 1 Ys 1Y5 1 Y6Y2

=

-1

= Y9YBY;Y;; 1Ys = YoY6Y2Y5 1Y:; 1 Ys 1 = 1,

Y11Y:; 1 Ys 1 Yu 1 Yg 1 Y5 1 Y6Y2

-1

-1

-1

Y11Y10Y1Y3Y4Y2 Y6 Y3 Y1 ·Yg 1Y0Y2YsY3Y4YJr}Y6Y3

-1

-1

-1

Y10 Yu Y3

-1

-1

Y11 Y10Y1Y2

-1

Y6

-1

Yg

-1

Y1

-1

Y6

= 1,

-1

-1

Y1

-1

-1

Y2

Y2

Y4

-1

Y5

-1

Y10Y1

-1

-1

-1

Y10

-1

Y0

-1

Yi

-1

-1

-1

Y10 Y11 Y3

-1

Y6

-2

Y1

-1

Y2

-1 Y1

-1

Y11Y3Y4YlOY4Y2Y1Y2YtYBY3YllYlOY7Y3Y4Y6Y2Y1

y:;

1

Ys 1 Ys Yo

-J

= 1,

-1

Y3

-1

Yo

=

1,

Y10

= 1, = 1,

Yt 1 Yto Y:; 1 Ys 1 Yto Y1 1 Y101 Yu 1 Y3 1 Y6 1 Y:; 1 Yg 1 Y101 Y5 1 Y6 Y2 Ys Y3 y:; 1

=

-1

Y1

Y2

Y11 Y3 Y11 Y10 Ys 1 Yto Y1 1 Y101 Yu 1 Y3 1 Ys 1 y:; 1 Y3 1 Y10 1 Yu 1 Yg 1 YB Y2

(y5y;)'

-1

Y2

= 1,

Y7 Y4 yg Y6 Y2 Y;- 1 y:; 1 y;- 1 Y:; 1 Y101 Y5 1 Y101 Y:; 1 Ys 1 Y2 Y1 Y2 Yl Y6 Y3

·Yg 1 Y5 1

-]

Y10 Y3

-1

Y10 Yu Y3

·Yg 1 Y10 1 Yu 1 Y3 1 Yg 1 Y5 1 Y6 Y2 Ys Y3 Yll Y10 Yo Y2 Y1 Y2 Yi Y6 Y3

Yll

-1

Y1

= 1,

Y11Y4Y10Y2Y1Y2Y1Y0Y3Y11Y10Y1Y2Y3Y2 ·Yg 1 Y10Y; 1YBY2Y5Y9

= 1,

= 1,

YoY; 1Y2YBY2Y0Y2Y; 1 y:; 1 y 6 1

=

=

1,

1,

Y11 Y3 Ys Y2 Yi Y2 Yi Y6 Y3 Yll Yto Y7 Y2 YB Y2 Y4 Y10 1 YB Y2 Y10 y:; 1 y;;1 yg

= 1,

Y11 Yto Y1Y2 Y10 1Y6 y; Y101 Y6 Y2 Y5 1 Y3 1 y:; 1 Ys 1 Y5 Yo Y4 Y10 Yo Y2 Y6 Y3

= 1.

(c) D has a faithful permutation representation PD of degree 1024 with stabilizer U = (Y1, y4, Y10). (d) D has a unique normal non-abelian subgroup Q of order 1024 with center Z(Q) = (z1,z2) of order 4, where z1 = Y9Y1Y2Y1Y2, z2 = yg Yn y4 Y1 . Furthermore, Q has a complement W = (w1, w2 , W3) in D of order IWI = 27·3·5, where W1 = Y11Yioii 1Y1Y6Yu}YsY10Ys Y2, W2 = Yn Yu} Y1 Y6 Y10 Y4 Ys Y10 Y7 and W3 = Yn Y10 Y3 Y4 Y1 Y6 Y10 Y4 YsY10Y7 · (e) Let a : D --+ D 1 = D / Z (Q) be the canonical epimorphism with kernel ker(a) = Z(Q). Let V = a(Q) and let u; = a(y;) E D 1 for i = 1, 2, ... , 11. Then V is an elementary abelian normal subgroup of order 28 of D 1 having a complement W1 = (k1, k2, k3) ~ W, where kJ = a(Wj) for j = 1, 2, 3. Furthermore, D1 has a faithful permutation representation of degree 256 with stabilizer W1 , and V = (qzll::; l::; 8), where -1 -1 = U5U9U4U10 U4 U1U5U3U1U10, q2 = U3U11U4U1U7U3, q3 = U9U7, q4 = q5 = UgU11U3U5U1U7U10UsU3U5, -1 q5 = U9U11 U3U11 U10U1 U5U3U1 U3U7,

ql

U11U1U5U4,

188

Fischer's simple group Fi22 -1 -1 U5U11 U3U9U11 U4 U5U3UsU10 U7,

q7

=

q3

-1 = U9U3U4-1 U1 U5U3UsU10 U7U1.

(f) The conjugate action of the three generators kj of W1 on V w. r. t. the basis B = {qz I1 :s; l :s; 8} of V is given by the following matrices: llOOOOOOJ 01000000 00110000 ( 00010000 00000010 • 01001111 00001000 00000001

M k3

M k2

1 0 1 0 0 0 0 OJ 11110000 11010000 ( 10010000 = 0 1 0 0 O1 OO , 00010011 01001011 10100010

llOOOOOOJ 10000000 10110000 ( 11100000 = 10000100 · 01001100 00001011 10000110

(g) The Fitting subgroup A of W1 is elementary abelian of order 16 and generated by a1 = k1 k3 k2 k1 k5, a2 = k2 k~ k2 kt a3 = k2 k1 k5 k1 k3, a4 = k1 k3 k2 k3 k2 k1 k2. It has a complement L = (k 1 , k2 ) in W1 which is isomorphic to the symmetric group S5. (h) A is a maximal elementary abelian normal subgroup of the Sylow 2-subgroup S = (A, k 1 , r) of W 1 with center Z(S) = (u), where r = k2k1k 3k§ k1 k2 k3 and u = k1 (k3 k2 )2 k 1k2 . The centralizer Cw 1 ( u) of u has order 27 · 3. It is generated by S and the element d = (k1k2)2(k2k1) 2 of order 3.

(i) Let MW1 be the subgroup of GL 8 (2) generated by the matrices of the conjugate action of the generators of W1 on V with respect to the basis B. Let Mai, Mu and Md be the corresponding matrix of ai, u andd, respectively. LetMX = CG1 8 (2)(Mu)nCG1 8 (2)(Md). Then M X has an abelian Sylow 3-subgroup MT of order 33 and MT contains a matrix 0 1 0 0 0 0 0 OJ 11000000 11010001 ( 10110111 Mx= 00011100 01101000 11111000 01110000

of order3 such that MC= (CMw 1 (Mu),Mx) has order2 7 ·3 2 .

5.1 Construction of the 2-central involution centralizers

189

(j) Let MK= (MC, MW1). Then the derived subgroup MK' of MK is a simple group of order jM K' I = 26 · 34 · 5, which is isomorphic to the unitary group U4 (2). (k) MK is an irreducible subgroup of GL8 ( 2) generated by the matrices m1 = Mk1, m2 = Mk2, m3 = Mk3, m4 = Mx of respective orders 2, 5, 3 and 3. With respect to this set of generators MK has the following set R(K) of defining relations: mi = m~ = m~ = m~ = 1,

(m3, m4) = 1, m 3 m2m1m 2 m3m1 = m 2 m1 )4 = 1 , -1

-1

(

(m 21mi) 4 = 1,

-1

m 3-1 m2m 3-1 m1m2m3m1m 2-1 = ( m2m 3-1 )4 = 1, (m4,m 21,m4 ) = l , m1m 4-1 m1m 4-1 m1m4m1m4 = 1, m 2-1 m 3-1 m1m 2-2 m3m4m1m 4-1 =1, -1 -1 1 m2m 3-1 m2m4m 2 2m 3 m 2 m3m4=.

(1) Let K be the finitely presented group constructed in (k). Then V is an irreducible 8-dimensional representation of K over F = GF(2) of second cohomological dimension dimp[H 2 (K, V)] = 0. Furthermore, K has a faithful permutation representation PK of degree 640 having a stabilizer which is the Sylow 3-subgroup ((m2m4)2, (m1m~m4)2) of K. (m) Let H 1 = (mi, qj jl ::; i::; 4, 1 ::; j ::; 8) be the split extension of K by V. Then H 1 has a set R(H1 ) of defining relations consisting of R(K), R 1 (V ~ K) and the following set R 2 (V ~ K) of essential relations: m1q1m 1 1 q1q2

m1q2m 11 q2

= 1,

m1q3m 11 q3q4

= 1,

= 1,

m1q4m 1 1 q4

= 1,

m1q5m 1 1 q7

= 1,

m1 q5m 11 q2q5q5q7qs

m1q7m 1 1 qs

= 1,

m1qsm 1 1qs

= 1,

m2q1m 21 q1q2q3

m2q2m 2 1 q3q4 m2q5m

m2q3m 2 1 q2q3

= 1,

21 q1q2q5q7 =

= 1,

m2q4m 2 1 q1q2q3q4

= 1,

m2q5m 21 q3q4q5

1,

= 1,

= 1,

m2q7m 21 q1qs

= 1,

m2qsm 21 q2q3q4q5qs

= 1,

m3q1m 31 q2

= 1,

m3q2m 31 q1q2

m3q3m 3 1 q1q4

= 1,

m3q4m 3 1 q1q2q3q4

= 1,

m3q5m 3 1 q1q5q5

m3%m 31 q2q5

= 1,

m3q7m 3 1 q5qs

m4q1m 4 1 q1q2

= 1,

m4q2m 4 1 q1

m4q4m 4 1 q1q2q5q7

= 1,

m4 q7m 4 1 q1 q2 q3 q4 q5 q7

= 1,

= 1,

m3qsm 3 1 q1%q7qs

= 1, = 1,

= 1,

41 q2q5q7qs = 1, = 1, m4q5m 4 1 q5q5qs =

m4q3m

m4q5m 4 1 q1q2q7qs

= 1,

= 1,

m4 qs m 4 1 q1 q3 q5 q7

= l.

1,

Fischer's simple group Fi22

190

(n) dimp [H 2 (H1 , F)] = 4 and there exists a unique central extension H of H 1 of order IHI = 217 · 34 · 5 whose Sylow 2-subgroups are isomorphic to a Sylow 2-subgroup of D. Also, H is a split extension of K by its Fitting subgroup Q, and H is isomorphic to the finitely presented group H = (hi I1 :S: i :S: 14) with the following set R(H) of defining relations:

= h~ = h~ = hl = h~ = h~ = h~ = h§ = h§ = 1, hi1 = hf2 = hf3 = hf4 = 1, hfoh1l = 1, (hi, h14) = 1 for l :S: i :S: 13, (hj, h13) = 1 for 2 :S: j :S: 12, (h3, h4) = (hs, h5) = (hs, h1) = (hs, hs) = (hs, hg) = (hs, h12) = 1, (h5,h1) = (hs,h10) = (h10,h12) = (h11,h12) = 1, h1 1h13h1h131h1l = 1, (h2h3 1)4 = 1, (h4, h:;1' h4) = 1, h3 1h2h1h2 1h3h1 = (h;; 1h1) 4 = h-;1h2h-;1h1h2h3h1h2 1 = 1,

hf

h1h4 1h1h4 1h1h4h1h4 = h2 1h3 1h1h2 2h3h4h1h4 1 = 1, h2h3 1h2h4h§h 3 1h 21h3h4 = h1hsh 1 1hsh5h 1 = 1,

J

J

J J

h1h5h 1 1h5h 1 = h1h1h 1 1h1hsh 1 = 1, h1hsh 1 1hs = h1hgh 1 1h11h 1Jh 1 = 1, h1h10h 1 1h5hgh1oh11h12h 131 = h1h11h 1 1hgh 1J = 1,

h1h12hth12h 1J = h2hsh;;1hsh5h1h 1J = 1, h2 h5h 2 1h1 hs h 1J h 1J = h2 h1h 21h5 h1 h 1J = l,

= h2hgh 21hsh5h1oh11h 1Jh 1J = 1, h2h1oh;;1 h1hshgh 1J = h2h11h 2 1hsh12h 1J = 1, h2 h12 h 2 1h5 h1 hs h10h12 = h3 hs h 3 1h5 h 1J = 1, h3h5h 3 1hsh5h 1} = h3h1h-;1 hshs = h3hsh 3 1h5h5h1hs = h3hgh 3 1hshgh1oh 1J = h3h1oh3 1h5h9 = 1, h3h11h3 1hgh12h11 = h3h12h3 1hsh10h11h12h1Jh11 = 1, h2hsh 21hsh5h1hs

J

J

h4hsh 4 1hsh5h 1}h1 = h4h5h 4 1hsh 1 = 1, h4h1h4 1h5h10h11h12h11 = 1,

J

h4hsh4 1hsh5h1oh11h 1Jh 1 = 1, h4hgh4 1h5h5h11h12h11 = 1, h4h1oh4 1hgh1oh12h 1Jh 1 = 1,

J

1,

5.1 Construction of the 2-central involution centralizers

l

l

h4h11h4 1h5h6h1hshgh11h1 = h4h12h4 1h5h1h10h11h1 h 5-lh101h 5h 10 h141 = h5 1h111h 5h 11 h141 = 1,

l

191 =

1,

l

(h6,hs) = h 61h 91h6hgh 1 = h 61h 1a1h6h1oh1 = 1, h 61h1/h6h11h1 = (h6,h12) = (h1,hs) = (h1,h9) = 1, 1hh 7-lh101h 7 h 10 h141 = h7 111h 7 h 11 h141 = 1, 1 1h- 1h h h- 1 1 1 h 7-lh12 h 7 h 12 h14 = h8 9 8 9 14 = , 1 h 8 h 1/hsh11h 1 = (hs,h12) = (h9,h10) = 1,

l

l

(h9,h11)

= (h9,h12) = h1a1h1/h1oh11h1l = 1.

Proof (a) By Table 5.5.3 the split extension E2 = ½ >, q_1). (k), (1) and (m) The systems ofrepresentatives of the conjugacy classes of H 2 , D 2 and D have been calculated by means of the permutation representations PH2 and PE, MAGMA and Kratzer's Algorithm 5.3.18 of [92]. The character tables of (n) have been obtained in [73] by means of PH2, PD and MAGMA. D Proposition 6.2.2 Keep the notation of Proposition 6.2.1. Using the notation of the character Tables DVD. 1. 4. 4, DVD. 1. 4. 3 and 6. 6. 2 of the groups H2 = (p2,q2,h), D2 = (P2,q2) ~ T = (p1,q1) and D = (T,x,y), respectively, the following statements hold.

(a) There is a compatible pair (x, r) E mf charc(H2) x mf charc(D) of degree 352 of the groups H 2 and D, (x,r) = (X4o +X41 +X57,T9 +r10), with common restriction given by

r1n

= XID = '¢26

+ '¢21 + 'l/J5g + '¢63 + '¢10 + '¢73

E mf chanc(T),

where irreducible characters with bold face indices denote faithful irreducible characters. (b) Let Q1 and ® be the up to isomorphism uniquely determined faithful semi-simple multiplicity-free 352-dimensional modules of H 2 and D over F = GF(l 7) corresponding to the compatible pair (x,r), respectively. Let ""'XJ : H2 ---, GL352 (17) and ""® : D ---, GL352 (17) be the representations of H2 and D afforded by the modules m and®, respectively. Let fJ = ""'XJ(h), P2 = ""'XJ(P2), q2 = ""'XJ(q2) in ""'XJ(H2) :::; GL352 (17). Then the following assertions hold.

(1) There is a transformation matrixT E GL3 52(17) such that

P2 = 7- 1,..,w(P1)T, ~

(2)

Sj

q2 = 7-i,..,w(q1)T,

= 7- 1,..,w(x)T and 1J = 7- 1,..,w(y)T.

= (fJ,~,IJ).

6.2 Construction of a 2-central involution centralizer

237

Proof (a) The character tables of the groups H2, D 2 ~ T and D are stated in the accompanying DVD and Section 6.6, respectively. In the following we use their notations. Using MAGMA and the faithful permutation representations P H 2 and PD of H 2 and D constructed in Proposition 6.2.l(e) and Lemma 6.1.2(a) H. Kim determined in [73] the fusion of the classes of D 2 in H 2 and Tin D, respectively. By means of the character tables of H 2, D 2 and D and Kratzer's Algorithm 7.3.10 of [92] he found two compatible pairs of degree 352. Only one of them provides an irreducible representation of degree 352 of the free product H 2 *D 2 D with amalgamated subgroup D 2 satisfying the Sylow 2subgroup test of Algorithm 7.4.8, Step 5(c), of [92]. It is stated in the assertion. (b) By Proposition 6. 2.1 there are well determined elements P2, q2 of D2, h of H2 and an isomorphism a : D2 _, T such that D2 = (P2, Q2), H2 = (D2,h) and T = (p1,q1), where P1 = a(p2) and Q1 = a(q2) are well determined words in x and y; see the proof of Proposition 6.2.l(j). Furthermore, D = (P1, Q1, x, y). The semi-simple faithful representation SU of H2 = (P2, q2, h) corresponding to the character x = X41 +X41 +X57 decomposes into the three non-isomorphic irreducible constituents SU(40), SU(41) and SU(57) corresponding to the irreducible characters X4o, T41 and X57, respectively. In order to get these three representations Kim used in [73] the faithful permutation representation PD of D = (p1, q1, x, y) and the MAGMA command LowindexSubgroups (PD, k) searching for conjugacy classes of subgroups Y of index ID : YI = k. He found three subgroups Yi, i E {40, 41, 57}, ofrespective indices k40 ~ 512, k4 1 = 512 and k 51 = 480, such that Xi is an irreducible constituent of the permutation character 1~. Kim constructed for each i E {40, 41, 57} the permutation matrices P(i, h), P(i,p2) and P(i, q2) of the three generators h, P2 and Q2 of H2 over the field F = GF(l 7), respectively. Then he applied the Meataxe algorithm implemented in MAGMA. Thus he obtained the three triples of diagonal block matrices P2 (i), Qz (i), 1i(i) E GLni (17) of the generators of H 2 = (p 2, q2, h) describing the three n;-dimensional irreducible constituents SU(i) corresponding to Xi of the semi-simple representation SU of H2, where (n40, n41, n57) = (96, 96, 160) by Table DVD .1.4. 4. The semi-simple faithful representation ® corresponding to the character T = Tg + Tio decomposes into the two non-isomorphic irreducible constituents ®(9) and ®(10) corresponding to the irreducible characters Tg and Tm, respectively. In order to get these two representations

Fischer's simple group Fi23

238

Kim used in [73] the faithful permutation representation PD of D = (p 1 ,q1 ,x,y) and the MAGMA command LowlndexSubgroups(PD, k) searching for conjugacy classes of subgroups X of index ID: XI = k. Thus he observed that the permutation characters (lx 9 )D and (lx 10 )D of the subgroups Xg = ((q1pf)3, (xq1p1q1x) 3, (xq1xp1q1) 2) and X10 = ( (p1xq1 ) 4, xq 1 x, (p 1q1xp 1 ) 2) of index k = 352 contain Tg and T 10 as irreducible constituents, respectively. Applying then the Meataxe program of MAGMA he obtained two sets of four matrices in GL 176 (17) of the generators of D = (p 1 , q1 , x, y) describing the irreducible representations 2!J9 and 2!J10 over F. Choose fixed bases H and V in the restrictions of QJ to D 2 and 2!J to T, respectively, such that they are unions of bases of their six direct summands. Calculate then the matrices b = t,,m(h), P2 = t,,m(P2) and t,,sm(P1), t,,sm(q1), t,,sm(x), 1,,sm(Y) with respect to the bases H and V, respectively. By Proposition 6.2.1 there is a fixed isomorphism C! between the two groups D 2 and T satisfying u(p 2) = P1 and u( q2) = q1. Therefore the two FT-modules QJIT and W1r described by the pairs (P2, q2) and (t,,sm(p1), t,,sm(q1)) of matrices in GL 352 (17) are isomorphic. Let Y = GL 352 (17). Applying now Parker's isomorphism test of Proposition 6.1.6 of [92] by means of the MAGMA command

Islsomorphic(GModule(sub), GModule(sub)) one obtains the transformation matrix q2 = K,5ID ( q1 )'Ti ·

Ti satisfying P2 = t,,sm (p1)'Ti and

By assertion (a) and Corollary 7.2.4 of [92] this transformation matrix

Ti has to be multiplied by a diagonal matrix V of GL 352 (17). In order to calculate its entries one has to deduce the composition factors of the restrictions Xi ID 2 and Tj IT to D2 and T, respectively. From the fusion and the three character Tables DVD. 1. 4. 4, DVD. 1. 4. 3 and 6.6.2 it follows that:

ID, = 'lp27 + 'lp53, X41 ID, = 'lp25 + 'lp59, X57 ID, == 'lp70 + 'lp73, and Tg Ir = 'I/J25 + 'lp53 + 'I/J73, T10 Ir = 'I/J27 + 'l/J5g + 'l/J70. X40

239

6.2 Construction of a 2-central involution centralizer

By Schur's Lemma and the degrees of these characters the following linear system in the variables a, b, c, d, e, f E F holds: 96 = 16b + 80d,

96 = 16a + 80c,

176 = 16a + 80d + 80f,

160 = 80e + 80f

and

176 = 16b + 80c + 80e.

It has the following solution: c = 8 - 7a, d = 8 - 7b, e = 7a - 7b + 1 and f = -7a + 7b + 1, where a and b run through all non-zero elements of F. Hence the diagonal matrix 'D has the form 'D(a,b)

= diag(a 16 , b16 , [8-7a] 80 , [8-7b] 80 , [7a-7b+1] 80 , [-7a+7b+1] 80 )

for suitable elements a, b E F. Let ~(b,d)

= 'D(b~d) ½-l / U with Z(H1) = Z(Q) ofodd index IH1 : UI. The target group H is an epimorphic image of the free product H 1 *u D with amalgamated subgroup U; H is realized as an irreducible subgroup of GL 128 (23). It has a faithful permutation representation of H of degree 276480 which is used to calculate the character table of H and a system of representatives of its conjugacy classes.

7.2 Construction of the 2-central involution centralizer

285

Proposition 7.2.1 Keep the notation of Lemma 7.1.1. Let E

(a, b, c, d, t, g, h, i, j, k, vi I1 :::::; i :::::; 11)

= E1 =

be the split extension of M2 4 by its simple module V1 of dimension 11 over F = GF(2). Then the following statements hold.

E = (x,y, e), where x = (cgjhi)7bt, y = (bv)2jkj and e = b have orders 7, 4 and 2, respectively. (b) z = v 1 = (xy 3 ) 14 is a 2-central involution of E with centralizer D = CE(z) = (x,y) of order 221 · 32 · 5 · 7. (c) D has a unique normal subgroup Q of order 29 . It is extra-special and generated by the following eight elements: (a)

q1 =

y2 ,

q5

(x 5 yx)7,

=

(xy)7,

q2 =

q7

=

q3 = (yx)7, q4 = (xy 2)7, (x 4yx 2 )7, q 8 = (x 4 yxy) 6 .

q5 =

(yxy)7,

Furthermore, the elements Qi satisfy the following set R(Q) of relations: 2 2 2 2 2 4 4 2 1 ~=½=½=~=%=~=~=~=' (q1,q2) =

(q1,q3)

=

z,

(q1,q4) = (q1,q5) = (q1,q5) =

( Q1 , q7) = ( q1 , qg) = Z,

(q2,Q6)=z, (q3 , q4)

Q6)

(Q6 , q7)

q3)

(q4,

q5) = (q4,

q7)

= Z,

=

= ( q3 , qg) = Z,

1,

(q4, qg)

=

1,

= ( q5 , qg) = Z,

( q7 , qg) =

Z.

=

= ( q2, q5) =

(q3 , q7)

(q5, qs) =

q7)

q4)

1,

z,

= (q5,

( q2,

1,

(q2,qs)=z,

(q2,q1)=l,

= ( q3 , q5) = ( q3 , q5) =

(q4, q5) = (q5,

( q2,

1,

z,

(d) Let a : D--+ D1 = D/Z(Q) be the canonical epimorphism with kernel ker(a) = Z(Q) = (z). Let V = a(Q) =(vi= a(qi) E D1 I 1 :::::; i :::::; 8). Then V is an elementary abelian normal subgroup of order 28 of D1 = (a(x), a(y)). With respect to B = {vi I1 :::::; i :::::; 8} the conjugate actions of a(x) and a(y) on V have the following matrices:

Mx

=

in GLs(2).

01 1 (0 0 0 1 1

0 1 0 1 0 1 0

01 0 0 0 0 1 0

01 1 1 0 0 1 1

01 1 0 0 0 0 0

0 0 0 1 0 1 1

0 1 0 1 1 0 1

0OJ 0 0 0 0 0 1

a nd

MY

=

01 0 0 ( 0 0 1 0

0 1 0 0 0 1 1

01 0 0 0 1 0 0

0 0 0 1 0 1 1

0 0 1 0 1 0 0

0 0 0 0 0 1 1

0 0 0 0 0 1 0

0OJ 0 0 0 1 1 0

Conway's simple group Co1

286

(e) The map cp: D--+ GLs(2) defined by cp(x) = Mx and cp(y) = My is a group epimorphism onto MD = (M x, My) with kernel Q. The group MD has an elementary abelian Fitting subgroup MY of order 26 generated by { cp( mj) 11 :::; j :::; 6}, where m1 m4

= (x 4 (yx)2y)5, = (xyx 4yxy)5,

m2 m5

= (x 3 (yx)3)5, = (yx 4yxyx)5,

= (x 2(yx)2yx 2 )5 , m 6 = (x 3yx 3yxy) 5.

m3

Furthermore, MC = (cp(x), cp(c)) and C = (x, c) are complements of MY in MD and of M = (Q, mj I 1 :::; j :::; 6) in D, respectively, which both are isomorphic to the alternating group A 8 . In particular, D has a faithful permutation representation PD of degree 215 with stabilizer C, where c = (x 2yx 2yxyxy 2xy) 3 has order 2.

(f) D1 has a faithful permutation representation P D 1 of degree 214 with stabilizer o:(C) = (o:(x),o:(c)). Furthermore, D 1 is a nonsplit extension of D/Q by V and Cn 1 (V) = V. (g) Dis a finitely presented group D = (Q, x, y) having a set R(D) of defining relations consisting of R( Q) and the following relations:

x 7 = y 4 = 1, z = (xy 3)14, z 2 = 1, (x,z) = (y,z) = 1, (yx- 1)7 = q1q2q4, (xyx- 1yx) 4 = q3q5q7, (xyx- 1yx- 1y) 4 = q4q5q7qs, (xyx- 1y) 6 = qi, (xyx- 1yx- 1yx) 4 = q5z, x- 1yx- 2yx 2yxyx- 1yx 3yx- 1yxyx 2yx- 2yx- 1y

= q2q7z, x- 2yx- 1yxyx- 2yx- 1yx- 3yx 2 yx 2yxyx- 1yx- 1yx- 1 = q2q4q5qs, yxyx-1yx-2yx2yx-1yx-2yxyx-2yx2yx-2yx2 x -1 q1x x- 1q4x

= = 1 x- q7x = qf = q1,

= q5q5qs,

q1q4q5q62 , x -1 q2x = q3, x -1 q3x = q1q2q4q5q7, q4, x- 1q5x = q2q5q7, x- 1q5x = q7, q1q3q2q4q5, x- 1qsx = q1q4q5qgq7,

Y- 1q2yq3 = (q1q2) 2 , Y- 1q3yq2 = 1, y- 1q4yq5 = 1, Y- 1q5yq4 = 1, Y- 1q5yq3q5qg = 1, y- 1q7yq1q2q4q5q7qg = 1, y- 1qgyq2q4q5 = (q1q2)2.

(h) MY contains a 2-central involution Mt= cp[(x 3yx 3yx 2 )3] of MD with centralizer CM D (Mt) of order 212 -3 2 . Furthermore, MY has a complement MC1 in Cl',rn (Mt) having an elementary abelian Fitting subgroup MW1 = (cp(f;) I 1 :::; i :::; 4) of order 24 which

7.2 Construction of the 2-central involution centralizer again has a complement MC2 order 22 · 32 , where

Ji

= (yx 3yxyx 2yxyxy 2 )2 ,

h

= (xyx2yx2yxyx5yx2)2'

=

(cp(rk), cp(tk)

I 1 :S

287

k :S 2) of

fz = (x 2yxyx 2yxyxyx 5y) 3 ,

14

= (x2yx4yx3yx3y2xy)3'

r1 = (yx 5 yx 5 yx 2yx 2)2, r2 = (x 3 yxyx 2yx 4 yxyx 2y) 3 , t1 = (xyx 3yx 2y 2 x 2 yxyx 3yx )5 (x 5 yx 3 yxyx 2yx 3yxy ) 3

. ( xyx3 yx2 y2 x2 yxyx3 yx) 1 o(x5 yx3 yxyx2 yx3 yxy) 3 '

t2 = ((xyx 3yx 2y 2x 2yxyx 3yx) 5 (x 5yx 3yxyx 2yx 3yxy) 3 ) 3 , and where r 1 and t 1 have order 3 and all other elements are involutions. Furthermore, r? = rr and 2 =

tt tr.

(i) The centralizer MX =

CcLs( 2 )(Mt)

of Mt in GLs(2) contains

the matrix 1 101 1 101] 10011100 00100000 ( 11011100 Mh= 10011001 11001000 01001111 01010111

of order 3 such that the subgroup MO = (MD, Mh) of GL 8 (2) has order 212 · 35 · 52 · 7 and the subgroup MH3

=

CMo(Mt)

=

=

(Mmj,Mfi,Mrk,Mtk,Mh)

(CMD(Mt),Mh)

ofGL8 (2) has order2 12 -33 , where l :S j :S 6, 1 :Si :S 4, 1 :S k :S 2. Furthermore, MO is isomorphic to the simple group (2).

ot

(j) The Fitting subgroup MY;i of M H 3 is extra-special of order 29 with center Z(Ya) = Mt, and

= Mm3, Mf5 = Mm4, Mh = Mm5 and Mfs = (Mm 5 Mfz) 2 • Furthermore, MY;i is isomorphic to the finitely presented group Ya = (Ji I 1 :S i :S 8) with the following set R(Ya)

where Mf5

of defining relations:

ff = Ji = fl = fl = J; =

= ff = fl = l, U1,fz) = U1,h) = (h,h) = U1,f4) = (fz,f4) = (h,14) = 1, fl

(fi,h)=(!z,h)=(h,h)=(Ji,h)=(h,h)=(h,h)=l,

Conway's simple group

288

Co1

(h,h)=(h,h)=(h,h)=(h,h)=(h,fs)=(h,h)=l, U1hh) 2 = (hf5fs) 2 = U1hh) 2 = Ui!sh) 2 = 1,

(hf5f4) 2 = U1h f4) 2 =

1,

f4f5f5f4f5f5 = f4h f5f4f5h = hfsf5faf5fs = l. MT = (M r 1 , Mt 1 , Ma 1 ) is an elementary abelian Sylow 3-subgroup of MH3 of order 27, where Ma 1 (Mt 1 ) 2 Mh(Mt 1 ) 2 Mh 2 (Mt1) 2 Mh. The matrix Ma2 = MhMm2Mm3Mm4Mm5MhMm1Mm2 M m 6 is an involution commuting with the subgroup (M r 1 , M r 2, M ti, M t2) and satisfying M af a 2 = M af. MK3 = (MT,Mr2,Mt2,Ma1,Ma 2) is a complement of MYj in MH3. M H3 is isomorphic to the finitely presented group

with the set R( H 3 ) of defining relations consisting of R(Y3) and the following relations: af = a~ =

rr = r~ = tf = t~ = 1,

(ai,rj) = (ai,tk) = (rj,tk) = 1 t t, _ t2 1 -

!? = hfsf4,

!? = hfsf4,

for all

l :::'.: i,j,k :::'.: 2,

1,

fi

1

= fif4,

fi

2

= fif4,

Jf = f5f5, ff = fif5f5, !? = f4, !? = h, JJ = fihfaf4, JJ = h, ff = fs, ff' = hfs, ff = fifaf4, !? = fif4, fi 1 = hh, Ji' = hfsf4, !;1 = f4f5f4fs, J;' = f5fafs, !I = hf4, JI' = hf4, J! = Ji, J!' = f4, ft = fdds, ft'= f4f5hfs, J; = f4!5f4h, J; = f5fs, JJ = h, JJ = f5fs, ft = fifd5fsfafs, !;' = f5, fi = f5fs, 2

1

1

1

2

1

1

1

1

1

2

1

1

!? = f4f5f4h,

fi 1 = f5fs,

ft' = f5,

!? = f5,

Ji"= f5,

J;

!? = fs,

Jg' =fs.

2

1

fi 2 = h,

!? = f4f5f4f5fs,

fg1 = hf5h, fi 1 = f4f5f4h,

= fahf4, J;' = h, ff 1 = f5hfs, JJ 1 = f4f5f4f5hfs, JJ' = fs, ft = his, 1

1

7.2 Construction of the 2-central involution centralizer

289

(k) The subgroup

U = (a2,r1,r2,t1,t2,fj,qi 11::::; i,j::::; 8) of D has center Z(U) = (z = q~) and satisfies the set R(U) of defining relations consisting of R( Q) and the following relations:

rr r~ tr

= = = = t~ = 1, JJ = 1, for 1::::; j::::; 8, 2 tt21 = t21, r1r 2 = r1, (a2,r1) = (a2,r2) = (a2,t1) = (a2,t2) = 1, (r1,t1) = (r1,t2) = (r2,t1) = (r2,t2) = 1, !? (hhf4)- 1 = ![2 (hfa!4)-l = Ji' (fif4)- 1 = !{ (fif4)- 1 = 1, Jf (fif5f6)- 1 = q5, JI' (!4)- 1 = !I (h)- 1 = JJ' (!ihhh)- 1 = JJ (h)- 1 = 1, 1; 2 (h!s)- 1 = JI' (!ihh)- 1 = !I 2 U1f4)- 1 = Ji' (hf4)- 1 = 1, fi 2 (hhh)- 1 = 1, ff 2 (!6fa!s)- 1 = q4, JI' (hf4)- 1 = !I 2 (hf4)- 1 = Ji' (!1)- 1 = !! 2 (14)- 1 = 1, !% 2 (!4f6h fs)- 1 = q1q2q4q6q7, !;' (f4f5f4h )- 1 = q2q4q7q6, !? (!5fs)- 1 = q2q4q6q1, JJ' (h )- 1 = qgq1qs, JJ 2 (!5fs)- 1 = q2q4q5q6q7, Jt 2 (!5)- 1 = 1, J;' (!5fs)- 1 = q2q5q6q1, J; 2 (f4fs!4h )- 1 = q2q4q5q7@, Ji' (!5fs)- 1 = q2q4q6q1, fi (h )- 1 = q5qsq1qs, Jt 2 (!6)- 1 = 1, J;' (!5)- 1 = qgq1qs, !? (!4f5f4f6!s)- 1 = q5, !~' (f4f5f4h )- 1 = q1, !~ 2 (!6)- 1 = q1q2q4q7q6, Jf 2 (h )- 1 = 1, J;' U6h!s)- 1 = q1qs, 1; Us)- 1 = 1, JJ'(!4f5f4fd1fs)- 1 = q1q4, JJ 2 Us)- 1 = 1, Jt 2 Us)- 1 = 1, (!1, h) = (!1, fa)= (h, h) = (!1, f4) = (h, f4) = (fa, f4) = 1, a~

2

2

2

2

2

2

(h,h)=(h,h)=(fi,h)=(h,h)=(h,h)=(h,h)=l,

= 1, Utfs) 2 = q1q4q5, (J4fs) 2 = q2q4q5q7q6, Us,fs) = (!6,fs) = (h,fs) = 1, (fih h) 2 = q1q4q5, (h!6h) 2 = q2q4q5q6q7, (fih h) 2 = q1q2q6q1, Ui!sh) 2 = q2q4q5q5q7, (hf5f4) 2 = q2q4q5q5q1, U1hf4) 2 = q1q2q6q1, (h, fs)

Conway's simple group Co1

290

= q2q4q5q7q5, hfsf5!3f5fs = 1, f4f5f5f4f5f5

qf'

= qi,

qr 1

f4h fsf4fsh

= q2q4q5q5q7,

= q1q5,

qi'

t1

= q1q2q4q5q5q7, qf' = q1q2q4q5q5q7, ql

/i_ ql - q1,

= q1q2q4q5q5q7,

h_ ql - q2@q7,

q1!4 -q 1,

h _ !B _ h _ fs _ ql - ql - ql - ql - q1,

q;

q~' = q1@q7, t1

q2

J,

q2

qi' q{

1

fs

q3

!B

q3

a,

=

q2q5,

q;' = q2q5,

t,

= q2q3q4q5q5q7, q2 = q3q7q5, = q4q5q7qs, q2fs = q1q7q5, q2f4

q2h = q1q4q5q7,

q{8

1

= = = = =

q2h = q1q5q5q7,

q5qsq2qs,

q1

q~'

/J

q2

= q4q5qgq5,

_

- q4q5qgq5,

q2h = q1q4q5q7q5,

qi = q3q5,

= q2q3q4q5q5q7,

1

= q1q3q4q5q5q7, qf' = q3, q2q3q4q5q5qg, q{' = q2q3q4q5q7q5,

q3q5,

1

f4 _

q1q3q4q7,

q3 - q3,

q2q3q5q5q7,

fs _

q3 - q2q3q5q7q5,

r1

q4 = q4,

fs _

q3 - q2q3q4q5q7,

h _

q4 = q5, h

r,

q4 = q4q5,

q3 - q3, t1-

q4

= q4,

h q4 = q4, q4 = q1q5, q4 - q2q5q5q7, q4 f4 _ fs _ f5 _ f7 _ fs _ q4 - q2q5q5q7, q4 - q4 - q4 - q4 - q4, t,

a,

r1

q5 = q5, q5h

=

q§4 =

q5 = q4q5,

t1

q4,

t2

q5 = q5 = q5 = q5,

q5J,

q~ 1 = q2q5qs,

q6 = q3q4q5q7,

= q5q5qg,

!B _

q6 - q1q4q5, 3

q~' = q5q5qs, q6/J = q1 q2qs,

t,

t1

a, q7

r,

=

= q5 , q5h = q1 q2 q5 q5 q7, q2q4q5q7, q§5 = q§5 = q{7 = q§8 = q5,

q1 q4,

qi' = q1q2q4q7, q{'

J, -

q6 = q2q7q3, q{3 h

= q1q2q7, 3 = q1q5,

q{4

= q5qgq7,

= q4q5,

fs

= q2q4q7, q;1 = q2q5q5qgq7, q? = q5q7qg, q6

q{5

q6

= q4q7, q7 = q1q5q2, q7t, -- q7, q7h -- q1q4q5q7, q7J, = q7, qf3 = q1 q5q2, qf4 = q2q4q5q5, qf5 = q1q2@, qf6 = q2q4q5q5, qf7 = q2q4q5q5, qf3 = q2q4q5q5, t1

7.2 Construction of the 2-central involution centralizer

291

U has center Z(U) = (qg) and a faithful permutation representation PU of degree 2048 with stabilizer Y generated by r2r1, (t2q6)2, (q6r1]4)4, (q6fsr1)4, (rrq6fs)3, (r1t1t2]4) 3 and (r1t1q6fs) 2 .

(1) The finitely presented group H3 has a non-split extension

by the elementary abelian group V = (v; I 1 :::; i :::; 8) with the following set R(H2) of defining relations: 3 a1

= a22 = r13 = r22 = t13 = t22=1 '

aa2 1

_ a2 1,

(ak,r1) (r1,t1)

rr2 _ r2 1 1,

tt2 _ t2 1 1,

= (ak,r2) = (ak,t1) = (ak,t2) = 1, = (r1,t2) = (r2,t1) = (r2,t2) = 1,

for

l:::; k:::; 2,

= l for l :::; i :::; 8, fJ = l for l :::; j :::; 8, (v;,vj) = 1 for l:::; i,j:::; 8,

v;

= 1,

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 al V1 a1 V1 V2 V4 V5 v6 Vs al V2a1 V4 V5 v6 Vs -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 al V3a1 V2 V3 V4 V5 v6 Vs al V4a1 V2 v6 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 a 1 v5a1 Vs a 1 v6a1 v 2 v 5 Vs a 1 vsa1 v 5 Vs -1 -1 -1 -1 -1 -1 1 al V7a1 V2 V5 v6 V7 Vs

= =

=

= , = ,

=

= ,

1 =

(a2,v 1 ) -1

-1 -1

a 2 v6a2v4 v 6 -1

2 =

(a2,v 1 )

-1

3 = (a2,v4 1 ) =

(a2,v 1 ) -1

-1

= a 2 V7a2v 4 v 7

=

1

(a2,v 5 1 )

,

= 1, = , = ,

-1 -1 -1 -1 -1 -1 -1 -1 a2 vsa2v5 Vs r 1 v1r1v1 v 5 r 1 v2r1v2 v 5 -1 -1 -1 -1 -1 -1 -1 -1 1 r1 V3r1V3 V5 r1 V4r1V5 r1 V5r1V4 V5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 r1 V6r1 V2 V5 Vs r1 V7r1 V2 V5 v6 V7 Vs

= =

=

=

= =

= 1,

-1 -1 -1 -1 -1 -1 -1 r 1 vsr1v 2 v 6 Vs r 2 v1r2v1 v 5 -1 -1 -1 -1 -1 -1 1 r2 V2r2V2 V5 = r2 V3r2V3 V5 = , -1 -1 -1 1 r2 V4r2V4 V5 -1 -1 -1 -1

= ' (r2, V5 1) = 1' r 2 v6r2v 5 v6 Vs = 1 , -1 -1 -1 -1 r2 V7r2V5 V7 Vs = 1, t 1-1 V1 t 1 Vl-1 V2-1 V4-1 V5-1 V6-1 V7-1 = l , t 1-1 V2 t 1 V2-1 V3-1 V4-1 V5-1 V6-1 V7-1 = 1 , t 1-1 V3 t 1 V1-1 V3-1 V4-1 V5-1 V6-1 V7-1 = 1,

=

1,

292

Conway's simple group Co1

7.2 Construction of the 2-central involution centralizer

293

Conway's simple group Co1

294

(fd6) 2 = (f5h )2 = (!5!7 )2 = (hfs)2 = 1, f f f -1 -1 -1 -1 -1 1 f 4J8 4J8V2 V4 V5 V5 V7 = ' (f5fs) 2 = (fds)2 = (h fs)2 = 1, fih hfih f2v1 1 V4 1 V5 1

= 1,

fihhfihhv 11v:; 1v;;1v7 1 = 1, f 1 f 8 f 3 f 1 f 8 f 3 V2-1 V4-1 V5-1 V5-1 V7-1 = l , f 2 f,6 f 4 f 2 f,6 f 4 V2- I V4-1 V5-1 V5-1 V7-1 = 1, (hf6h) 2 =

fih f4fih f4vi 1 v:; 1 v5 1 V7 1 = f4f5f5f4f5f5 = 1, f4h f5f4f5f?v:; 1v;; 1v5 1V5 1V7 1 = hfsf5!3f5fs = l.

(m) H2 has an extension H 1 by Z(Q) = (z) which is isomorphic to the finitely presented group

H1 = (a1, a2, r1, r2, t1, t2,

Jj, q; 11 .;), 1 :::; i:::; 8, of semi-simple characters of G 1 S" Fi 23 and A 1 of minimal degree 8671. The irreducible constituents of their 372

Fischer's simple group Fi; 4

373

restrictions to the subgroup H 1 are described by well determined characters of Table 6.5.2. Each semi-simple character ))

one obtains the transformation matrix 'Ti satisfying u = fl:®(u)7i and tJ = fl:® (V) 7i . (a) Let m'.D = (u,tJ) and mSj = (u,tJ,r). Let D = Cy(m'.D) and 1i = Cy (mSj). Let 812a, 812b and D23a, D23b be two distinct copies of the irreducible characters 812 and 823 of mD, respectively. Using PG 1 and MAGMA it can be checked that the irreducible characters ?rs and n 16 of mH have the following restrictions to mD = (u, v): ?rs lmD

= D12a + 023a + 027 + 039,

?r15 lmD

= D12b + D23b + 029 + 054 + 059,

where the irreducible characters 812, 823, 827, 829, 039, 054 and 059 of mD ~ G 2(3) x 83 have degrees 78, 156, 182, 182, 364, 728 and 1664, respectively. Since WlmD is a semi-simple FmD-module Theorem 2.1.27 of [92] implies that there is an isomorphism o: : D--+ D1 = GL2 (13) x GL2 (13) x F*5. (b) Furthermore, Schur's Lemma asserts that 1{ 1 = o:(1-i) is generated by the two blocked diagonal matrices given in the statement because 2 is a primitive element of the multiplicative group F* of F = GF(13). (c) Let .s1 = 7i- 11,;w(s)7i. Let m(!; = (m'.D,.s1) and£ = Cy(mQ;). Let D12a, D12b and J23a, J 23b be two distinct copies of the irreducible characters 812 and 823 of mD, respectively. Using PG1 and MAGMA it can be checked that the irreducible characters 'lj;4, 'l/J10, 'lj;19, 'lj;35 and ?r51 of mE have the following restrictions to mD = (u, v):

= D12a, '!/J10 Im D = D23a, 'l/J19 Im D = D12b + 029, = ()27 + ()54, '!/J6llmD = ()23b + ()39 + ()59. Lemma implies that £ 1 = o:(£) is generated by the

'!/J4 Im D

'!/J35lmD

Now Schur's five blocked diagonal matrices bj given in the statement. (d) An 1-i-£ double coset representative is of the form diag(A, B, 1, 1, 1, 1, v) for some A, B E Y and v E F*. By multiplying from left and right, we observe that diag(A, B, 1, 1, 1, 1, v) and diag(A', B', 1, 1, 1, 1, v) represent the same double coset if and only if the first columns of A and B are each a scalar multiple of the first columns of A' and B', respectively. So, we have 12 choices for v, and I GL(2, 13)1/12) = 2184 choices for A and B. Thus there are 2184 2 · 12 = 57238272 1-i-£ double cosets. (e) By Theorem 7.2.2 of [92] the irreducible representations of the free product mH *mD mE of the groups mH and mE with amalgamated

Fischer's simple group Fi; 4

384

subgroup mD are described by the 7-{-£ double coset representatives T of 'D. The elements rand f = (u 3 vsv) 9 are two commuting involutions of G 1 ~ Fi 23 by Lemma 9.2.l(o). Let u and f be their matrices in m.5) and m 1 = (IJ, q), Ql1 = (S:> 1, t) and 18 1 =

(S:>1,tu).

sis

Let E = (a,b,c,d,t,g,h,i,j,k,vi 1 11) be the non-split extension of the Mathieu group M2 4 by its simple GF(2)-module V2 constructed in Lemma 7.1.1 (g), and let E 23 = (a, b, c, d, t, g, h, i,j). Let J: = [(1Jq21Jq1Jq2)11 (q21J2q1Jq1J)11 (qtJ2qtJqtJqtJq)4]12, ui = (J:IJJ:)7, u2 = (J:IJJ:IJJ:IJJ:)4, U3 = (J:IJJ:1) 2J;l) 2J:IJJ:1J)2, U4 = (J:IJJ:1J 5J:tJ 4)2 andu5 = [tJ(.stu)21)tu]7. Then the following assertions hold. J

(a) The subgroup '.1'1 = (U; J 1 S i S 4) of '.D = (J:, tJ) is a Sylow 2-subgroup of 181 = (q, IJ, tu) of order 218 . (b) '.1'1 has a unique maximal elementary abelian normal subgroup SB of order 211 . It is generated by the 11 involutions u1,

ut

(u1u2)2,

(U1 U2U3 )4,

(u1u3) 2,

(U1 U2U4 )4,

(u1u4) 2,

(U1 U3U4)4,

(u2u4)4,

(U~U3 )2,

(U1 U2U4 U2 )2.

(c) .s = (tJ 5 t) 7 is an involution ofl.2l1 such that l.2l1 = (S:>1,.s), '.1'1 = '.1'1 and SB 5 = SB, and 6 = ('.1'1,.s) is a Sylow 2-subgroup of Ql1 of order 2 19 . (d) 911 = N,s 1 (SB) = (J:, IJ, tu) is isomorphic to a non-split extension of M23 by½ JM23 and :Di = Nw. 1 (SB)= (J:, IJ,.s) (e) There is an isomorphism p between 911 and the subgroup E 23 of E such that p(tJ) = (y~y3y2y3)3(w3W1W2W1W2W1W3WfW3W2W3W2W1 w3) 20 , p(J:) = (x1x2X4X5X4X2X5) 3 and p(tu) = (e2e3e2e~)7, where x1

= (ij) 3 ,

x2

=

X4 = (jhighaji)4,

(gahigai)2,

X3

=

(aghijagh)4,

x 5 = (aighjigai)4,

= i, Y2 = ag, Y3 = (ahj)3, W1 = (Y2Y~) 2 , W2 = (Y1Y2Y1Y2Y3)3, W3 = (Y1Y2y3y~)3, e1 = (agijih)4, e2 = (ag3ihj)7, e3 = ghghiai. Y1

(f) There is an isomorphism µ between :D 1 = Nw. 1 (SB) and the centralizer CE(u) of the involution u = (p(J:)p(tJ)2)7 of E such that

9.4 ® is isomorphic to Fischer's simple group Fi; 4

399

µ(r-) = P(F), µ(tJ) = p(tJ) and µ(s) = (mfm2m1m2)2, where m1 = agahj, m 2 = (ijhkj) 2 and m 3 = (ahjagk) 5 . (g) The subgroup Q: = (F, IJ, tu, s) of ® has a faithful permutation representation PQ: of degree 1518 with stabilizer ( (t)E) 7, (tut)EIJ )3, (EtJ3)2, (tJ2tl11)2)3). (h) The groups Q: and E are isomorphic. (i) 3 = (FtJtu )8 is a 2-central involution of Q: with centralizer CIE (3) of order 221 · 33 · 5 generated by the elements t 1 = (stJ 3)3, t 2 = (tJ 2tut)E)6, t 3 = (EtJtl1tJE) 2 and t 4 = (stut)Etu )6 with respective orders 2, 4, 4 and 2.

(j) C® 1 (3) has order 218 · 35 · 5. It is generated by (FtJFtu )7, fa = (Ftut)tut) 2) 7 and tJ = (tuqtut)qt))7.

h = r-tJtu, fa =

(k) The subgroup (tifati,(fifafitifi)4,(fafitifiti) 4 ) ofC® 1 (3) has index 512. Furthermore, it does not contain 3. Q:. It is also the unique maximal elementary abelian normal subgroup of the Sylow 2-subgroup 6 = (u; 11 :S i :S 5) of Q: contained in CIE (3).

(1) 23 is the Fitting subgroup of

Proof In order to simplify the notation of the proof we replace the German letters by Roman letters. In particular, we let ME= (x, y, w, s) be the subgroup CE. (a) Let PA 1 be the faithful permutation representation of A 1 of degree 56320 constructed in Lemma 9.1.l(c). By Lemma 9.1.2 and Lemma 9.2.l(f) we know that H 1 = (y, q) = (x, y, q). Now Corollary 6.2.4(b) asserts that D = (x, y) has odd index in H 1 and therefore in G 1 . Thus D contains a Sylow 2-subgroup of G 1 . The given Sylow 2-subgroup T1 of G 1 and its generators t; have been found by using MAGMA, the permutation representation P A 1 and the program GetShortGens (H_1, T_1). (b) Applying the MAGMA command

Subgroups(T_1: Al:=Normal, IsElementaryAbelian := true) we observed that T 1 has 44 elementary abelian normal subgroups. Exactly one of them is maximal and has order 211 . It is denoted by B. Its given generators have been calculated by means of the program GetShortGens(T_1, B). (c) Since IA1: H 1 = 2 a Sylow 2-subgroup of A1 has order 219 . Applying P A 1 and MAGMA it is easy to verify that s = (y 5 t) 7 is an involution of NA 1 (Ti) not contained in H 1 and that S = (T1 , s) has order 219 . Hence B 3 = B holds trivially by (b ). 1

Fischer's simple group Fi; 4

400

(d) It can easily be verified by another application of PA 1 and MAGMA that NA 1 (B) = (x, y, s). Similarly one establishes that N1 = Nc 1 (B) = (x, y, w) by means of the faithful permutation representation PG 1 of degree 31671 with stabilizer H 1 constructed in [73]. Hence N 1 is a non-split extension of M23 by Lemma 6.1.2 and Kim's Theorem 6.3.1. (e) By Lemma 7.1.l(h) E = (a, b, c, d, t, g, h, i,j, k) has a faithful permutation representation PE with stabilizer U3 = (g, h, i, (dg )5 , ( dhj k ) 3 , (ijkj) 2, (dhjidg) 3). Lemma 8.2.2 of [92] states that its subgroup E 23 = (a, b, c, d, t, g, h, i, j) has index JE : E 23 J = 24. Applying the command Isisomorphic(N_1,E_{23}) MAGMA establishes an isomorphism p: N 1 ----+ E 23 . The words of the images p(x), p(y) and p(w) of the generators x, y and w of N 1 are constructed as follows. Let C = CE 23 (p(x)). Using PE and MAGMA one sees that JCJ = 214 . The five generators

(ij)3, x2 = (gahigai) 2, X3 = (aghijagh)4, X4 = (jhighaji)4, x 5 = (aighjigai) 4 of C were obtained computationally by means of the x1

=

program GetShortGens(E_ {23} ,C). Using the program LookupWord(C, \rho(x)) MAGMA returned p(x) = (x1X2X4X5X4X2X5) 3. The expressions for p(y) and p( w) are obtained similarly. (f) By Kim's Theorem 6.3.1 we know that z1 = (xy 2 )7 is a 2-central involution of G 1 . Clearly u = (p(x)p(y) 2)7 is an involution of E 23 . Applying MAGMA and PE the reader can verify that Cu= CE 23 (u) has order 219 . 32 . 5. 7 · 11. Similarly one observes that D1 = NA 1 (B) = (x, y, s) has the same order. Using P A 1 , PE and the command Is Isomorphic (D_1, C_u) MAGMA establishes an isomorphism µ : D1 ----+ Cu. Applying the MAGMA command IsConjugate(C_u, \mu(y), \rho(y)) we found an element c1 E Cu such that µ(yl 1 = p(y). Using the command exists(c){q: q in C_{C_u}(\rho(y)) I (\mu(x)-{c_1})-q

= \rho(x)} one gets an element c2 E Cu such that µ(xl 1 c2 = p(x). Henceµ' : D1 ----+ Cu defined by µ' (d) = µ( dl 1 c 2 , d E D 1 , is an isomorphism between D1 and Cu such that µ'(x) = p(x) and µ'(y) = p(y). It has been checked that CE ( (p(y), p(x))) = (u). Furthermore, µ'(s) has a centralizer CM 23 (µ'(s)) of order 217 which is generated by the three elements m 1 , m 2 and m3 of M 23 given in the statement. Another application of the Lookup command yields the wordµ' (s) = (m1m2 m1 m2 ) 2 . Hence the map µ' : D1 -. CE (u) satisfies all conditions of (f). (g) Using (e), (f), PE and MAGMA it has been verified that E = (p(x), p(y), p(w), µ(s)). Applying the program GetShortGens CE, U_3)

9.4 . Then CM: =GModule (PU24, FEalg) denotes the cohomology module. Applying the command

Cohomologica1Dimension(PU_(24}, CM, 2) MAGMA establishes that the second cohomological dimension is 3. Thus there are seven non-split 2-central extensions U; of 1/24. The command P : = ExtensionProcess (PU24, CM, U24) was used to establish a presentation for each of them. We checked computationally that the Sylow 2-subgroups S; of the extensions U; are pairwise non-isomorphic. (k) Using the MAGMA command Is Isomorphic (S_i ,D) it has been verified that D is isomorphic to the Sylow 2-subgroup S1 = (Ji, h, h, f4, f 6 , h) of the extension U1 . Therefore only the presentation of U1 is given. (1) Clearly, S = (!1, h, h, f4, f5, h) is a Sylow 2-subgroup of U1. Using PU1 and the MAGMA command

Subgroups(S: Al:=Normal, IsElementaryAbelian := true) one verifies that S has exactly two maximal elementary abelian normal subgroups A and B of order 32. Another application of the short generator program provides the generators of A and B given in the statement.

Tits' group 2F 4 (2)'

468

Furthermore, routine MAGMA calculations verify that Nu 1 (A) = S

~

CE,· (m) MAGMA also confirms that Q = (!1, h, h, f4) is the Fitting subgroup of U1 and that IQI = 29 . (n) It has been checked with MAGMA that the derived subgroup Q' of Q equals the normal subgroup B of S. In particular, Q is not extraspecial. D The following subsidiary result is due to Wang and the author [96]. Lemma 10.1.2 Let E = (e; I 1 s:; i s; 8) be the finitely presented group constructed in Lemma 10.1.l(f). Let s = e 2 (ere 2) 4 (efe2e 1e2) 2 (er e2e1 e2e1 ) 2 , h = (e2 e1 e2e1 e2ef e2 )2 e1 e2ef e2et e2e1 e2 and r = e3. Then the following assertions hold.

(a) E = (s, h, r) has the following set of defining relations R(E):

= h4 = r 3 = 1, hr- 1 sh- 1 shr = 1,

s2

(sr- 1) 4 = 1, (h- 1r- 1h- 1rs) 2 = 1, h- 1 rh- 1 sh- 2 r- 1 h- 1 r- 1 sh- 1 = 1. (b) E has a faithful permutation representation of degree 2048 with stabilizer (r). (c) S = (s,h) is a Sylow 2-subgroup of E. Proof (a) Let X = (s 1 , h 1 , r 1) be the finitely presented group with defining set of relations R(X) equal to R(E) except for the indices of its generators. Then X has a faithful permutation representation PX of degree 211 with stabilizer (r 1 ). Let PE be the faithful permutation representation of E = E2 given in Lemma 10.1.l(f). Applying the command Isisomorphic(PX,PE) MAGMA establishes an isomorphism a : X ----+ E. Calculating sets of short generators x;, Yj and rk of the centralizers CE(a(s)), CE(a(h)) and CE(a(r) and applying then the LookupWord program for the three images in their centralizers in Ethe author found the presentations of the elements s = a(s 1 ), h = a(h 1 ) and r = a(r 1 ) stated in the hypothesis. (b) This assertion is trivial by the proof of (a). (c) This statement has been checked with MAGMA. D We now show that the constructed group U1 is isomorphic to the centralizer Cr(z) of a 2-central involution z of the Tits group 2F 4 (2)'. For that proof we restate Parrott's finitely presented group H 1 of [105]

10.1 Construction of the 2-central involution centralizer

469

and the finitely presented group H constructed by Wang and the author in [96]. Parrott derived the presentation of H 1 from Tits' original presentation of the group of Lie type 2F 4 (2)'; see [132]. Proposition 10.1.3 Let U1 = U1,h,h,f4,f5,f5,h) be the finitely presented group of order 211 · 5 constructed in Lemma 10.1.1. Then the following statements hold.

(a) U1 is isomorphic to the finitely presented group H = (hi I 1 ~ i ~ 11) of {96} with the following set R(H) of defining relations: h~ = ht = h? = h§ = h~ = hio = hi 1 = 1, (h5, h1a1) = (h5, h1/) = (h5, h7 1) = 1, (h1, h8) = (h1, h9) = (h8, h9) = (h1, h10) = 1, (h8, h10) = (h9, h10) = (h1, h11) = (h8, h11) = 1,

(h9,h11) = (h10,h11) = (h1,h1i1) = (h2,h1/) = 1, (h3,hs 1) = (h3,hr;1) = (h3,hi/) = (h4,hs 1) = 1, (h4) hg 1) = (h4' h101) = (h4' h1/) = (h5' h7 1) = 1, (h5,h1a1) = (h5,h1i1) = (h5,h9 1) = 1, 1, h 32h-l 11 = h2h-l 4 11 = h2h-1 5 11 = h2h-l 6 '11 = h2 1h1 2 h2h1 = h2 1h5h2h5h1l = h1h3h--;-1h4h1/ = 1, h1h4h1 1h5h1/ = h2 1h4h2h5h1/ = h2 1h8h2hs 1hg 1 = 1, 1 h- 1h 858 1 l 1 h- 1h 747 h h- 1hh h- 1hh h- 1hh 1-lh918 9=4 11=5 11=,

h 6 1h8h5h 8 1h 1/ = h 61h9h5h 9 1h 1/ = h3h4h3h4h81h 1/ = 1, h5h5h5h5h7 1h1/ = h1 1h1oh1hs 1h1/ = h2 1h9h2h1l h1/ = 1, h 2-lh 10 h 2h7 1h111 = h3 1h 7h 3h7 1h111 = h3 1h 10 h 3h101h111 = l , 1 1 h- 1 h h h- 1 hh- 1 h817 h h- 1 hl h 2-lh3248 11=1 1011=, 1 1h- 1h- 1 l 1hh 1-lh3h 1h 6 h8 1h9 101 = h2 1h 7h 2h7 h9 10 11 = , 1 1 1 1 h 1 h7h1h 9 h 10 h 1/ = h5h4h3h 1 h3h5h1h/h91h 1a1h 1/ = 1, (b) U1 is isomorphic to Parrott's group H 1 = (z, t, v, u, w, a, b, c, d, x, y, r) of (105} with the following set R(H1) of defining relations: z2 = t2 = v2 = u2 = w2 = a2 = r2 = 1, x4 = 1, (ry) 5 = d- 1u- 1du = y- 1a- 1ya = d- 1w- 1dw = c- 1r 1ct = 1, c- 1u- 1cu = c- 1w- 1cw = a- 1x- 1ax = x- 1txt = b- 1wbw = l,

Tits' group 2F 4 (2)'

470

d- 1 t- 1 dt = c- 1 v- 1 cv = z, a- 1 b- 1 ab = x- 1 vxv = t, c- 1 d- 1 cd = c2 = wu,

b- 1 x- 1 bx = a,

r- 1 ar = du, r- 1 br = dcbvt,

a- 1 c- 1 ac

y- 1 d- 1 yd = bw,

x- 1 c- 1 xc = abuv,

d- 1 b- 1 db = b2 = v,

r- 1 vr = uv, r- 1 dr = au, x2

= vt,

b- 1 c- 1 bc = uv,

y- 1 c- 1 yc = atz,

x- 1 d- 1 xd = abcuv, r- 1 tr = wuvz,

a- 1 d- 1 ad = x- 1 wxw = u,

= yz,

r- 1 ur = u,

x- 1 uxu = v, r- 1 wr = vtz,

= cdau, rxr = (yr) 2 x, a- 1 waw = b- 1 ubu = z. r- 1 cr

In particular, any of the three groups U1 , H and Hi is isomorphic to the centralizer Cr (z) of a 2-central involution z of the Tits group T = 2F 4 (2)'. Proof (a) The group H has order 211 · 5, as can be checked by means of the MAGMA command CosetAction(H, sub). In particular, H has a faithful permutation representation PH of degree 211 · 5. By the proof of Lemma 10.1.1 we know that U1 has a faithful permutation representation PU1 of degree 211 . Now we apply the MAGMA command Islsomorphic(U_1,H). MAGMA confirms the assertion. (b) Both groups H and H 1 have a regular permutation representation PH and PH1 with stabilizer 1. Using the above isomorphism test program of MAGMA we checked in [96] that H ~ H 1 . Now (a) completes the proof. The statement H1 ~ Cr (z) holds by [105]. D Definition 10.1.4 A finite simple group G is said to be of Tits-type if it has a 2-central involution z such that its centralizer Cc ( z) is isomorphic to the finitely presented group H of Proposition 10.1.3. Proposition 10.1.5 Let H = (h1,h2,h3,h4,h5,h5,h7) be the finitely presented group of order 211 · 5 constructed in Lemma 10.1.1 (k) with the set of defining relations R(H) adapted according to the notations of the generators of the groups H and U1 . Then the following statements hold. (a) H = (s,h,x), wheres= h 1 , h = h 6 and x = h 5 have respective orders 2, 4 and 5. (b) S = (s, h) is a Sylow 2-subgroup of H with center Z(S) = (z) = Z(H), where z = (sh) 8 . (c) V = (z,t) is the unique normal Klein 4-group of S, where t = (shsh 2 ) 4 .

10.2 Fusion

471

(d) A = (z, t, a 1 , a 2 , a 3 ) is the unique normal elementary abelian subgroup of order 32 in S which is not normal in H, where (h 2 sh 2 sh 3sh)2, a2 = (sh 2 ) 4 and a3 = h 2 sh 2 shsh 2 sh. B = (z,t,a 1 ,a 2 ,b3) is the unique normal elementary abelian subgroup of order 32 in S which is normal in H, where b3 = (sh 3 sh) 2 . Both maximal elementary abelian normal subgroups A and B are self-centralizing in H. D = NH(A) = S. C = CH(V) has order 210 and center Z(C) = V. H has a faithful permutation representation PH of degree 29 with stabilizer (x, h), the Frobenius subgroup. A system of representatives of the 28 conjugacy classes of H = (h,s,x) is given in Table 10.7.1. A system of representatives of the 35 conjugacy classes of D = (h,s) is given in Table 10.7.3. The character table of H is Table 10.8.1. The character table of D is Table 10.8.2.

a1 =

(e)

(f)

(g) (h) (i) (j) (k)

(1) (m)

Proof All these statements have been checked with MAGMA and the faithful permutation representation of H given in (i). The various generators of the subgroups of H have been found by the methods used already in the proofs of the previous results. D

10.2 Fusion

In this section the conjugacy classes of elements of even order of a finite simple group G of Tits-type are determined. Furthermore, it is shown that the normalizer Na(A) of the maximal elementary abelian normal subgroup A of a fixed Sylow 2-subgroup S of H is uniquely determined up to isomorphism in all such simple groups G. Lemma 10.2.1 Let G be a finite simple group of Tits-type having a 2central involution z with centralizer H = Ca (z) = (h, s, x) defined in Proposition 10.1. 5. Let u = (h 2 s )4 . Then the following statements hold.

(a) u is an involution of the Sylow 2-group S defined in Proposition 10.1. 5 with centralizer CH (u) of order 29 . (b) T = CH (u) is a self-normalizing subgroup of Ca ( u). (c) T is a Sylow 2-subgroup of Ca ( u). (d) z and u are not conjugate in G.

Tits' group 2F 4 (2)'

472

Proof (a) Using MAGMA and the faithful permutation representation of H of degree 512 described in Proposition 10.l.5(g), it is easy to see that u E S, and ICH ( u) I = 29 .

(b) Suppose that a subgroup R of U = Ca(u) normalizes T. By MAGMA Z(T) n T' = (z), so (z) 7. Therefore one can replace the presentation of H stated in Lemma 11.1.l(g) by Schur's original presentation of 2A8 • This is done in [80]. Remark 11.1.3 In view of Lemma 11.1.1 all conditions of Steps 4 and 5 of Algorithm 1.3.8 are satisfied. Thus one could immediately apply all steps of Algorithm 7.4.8 of [92] to provide an existence proof of a simple group of Mel-type. This has been done in the author's joint paper [80] with H. Kratzer, W. Lempken, and K. Waki, where Mel is realized as a simple subgroup of GL 22 (11). In the following we instead realize Mel as a finitely presented group.

11.2 Structure of the given centralizer H

= 2A8

In this section we construct another presentation of the given centralizer H = Ca(z) of any finite simple group G of Mel-type which will be extended to a unique presentation of the simple target groups G in Section 11.3.

sis

Proposition 11.2.1 Let H = (hi 11 3) ~ 2As be the finitely presented group defined in Lemma 11.1.1 (g). Then the following assertions hold.

McLaughlin's group Mel

506

(a) H = (h,n), where h = h 1 and n = h 1h§ have orders 6 and 7, respectively. (b) The involution z = (hn) 15 generates the center Z(H) of H. (c) The involutions t = h 2 and z generate the unique normal Klein 4group V of the Sylow 2-subgroup S = (a1, a2, c1, c2, r) of H generated by the five involutions a 1 = (nh 2n 3 hn) 3 , a 2 = (hnhn 2h 3 n) 3 , c1 = (nh 3 nhnz)2, c2 = (hnh 3 n 2hn) 3 andr= (h 2nhnh 2n) 3 . (d) A= (z,t,a1,a2) and A 2 = (z,t,c1,c2) are the only elementary abelian subgroups of order 24 of S. They are both maximal elementary abelian normal subgroups of S. Furthermore, V = An B. (e) C = Cs(V) =AB= (z, t, a1, a2, c1, c2) has the following setR(C) of defining relations: z 2 = t 2 = ar = a~ =

er = ~ = 1,

= (c1,c2) = (z,t) = 1, (z,a1) = (z,a2) = = 1, (t,a1) = (t,a2) = (t,c1) = (t,c2) = 1,

(a1,a2) (z,c2)

(z,c1)

=

In particular, V is the center Z (C) of C. (f) C has a complement W = ( w, r) in M = NH (V) generated by r and w = (h 2nh4 nh) 2. M = (z,t,a1,a2,c1,c2,w,r) has a set R(M) of defining relations consisting of R( C) and the following relations:

= w2, (z,w) = (z,r) = (t,w) = a'f = za1a2, a1 = a1, a~ = za1, a 2 = a1a2, Cl = ZC1C2, c1 = c1c2, ~ = ztc1, C2 = C2.

r 2 = w3

= 1,

wr

(g) N1 = NH(A) =A: Ki, where K1 = (w,r,u1) u1 = [(nh) 3 nh 2 n 2 ] 4 • (h) N2 = NH(B) = B: K2, where K2 = (w,r,u2) u2

=

1,

tr

= zt,

~

GL3(2), and

~

GL3(2), and

(nhn 2h4 ) 4 •

(i) H = (z, t, a1, a2, c1, c2, w, r, u1, u2) has a set R(H) of defining relations consisting of R( M) and the following relations:

= U23 = 1, (u1c1) 4 = (u1c2) 6 = (u1r) 3 = 1, (wu1w) 3 = urwu1r, (z,u1) = 1, tu 1 = zta1, af 1 = zt, a~ 1 = zta1a2, 3

U1

11.2 Structure of the given centralizer H

= 2As

507

(uw)2 2 -- u2wu2 2 2,

(j) H has a faithful permutation representation of degree 240 with stabilizer (w,r,u1). Proof Let PH be the faithful permutation representation of H = (h 1 , h2, h 3 ) of degree 240 defined in Lemma 11.1.l(k). The Sylow 2subgroup S of Hand its elementary abelian normal subgroups V, A and B have been found by means of MAGMA and PH after having checked that H = (h, n), where h = h 1 and n = h1h§ in H. Using the program LookupWord(C,xAy) we established the relations of C = CH(V) = AB. The complement (w,r) of C in M = NH(V) was calculated by means of PH, MAGMA and the command HasComplement(M,C). The new relations of M were established by another application of Algorithm 1.4.3. Using the MAGMA command CosetAction (M, sub) the author verified that R(M) is a set of defining relations of M. The complements K 1 and K 2 of the normal subgroups A and B in N 1 = NH(A) and N 2 = NH(B) were again found by MAGMA. Their generators were calculated using the program GetShortGens (H,K_i). Employing the MAGMA command FPGroup (K_i) we obtained their relations with respect to the given generators. The same method was used for various small subgroups of H in order to get the additional relations of H given in (i). Employing the Todd-Coxeter Algorithm implemented in MAGMA it has been verified that H = (z, t, a1, a2, c1, c2,w, r,u1,u2) is a finite group of order IHI = 40320 satisfying R(H). In particular, it has a faithful permutation representation of degree 240 with stabilizer (w, r, u1). D

Corollary 11.2.2 Let H = (z, t, a1, a2, c1, c2,w,r,u1, u2) be the finitely presented group constructed in Proposition 11.2.1 {i). Let E be the finitely presented group constructed in Lemma 11.1.1. Let d = tu 1 . Then the following assertions hold.

(a) d has order 6, and H = (d,u1,u2). (b) Representatives hi and corresponding centralizer orders oflCH(hi)I of the 23 conjugacy classes

hf!

of H are stated in Table 11.4.2.

(c) The character table of H is given in Table 11.5.3.

508

McLaughlin's group Mel

(d) The normalizer D of the maximal elementary abelian normal subgroup A= (z, t, a1, a2) of the Sylow 2-subgroup S of H defined in Lemma 11.2.l(c) is generated by d and u 1 . Representatives di and the corresponding centralizer orders of ICH(di)I of the 16 conjugacy classes df of D = (d,u1) are stated in Table 11.4.3. (e) The character table of D is given in Table 11.5.2. (f) There is exactly one compatible pair (X,T) E mfchanc(H) x mf charc(E) of degree 22 of the amalgam Ht- D-----+ E:

(x, T)

= (xa + X4, T1 + T2 + Ts)

with common restriction TID = XID = 'lp1

+ 'lp4 + 'lp7 + 'lps,

where irreducible characters with bold face indices denote faithful irreducible characters. Proof (a), (b) and (d) By Proposition 11.2.l(j) H has a faithful permutation representation PH of degree 240 with stabilizer (w, r, u 1 ). Using it, the program GetShortGens(H,D) and MAGMA the author found the two generators d = tu 1 and u 1 of D = NH (A). The equation H = (D, u 2 ) is easily verified by means of MAGMA. The systems of representatives of the conjugacy classes of H and D have been computed by means of PH, MAGMA and Kratzer's Algorithm 5.3.18 of [92]. (c) and (e) The character tables of H and D were calculated by means of PH and MAGMA. (f) Let E be the finite group defined in Lemma 11.1.1. Then E has a 2-central involution z such that CE(z) ~ NH(A) by Lemma 11.1.l(h) and Proposition 11.2.1. Thus the amalgam Ht- D-----+ Eis well defined. The authors of [80] found the unique compatible pair by an application of Kratzer's Algorithm 7.3.10 of [92], the character Tables 11.5.1, 11.5.2 and 11.5.3 and the fusion of the classes of D = NH(A) in Hand D = CE (z) in E. It has been adjusted to the new character tables of the three local subgroups. D

11.3 Existence and uniqueness proof

In this section a new existence and uniqueness proof for McLaughlin's simple group Mel is given. It simplifies the one by Kratzer, Lempken,

11. 3 Existence and uniqueness proof

509

Waki and the author given in [80]. Both proofs realize a simple group G of Mel-type as a uniquely determined finitely presented group having a faithful permutation representation with stabilizer H ~ Ca ( z). Whereas the proof in [80] exploits the existence of an automorphism¢ of H ~ 2A 8 fixing a given Sylow 2-subgroup S of H and permuting the two maximal elementary abelian normal subgroups A and B of S, the new proof is independent of such an assumption, which can be derived from I. Schur's presentation of 2A8 ; see [116] and [80].

Theorem 11.3.1 (Kratzer~Lempken~Michler~Waki) Let G be a finite simple group of Mel-type. Then the following assertions hold.

(a) G has a 2-central involution z such that Ca (z) is isomorphic to the finitely presented group H = (z,t,a1,a2,c1,c2,w,r,u 1,u2) with the set R( H) of defining relations given in Proposition 11. 2.1 (i). Thus we may assume that H = Ca (z).

(b) S = (z, t, a1, a2, c1, c2, r) is a Sylow 2-subgroup of G having a unique normal Klein 4-group V = (z, t). (c) The normalizer Na (V) of V contains an element x tJ. NH (V) of order 3 such that G = (z, t, a1, a2, c1, c2, w, r, u1, u2, x) has a set R( G) of defining relations consisting of R( H) and the following relations: x 3 = 1,

af =a1, 1 cf= c1c2, c~ = c1, = 1, (x- r) 2 = 1, xzc2wx- 1zc2w 2 = 1, (zc2x- 1)3 = 1, (u;- 1x- 1)2u1x = 1, ( x, za2 ) = 1, U2X -1 U2W 2X-1 U2-1 XU2-1 X = 1. zx =t,

tx =zt,

a~ =zta2,

(x,w 2 )

(d) Ghasorder2 7 -3 6 -5 3 -7-11. (e) G is uniquely determined up to isomorphism. (f) G has a faithful permutation representation PG of degree 22275 with stabilizer H. (g) G = (u 1 , d, u 2 , x) has 24 conjugacy classes gf whose representatives gi and orders of the corresponding centralizers ICa (gi) I are stated in Table 11.4-4. (h) The character table of G coincides with the one of Mel in the Atlas /19}, p. 101.

McLaughlin's group Mel

510

Proof (a) holds by Lemma 11.1.1 and Proposition 11.2.1. (b) This is a restatement of Proposition 11.2.l(c). (c) By Table 11.4.2 H = Ca(z) has two conjugacy classes of involutions which can be represented by the generators z and t of the Klein 4-group V. Since G is simple, Glauberman's Z*-theorem asserts that z and t fuse in G; see Theorem 4.7.3 of [92]. Furthermore, Table 11.4.2 implies that Ca(V) = CH(V) = CH(t) has order 26 ·3. Hence Q = Cs(t) has index 2 in the Sylow 2-subgroup S of H given in statement (b). Furthermore, V = Z(Q) by Proposition 11.2.l(e). Ast is conjugate to z we also know that Q is properly contained in a Sylow 2-subgroup of Ca(t). Thus Lemma 1.4.4. of [92] implies that N = Nc(V) > NH(V) and N /Cc (V) ~ GL 2 (2). In particular, there is an element x E N of order 3 such that zx = t, tx = zt and x (/; NH(V). Moreover, N = (NH(V),x). A presentation of C = Cs(V) = (z,t,a1,a2,c1,c2,w) is given in Proposition 11.2.l(e). Using it and the faithful permutation representation PH of H stated in Proposition 11.2.l(j) MAGMA is able to calculate the automorphism group AC of C and determine generators of the group IC of its inner automorphisms by means of the commands AutomorphismGroup(C), PermutationRepresentation(AC) and InnerGenerators (AC). Note that IC is an elementary abelian normal subgroup of order 16 in AC. By Proposition 11.2.l(f) and Theorem 1.4.15 of [92] we know that NI= Nc(V)/V ~ NH(V) is isomorphic to a semidirect product of IC and a subgroup T of AC of order ITI = 18 having a normal Sylow 3-subgroup of order 9 and exponent 3. Using the subgroup command of MAGMA and performing the necessary isomorphism tests the author verified that up to isomorphism there is only one subgroup NI of AC which has all the required properties. Moreover, NI = (cl, c2, c3, c4, kl, k2, u) has the following set R(N I) of defining relations:

Ci = c~ = c~ = c~ = kf = k~ = u 2 = 1, (q,~)=01,~)=(q,~)=(~,~)=02,~)=0a,~)= (k1, k2) = 1,

k'{ =ki,

k~ = kt

C4k1 = C3,

Clk2 = C1,

Cl = C1C2,

c~ = c2,

Clk1 = C1C2, ~1 = c1, l1 3 = C3C4, k2 k2 C2 = C2, C3 = C4, c!2 = C3C4, C3 = C3,

C4 = C3C4.

Furthermore NI has a faithful permutation representation P NI of degree 16 because T = (k1, k2, u) is a complement in NI. Therefore Holt's

11. 3 Existence and uniqueness proof

511

Algorithm can be applied to build the extensions of NI by the F(NJ)module V over F = GF(2). Let W = NH(V) = (C,w,r). Then W/V ~ JC : (k 1, u). Hence we may identify x with k 2 . From Proposition 11.2.l(f) follows that w = k1 operates trivially on V, and r = u operates by zr = z and tr = zt. Let Ml be the identity matrix in GL 2 (2). Let Mkl, Mu and Mk2 be the matrices in GL 2(2) corresponding to the actions of k1, u and k 2 = x on V. The MAGMA commands FEalg :=MatrixAlgebra and CM: =GModule (PNI, FEalg) implement the F(N !)-module structure. Applying then the MAGMA command CohomologicalDimension (PNI, CM, 2) one sees that this dimension is 1. Since C does not split over V by Proposition 11.2.1(e) it follows that Na (V) is the uniquely determined non-split extension El of NI by V. By means of the MAGMA commands P := ExtensionProcess(PNI, CM, NI) and E1 := Extension(P, [1]) we obtained the following set R(Ei) of defining relations of the extension E 1 = (e; 11 ~ i ~ 9): e 12 = e 22 = e 32 = e 24

1, = e53 = e36 = e72 = es2 = e2 9 =

= (eJ,e 9 ) = 1 for 1 ~ j ~ 5, (e1 , e2) = (e3, e4) = (es, eg) = (e7, es) = 1,

(eJ,es)

(e5,e1) = (e5,e2) = (e5,e5) = (e7,e2) = (e7,e3) = 1, e 6-1 ege5eseg = 1 , e7ege7eseg = 1, e 6-1 ese5e 9 = 1,

= 1, e2e3e2e3es = 1, 1 e2e4e2e4eg = 1, e7e5e7e 5 = , e7e5e7e 6-2 = 1 , -1 e1e5e2e1 = 1 , e -1 e2e5e1 = 1 , e5-1 e3e5e4e3 = 1 , e5 5 = 1,

e1e3e1e3eg

e1e4e1e4eseg -2

e 5-1 e4e5e3

= 1,

e7e1e7e2e1

= 1,

e6-1 e3e5e4

= 1,

e6-1 e4e5e4e3

= 1,

e7e4e7e4e3 = 1.

Let Yi = (e1, e2, e3, e4, e5, e7, es, eg). By means of the MAGMA command Islsomorphic (C, Y_1) we established an explicit isomorphism ¢ : Yi ---+ C. It has the following images: ¢(e1)

=

¢(es)

= w2,

za2,

¢(e2)

= a1,

¢(e7) = r,

¢(e3)

=

¢(es)= z,

zc2,

¢(e4)

¢(eg)

= tc1,

= t.

The subgroups A1 = (es,eg,e1,e2) and B1 = (es,eg,e3,e4) of E1 are the unique maximal elementary abelian normal subgroups of the Sylow

512

McLaughlin's group Mel

2-subgroup 81 = (e1,e2,e3,e4,e7) of E1. Thus ¢(A1) =A= (z,t,a1,a 2) and ¢(B1) = B = (z,t,c1,c2) are the unique maximal elementary abelian normal subgroups of the Sylow 2-subgroup S of G by Proposition 11.2.l(d). Using MAGMA and the permutation representation PH it has been checked that A and B are the unique maximal elementary abelian normal subgroups of C = Cs(V). Hence both are normal in Nc(V). In particular, they are invariant under conjugation by x E Nc(V). The action of x = e5 on the bases {es,e9,e1,e2} and {es, eg, e3, e4} of A1 and B1 has been determined by application of the program LookupWord(U, e_j ~{e_6}), where ej runs through the basis of U E {A1,Bi}. The same method has been applied to determine the action of the remaining generators e5, e7, e1 and e3. Similarly, one finds the matrices of u1 E NH(A) and u2 E NH(B) w.r.t. the bases {¢(es), ¢( eg ), ¢( e1), ¢( e2)} and {¢(es), ¢( eg), ¢( e3), ¢( e4)} of A and B, respectively. By Proposition 11.2.l(g) and (h) the subgroups K 1 = (w,r,u1) and K2 = (w,r,u2) are complements of A and Bin NH(A) and NH(B), respectively. Therefore L1 = (¢- 1 (K1),e5) = (e3,e5,e7,¢- 1 (ui),e5) andL2 = (¢~ 1 (K2),e5) = (e1,e5,e7,¢- 1(u2),e5) are complements of A1 and B1 in Q = NE, (Ai) and R = NE, (B1), respectively. The generating matrices of L 1 and L 2 are denoted by Qej, Qui and Rek, Ru1 , respectively. They are: 1000) (1000) (0100) _ 0100 _ (1000) 0100 Qe3 - ( o 1 1 o , Qe5 o o 1 1 , Qe7 -_ 1100 o o 1 1 , Qe5 -_ 1100 oo 1 o , 1001 0010 0001 0001 _ 1000) 0100 _ 1000) 1101 _ 1000) 0100 Qui - ( 1111 , Rei - ( o 11 o , Re5 - ( o o 11 , 00 10 1 101 1100 _ 0100) 1100 _ 1000) 1100 Re7 - ( o o 1 o , Re5- ( 0001, 0 0 11 0011

_ 1000) 1010 Ru2 - ( 111 o . 1111

Using the MAGMA command FPGroup(X) for X E {L1,L2} one verifies that the matrix groups M L1 = (Qe3, Qe5, Qe5, Qe7, Qui) and M L 2 = (Re1, Re 5, Re 6 , Re7, Ru 2) are isomorphic to the finitely presented groups FL1 = (g3,g5,g5,g7,g10) and FL2 = (g1,g5,g5,g7,gu), respectively. Here g10 and gu correspond to Qui and Ru2, respectively, and F L 1 has the following set R( L 1) of defining relations: 3 = g53 = g63 = g72 = glO = 1, (g5 1g7 ) 2 = (g7g3) 2 = (gr: 1g7 ) 2 = (g7g3) 2 = g3gu}g7g10 = 1, -1 -1 -1 -1 -1 1 g3glO g7g10 = glO g3g7g1og7 = g5glO g7g5 glO g5g10 = , (g5,g5) = (g6 1g7) 2 = (g3g6 1)3 = 1, -1 -1 )2 ( glO g6 g10g5 = 1 ·

g32

11.3 Existence and uniqueness proof

513

F L2 has the following set R( L 2) of defining relations: 3 = 953 = 963 = 912 = 911 = 1' (95 197 ) 2 = (95 19197 )2 = (g195 1) 3 = 1, -1 -1 -1 -1 1 911919791191 = 1 ' 911 95 911 9591195 = ' (95,91) = (95,95) = (95 191) 2 = 1, 91195 19119595 1(91/96) 2 = 1. 2

91

Since V = (z, t) is a maximal characteristic elementary abelian normal subgroup such that U = (H, Ne (V)) has a unique conjugacy class of involutions, Corollary 4.8.10 of [92] asserts that G = U. As Ne (A) n Ne (B) = Ne (V) it follows that G satisfies all the relations of the respective complements T1 and T 2 of Ne(A) and Ne(B). The groups L 1 and L2 are isomorphic to T1 and T2 under the isomorphisms ¢1 : L 1 -+ T1 and ¢2: L1-+ T1 defined by 'l/J1(9k) = ¢>(ek) fork E {3,5, 7}, ¢1(910) = u1, ¢1(95) = x and 'l/J2(9j) = ¢>(ej) for j E {1,5,7}, ¢1(911) = u2, ¢1 (95) = x, respectively. Inserting the values of ¢( e8 ) in the basis A 1 = {¢( e 8 ), ¢( e 9 ), ¢( e1 ), ¢( e2)} of the maximal elementary abelian subgroup A of S, and using the action of x = e 6 on the corresponding basis of A 1 given by the matrix Qe5, we obtain the following equations:

Similarly, the coefficients of matrix Re6 imply the following relations:

Furthermore, the relations of R(L 1 ) and R(L 2 ) involving 96 the following additional relations: (x, w 2)

=

(x- 1r) 2

(u1 1x- 1)2u1x

=

= 1,

(zc2x- 1) 3 = 1, (x,za2)

= 1,

=

x yield

= 1, u2x- 1u2w 2x- 1(u 21x) 2 = 1. XZC2WX-l zc2w 2

Hence G = (H,x) satisfies the set R(G) consisting of R(H) and these additional relations. Using the Todd-Coxeter Algorithm implemented in MAGMA by means of the command CosetAction(G,H) we checked that R( G) is a defining set of relations for the finitely presented group

G. Thus (c) holds. Furthermore, this application of MAGMA also provides a faithful permutation representation PG of G of degree 22275 with stabilizer H. In particular, (d) and (f) hold.

514

McLaughlin's group Mel

(e) Since N = Nc(V) = (NH(V),x) is the uniquely determined extension group of the uniquely determined subgroup NI in the automorphism group of the normal subgroup C = Cs(V) of NH(V), and since A and B are the uniquely determined maximal elementary abelian normal subgroups of C, the finitely presented group G = (z, t, a1, a2, c1, c2, w, r, u 1 , u 2 , x) is uniquely determined up to isomorphism by H and the unique action of the Glauberman element x of Nc(V) on V. (g) Let d = tu 1 • Then H = (d,u1,u2) by Corollary 11.2.2. Hence G = (d, u 1 , u 2 , x) by (c). The system ofrepresentatives of the conjugacy classes of G has been computed by means of PG, MAGMA and Kratzer's Algorithm 5.3.18 of [92]. (h) The character table of G was computed by means of PG and MAGMA. It agrees with the one of the Atlas [19]. In particular, the finitely presented group G is simple. D Corollary 11.3.2 Let G = (d, u1, u2, x) be the finitely presented group constructed in Theorem 11.3.1. Then the following assertions hold.

(a) G has a faithful permutation representation of degree 275 with stabilizer Q generated by the elements q1 = u1x 2, q2 = (u2xu1u2) 15 and q3 = (u2xu1x) 7 of orders 3, 2 and 2, respectively. Q is a maximal subgroup of G. (b) The permutation character A = lQ G has three irreducible constituents of degrees 1, 22 and 252, and Q ~ U4(3). (c) G has an irreducible 22-dimensional irreducible representation over the prime field GF(ll). The four generating matrices il, u 1, u 2 and J in GL 22 (11) are documented on the accompanying DVD. (d) G has an irreducible 22-dimensional irreducible representation over the prime field GF(2). Proof (a) The stabilizer Q of the 275-dimensional permutation representation was found by means of the permutation representation PG of Theorem 11.3.l(d) and the MAGMA command LowindexSubgroups(PMcL, 275). In particular, Q is a maximal subgroup of G. The three generators q; of Q were calculated by means of the program Get Sha rt Gens ( G, Q). The isomorphism type of the simple group Q ~ U4(3) was established by applying the command CompositionFactors(Q). (b) The permutation character A = lQ G has non-trivial inner products with the irreducible characters X1, X2, X4 of the Atlas [19], p. 101.

11.4 Representatives of conjugacy classes

515

(c) Let QG be the permutation representation of G = (d, u 1 , u 2 , x) of (a). Using the MAGMA command PermutationMatrix(FiniteField (11), y), y E {d,u 1 ,u2,x}, we define the subgroup MQ275 = (Md, Mui, Mu2, Mx) of GL21 5 (11). MAGMA's implementation of the Meataxe Algorithm provides four composition factors of degrees 22, 1, 251 and 1 by running the command MQirr275 : =Irreducible Representations (MQ275). The four generators of the 22-dimensional 11-modular irreducible representation of G are the matrices () = MQirr275[1].l, u1 = MQJrr275[1].2, u2 = MQirr275[1].3 and ~ = MQirr275[1].4 documented on the accompanying DVD. (d) This statement is proved as (c). The resulting generating matrices can be calculated by the same commands after replacing the prime 11

~2.

D

11.4 Representatives of conjugacy classes 11.4.1 Conjugacy classes of E = (e1,e2,e3,e4,e5) Class Revresentative l Centralizeil 1 1 21 · :P · 5 · 7 2, e5 27 • 3 · 7 2o el 25 · 3 22 . 32 3, e3 22 . 32 3o ele2 25 4, ele5 23 4o ele4 5 ele2e3 5 6, e4e5 22 · 3 60 ele2e3e4 22 · 3 7, ele3e4 2-7 ( ele3e41 3 2·7 72 23 ele4e5 8 14, ele4e5e3 2•7 ( ele4e5e3) 3 142 2·7

2P 3P 1 1 2, 1 2o 1 3, 1 3o 1 2, 4, 2o 4o 5 5 3, 2, 3o 2o 7, 7o 72 71 4 8 7o 140 71 141

5P 1 2, 2o 3, 3o 4, 4o 1 6, 60 7o 71 8 140 141

7P 1 2, 2o 3, 32 4, 4o 5 6, 60 1 1 8 2 21

516

McLaughlin's group Mel

11.4.2 Conjugacy classes of H

= (d, u 1, u1)

Class Representative I Centralizer! 2P 1 1 1 27 · 32 · 5 · 7 ( du 11 7 2 7 -3 2 -5-7 1 21 (d)3 1 26 • 3 22 (ulu2) 4 2 3 · 3 2 · 5 31 31 (d)' 2 2 · 3 2 32 32 (ulu2) 3 2 5 · 3 21 41 24 2, d 2 ul 42 (d 2 ulu2) 2 5 5 2 ·3 ·5 (ulu2) 2 61 2 3 · 3 2 · 5 31 d 3 u2 62 2 2 · 3 2 32 6 d 2 2 · 3 32 (d)5 2 2 · 3 32 64 (dul ) 2 2 · 7 71 71 (dul) 6 72 2 · 7 72 23 41 8 dul 2 2-3-5 5 10 d 2 ul u2 12 ulu2 2 2 · 3 61 141 dul 2-7 71 (dul ) 3 2-7 72 142 (d 2 u2ul) 2 2-3-5 151 151 (d 2 u2v.1) 14 2-3-5 15, 152 30 d 2 u2ul 2 · 3 · 5 151 (d 2 u2ul) 7 2-3-5 15, 302

11.4.3 Conjugacy classes of D

3P 1 21 22 1 1 41

42 5 21 2 22 22 72 71 8 10 41 14, 141 5 5 10 10

5P 1 21 2, 31 32 41 42 1 61 6, 64 63 72 71 8 21 12 14, 141 31 31 61 61

= (d, u 1 )

Class Representative I Centralizer! 1 1 27 • 3 · 7 (dul) 7 21 27 · 3 · 7 (d)3 26 • 3 22 25 (d 2 ul) 2 23 (d)2 3 22 · 3 25 (dul 2 ) 2 41 24 4, d 2 ulduldul 23 4, d 2 ul d 61 22 · 3 (d)5 22 · 3 62 6, d 3 uldul 2 22 · 3 (dul) 2 2-7 71 (dul ) 6 2 ·7 72 23 dul 2 8 dul 2-7 141 (dul) 3 2-7 142

2P 1 1 1 1 3 21 2, 2, 3 3 3 71

72 41 71 72

3P 1 21 22 23 1 41 42 4, 2, 22 21 72 71 8 14 141

7P 1 21 22 23 3 41

42 4, 61 62 6 1 1 8 21 21

7P 1 21 22 31 32 41 42 5 61 62 6, 64 1 1 8 10 12 21 21 152 151 30, 301

517

11. 5 Character tables of local subgroups 11.4.4 Conjugacy classes of G

=

(d,u 1 ,u2 ,x)

Class Representative I Centralizer! 1 1 2 7 • 36 • 53 . 7 . u (d)3 21 . 3 2 . 5 . 7 2 (ulu2) 4 31 23 · 36 • 5 22 . 35 (d)2 32 (ulu2) 3 4 25 · 3 (ulu2x) 6 51 2 · 3 · 53 52 5, dulx ( ul u2) 2 23 · 3 2 · 5 61 22 . 32 6, d (dul) 2 2-7 71 (dul ) 6 2·7 72 23 dul 2 8 33 dulxu2 2 9 33 (dulxu2 2 ) 2 92 (ulu2x) 3 10 2·3·5 11 ulxu2 111 (ulxu2) 2 11 112 22 . 3 12 ulu2 14, 2•7 dul (dul) 3 2·7 142 (ulu2x) 2 2·3·5 151 (ulu2x) 14 2·3·5 152 2.3.5 ulu2x 301 2.3.5 (ulu2x) 7 302

2P 1 1 31 32 2 51 5, 31 3, 71 72 4 9, 91 51 llo 111 6, 7, 72 151 152 151 152

3P 1 2 1 1 4 51 5, 2 2 72 71 8 3 31 10 11 112 4 14, 141 51 51 10 10

5P 1 2 31 32 4 1 1 61 6, 72 71 8 9, 91 2 ll 112 12 14, 141 31 31 61 61

7P llP 1 1 2 2 31 31 32 32 4 4 51 51 5, 5, 61 61 6, 6, 1 71 1 72 8 8 9, 9 92 91 10 10 llo 1 1 111 12 12 2 141 2 142 152 152 151 151 302 30, 301 301

11.5 Character tables of local subgroups 11.5.1 Character table of E = (e1, e2, e3, e4, e5) 2 3 5 7

7 1

5 1

2 2

2 2

i

3P 5P 7P 1 X.2 6 6 2 3 X.3 10 10 -2 1 1 X.4 10 10 -2 1 1 X.5 14 14 2 2 -1 X.6 14 14 2 -1 2 X.7 15 15 -1 . 3 X.8 15 -1 3 3 . X.9 21 21 1 . -3 X.10 35 35 -1 -1 -1 X.ll 45-3-3 . X.12 45-3-3 . X.13 90-6 6 . . X.14105-7-3 3 X.15120-8 .-3

where A= -½(1- iy7).

5

3

i

2 1

2 1

1

1

i

i

3

1

1

i

i

a a a 2b 7b 7a 8a 14b 14a 6b 7b 7a 8a 14b 14a 6b la la 8a 2a 2a 2 1 . -1 -1 -2 . 1 1 A -2 . . 1 1 .A 2 . -1 2 -1 2 . -1 -1 2 -1 -1 . . -1 1 -1 1 . -1 . 1 1 -1 1 . 1 -1 1 . -1 -1 A 1 1 . A 1 1 . -1 -2 . 1-1 .-1 . 1. 1

-1 -1

-1

A A A A

A A

1 -1 1 1 1 -1 -1 -1 . -1 . 1 .

A-1-A -A A-1-A -A

-1

1 1

1 .

1 .

. -1 -1

518

McLaughlin's group Mel

11.5.2 Character table of D

= (d, u1)

652543222 1 1 1 1 1

3

7 a a 7a Sa 14b 14a la Sa 2a 2a 1

X.2

3 3 X.4 6 X.5 7 X.6 7 X.7 7 X.8 8 X.9 8 X.10 8 X.11 8 X.1214 X .13 21 X.1421 X.15 24 X.16 24

X.3

3 3 -1 . 3 3 -1 . 6 6 2 . 7 7 -1 1 7 -1 -1 1 7 -1 3 1 -8 2 8 8 . -1 -8 . -1 -8 . -1 14-2 2-1 21 -3 1 . 21-3-3 -24 -24

-1 A A A A -1 A A A A -1 -1 2 . . . -1 -1 -1 -1 1 1 1 . -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 . -1 . -2 1 -1 -1 . -1 -1 -1 1 1 1 B B 1 1 -1 -1 -1 -1 B B 1 1 2-2 1 1-1 -3 1 -1 1 1 1 1 . -1 -1 -1 2 -1 3 -1

A A A A

. -A -A . -A -A

where A= -½(1- iv?) and B = -i)3. 11.5.3 Character table of H 27 32 5 1 7 1

763254 21221 1 1 1

3 2 1

=

(d,u1,u2) 2 2

2 1

2 1

3

2 1

2a 2b 2b 6b 6d 6c 6b 6c 6d X.2

X.3 X.4 X.5 X.6 X.7

X.8 X.9 X.10 X.11 X.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19 X.20 X.21 X.22 X.23

7 8 14 20 21 21 21 24 24 28 35 45 45 48 56 56 56 56 56 64 64 70

7 -8 14 20 21 21 21 -24 -24 28 35 45 45 -48 56 -56 -56 -56 -56 -64 64 70

-1 . 6 4 -3 -3 -3 . . -4 3 -3 -3 8 . .

-2

4 1 3 -1 -4 2 . -1 2 2 2 5 -1 4 6 1 -3 1 -3 1 1 -6 . -6 . 1 i 4 . 5 2 -5 -1 . -3 1 . -3 1 . 6 -4 -1 -4 -1 -4 -1 2 2 2 2 4 -2 . 4 -2 . -5 2 -2

2 4 1 -1 -1 -2 4 -2 i -1 -1 2 5 -1 1 -1 -1 . -1 i 6 1 -3 . -1 1 -3 . -1 -1 6 B B -1 6 B B -2 1 1 -1 -1 . -1 5 2 B B 1 B B 1 -2 -6 . -1 -1 1 -4 -1 -1 -1 1 4 1 C 1 4 1 C 1 -2 -2 1 -2 -2 -1 -4 2 -1 4 -2 . -5 1

where A= -½(1- ivl5), B

c

c

-1 -1 -1 -1 2 -1 -1 2 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 i -2 i 1 i 1 1 1 A A A A 1 1 A A A A . -B -B -1 -1 1 1 . -B -B -1 -1 1 1 -2 1 1 1 1 1 B B B B 2 -1 -1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 A A-A-A -1 A A-A-A -1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 -1

= -½(1 - iv?) and C = -iv'3.

12 Rudvalis' group Ru

In this chapter we give a new existence and uniqueness proof for the sporadic simple Rudvalis group Ru using Algorithm 1.3.15. This group was originally discovered by A. Rudvalis [115] as a rank 3 permutation group. The first existence proof for Ru was given by Conway and Wales in [20]. D. Parrott characterized the Rudvalis group by means of the group structure of the centralizer H = CRu (z) of a 2-central involution z in [106]. In the course of his uniqueness proof Parrott established a presentation for the Sylow 2-subgroups S of Ru. Using it M. Kratzer [79] determined a presentation of the centralizer H = Ca(z) of a 2central involution z of G = Ru. Applying then Algorithm 7.4.8 of [92] he gave a self-contained existence proof of a simple group G of Ru-type. He realized G as a simple subgroup of GL 133 (5) and calculated a faithful presentation and its concrete character table; see [79]. These results have inspired the following existence and uniqueness proof due to the author. In Section 12.1 Algorithm 1.3.15 is applied to an irreducible subgroup T ~ GL 3(2) of GL 8 (2). We construct an iterated extension E of T by its irreducible FT-modules W and V of dimensions 8 and 3 over F = GF(2), respectively. Then we determine the centralizer DE= C = E(z) of a 2-central involution z of E. Now Step 3 of Algorithm 1.3.15 is applied to build a group H with center Z(H) of order 2 having a special subgroup C of order 211 with center Z(C) ~ V such that DH= NH(C) ~ DE. In Section 12.2 it is shown that the free product P = H *D E of H and E with amalgamated subgroup D has an irreducible representation (8 in GL37s (17). It is a simple group, having a 2-central involution 3 such that C®(J) ~ H; see Theorem 12.2.1. In Section 12.3 it is shown that each simple group G of Ru-type has two conjugacy classes of involutions zG and sG. The structure of the centralizer Ca (s) of the non-2-central involution s is determined in 519

520

Rudvalis' group Ru

Proposition 12.3.4. Furthermore, the group order of any simple group of Ru-type is calculated, and all its conjugacy classes are classified. The group structure of the normalizers of the elements of odd prime orders and the fusion of their conjugacy classes in G are also determined. The uniqueness proof is given in Section 12.4 by means of these results and Theorem 7.5.1 of [92].

12.1 Construction of the 2-central involution centralizer In this section we apply Algorithm 1.3.15 to an irreducible subgroup T ~ GL 3(2) of GL 8 (2) and construct a presentation of a group H with center Z(H) of order 2, which finally will be proved to be isomorphic to the centralizer Ca (z) of a 2-central involution z of the simple sporadic group G ~ Ru of Rudvalis [115]. It is well known that GL 3 (2) has exactly three faithful irreducible representations over F = GF(2). Two of them are dual to each other and have dimension 3, and the third has dimension 8. It occurs as the unique non-trivial composition factor of the tensor product of the two nonisomorphic 3-dimensional irreducible GL 3(2)-modules over F. Let V and W be the natural F-vector spaces of dimensions 3 and 8, respectively. By means of Algorithm 1.3.15 we now construct the split extension E 1 of T by W, and then a suitable non-split extension E of E 1 by V.

Lemma 12.1.1 Let T be the subgroup of GLs(2) generated by the following matrices: 10010000) 11011000 00100100 ( 00010000 Mkl = 00011000 ' 00000100 00000010 00000111

00001000) 00000010 00000001 00100000 ( M k2 = 10001 0 0 0 · 01000000 00000100 00010000

of orders 2 and 3, respectively. Then the following assertions hold.

(a) The natural 8-dimensional vector space W over F is an irreducible FT -module. (b) Tis isomorphic to L = GL3(2) generated by the matrices Lkl

= ( 0110) 10 001

,

Lk2

= ( 001) 100 . 010

The isomorphism 'ljJ : T --; L is given by '1/J(Mki) i = 1,2.

Lki for

12.1 Construction of the 2-central involution centralizer

521

(c) Both groups T and L are isomorphic to K = (k1 , k 2 ) having the following set R(K) of defining relations:

kf

= k~ =

1,

(k1k; 1)7

= 1,

(k2k1k2 1k1) 4 = 1.

(d) The split extension E1 of T by W is isomorphic to the finitely presented group E1 = (ti, t2, t3) with the following set R(E1) of defining relations:

tr = t~ = t~ = 1, (f21t3t2t1) 2 = (t2t3t21t3) 2 = 1, (t1t3) 4 = 1, (t-:; 1t1t3t1t2t1) 2 = (t2t3t-:; 1t1t3t1) 2 = 1, t-:; 1t1t2t3t-:; 1t1t-:; 1t1t3t1t3t-:;_- 1t3 = 1, (t1t-:; 1)7 = 1, (t2t1t-:; 1t1) 4 = 1, t-:; 1t1 t2 t1 t2t1 t3 t1 t-:; 1t1 t-:; 1t1 t2t1 t2t1 t3 t1 t3 t-:; 1t1

=

1.

(e) The co homological dimension of E 1 with coefficients in the F E 1 module Vi is 3, and there is a non-split extension E of E 1 by Vi which is isomorphic to the finitely presented group E = (e 1, e 2, e3, e4, e5, e5) with the following set R(E) of defining relations: 2 3 2 2 2 21 e1 = e2 = e3 = e4 = e5 = e5 = ' (e1e 21)7 = 1, (e 21e3e2e1) 2 = 1, (e4,e5)

= =

(e4,e6)

= =

(e5,e6)

(e1e3) 4 = 1,

= (e1,e5) = = 1,

(e1,e5)

=

(e3, e4) (e3, e5) (e3, e6) -1 -1 -1 -1 e1 e4e1e4 e5 = e 2 e4e2e6-1 = 1 , -1 -1 -1 .-1 1 e2 e5e2e 4 = e 2 e6e2e5 = , e2e3e 2-1 e3e2e3e 2-1 e3e 4-1 = 1, (e 21e1e3e1e2e1) 2 = 1, (e2e1e21e1) 4 = 1, e2e3e 2-1 e1e3e1e2e3e2-1 e1e3e1e 5-1 e6-1 = 1, e 2-1 e1e2e3e 2-1 e1e 2-1 e1e3e1e3e 2-1 e3 = 1 , e2-1 e1e2e1e2e1e3e1e2-1 e1e 2-1 e1e2e1e2e1e3e1e3e2-1 e1

1,

= 1.

Furthermore, E has a faithful permutation representation of degree 2048 with stabilizer (e1, e 2). (£) z = (e1e3(e1e2e1e2e1e§) 2)8 is a 2-central involution of E and D = CE(z) = (x, y) is a subgroup of E of order 2 14 · 3, where

= e3e22e1e2e1e3e22e1e2 (e1e2e1e22)2 an d y = e3e22e1e2. Furthermore, E = (x, y, e), where e = e2 . x

Rudvalis' group Ru

522

(g) The Fitting subgroup C of E has order 211 • It is generated by c1 c5

= x4, c2 = (y)2, c3 = (x 2 y)4, c4 = (xyx)4, = (x 2 yx)3, c7 = (x2 y2 ) 2 and cg = (x 3 yx)2.

c5

=

(x 3 y)3,

(h) C is the unique maximal special normal subgroup of any Sylow 2-subgroup S of E with center Z(C) ~ V and C/Z(C) ~ W. (i) A system of representatives of the 35 conjugacy classes of E = (x,y,e) is given in Table 12.5.3. (j) A system of representatives of the 52 conjugacy classes of D = (x,y) is given in Table 12.5.2. (k) The character table of E is given in Table 12.6.2.

(1) The character table of D is given in Table 12.6.3. Proof The first three assertions are easily verified using MAGMA. (d) Since E 1 is a split extension of the irreducible subgroup T of GL 8 (2) by W ~ F 8 one can apply the MAGMA commands Q: =Getvecs (T) and E_ 1 : =Semidir (T, Q) . They yield the presentation of the split extension E 1 of T by W stated in the assertion. (e) Using the data of (b) and (d) an application of Holt's algorithm [54] in MAGMA shows that the second cohomological dimension of E 1 in V = F 3 over F is 3. Hence there are seven non-split extensions of R by V. For each non-trivial cocycle MAGMA provides a presentation of a non-split extension E of E 1 by V. One of these seven non-split extensions E has the presentation given in the statement. The six other extension groups are not isomorphic to E. From (c) and the presentation of E it follows immediately that (e 1 , e2 ) L is a stabilizer of a faithful permutation representation PE of E of degree 211 . (f) All statements of (f) can be verified by means of PE and MAGMA. (g) The group Eis a split extension of L ~ GL 3 (2) by a special normal subgroup C = CE(V). The author found its generators c;, 1 Si S 8, by means of PE, MAGMA and the program GetShortGens(E,C). ~

(h) Since L ~ GL3 (2) is a complement of C in E, the subgroup C is a maximal special normal subgroup of E. Using PE and the MAGMA command Subgroup(S: Al:=Normal, IsElementaryAbelian := true) one checks that C is the unique maximal special normal subgroup of order 211 in any Sylow 2-subgroup S of E.

12.1 Construction of the 2-central involution centralizer

523

(i) and (j) The systems of representatives of the conjugacy classes of E and D have been calculated by means of PE, Kratzer's Algorithm 5.3.18 of [92] and MAGMA. (k) and (1) The character tables of D and E have been computed by means of the permutation representation of PE described in (e) and MAGMA. D

Proposition 12.1.2 Let E = (e1,e2,e3,e4,e5,e5) = (x,y,e) be the finitely presented group constructed in Lemma 12.1.1 ( e). Then the following assertions hold.

= (x, y) has a center Z(D) of order 2 which is generated by = x8 • (b) D = (x, y) has exactly one normal subgroup N of order 211 with center Z(N) = (z) such that IAut(N)I = 219 · 3 · 5. (a) D Z

N is generated by n1 = x 4 , n2 = (x 2y) 2 , n3 (xy2)2' n5 = (yx2)2 and n5 = (y2x)2.

=

(xyx) 2 , n4

=

N has a complement K = U1, j2) ~ 84 in D, where j1 (y3xy2x2yx)4, j 2 =x2yx2y2x2yx2y.

=

(c) The normal subgroup I of inner automorphisms of N has a complement T in Aut(N) such that its Fitting subgroup F(T) has a complement R in T isomorphic to 85 . The subgroup IR has order 213 · 3 · 5. It is isomorphic to the group H1 = (Y1, Y2, y3, Y4, y5, Y6, Y1, Ys) having the following set R(H1) of defining relations:

2 2 2 4 4 4 2 4 1 ½=~=~=¼=~=~=~=~=' (y1y2) 2 = 1, (Y1Y3) 2 = 1, (Y2Y5 1)2 = 1, Y5 1Ys 1ir:1Ys = 1, (yd;;1Y2) 2 = 1, (Y1Yi 1Y3) 2 = 1, -1 -1 1 Y2Y4 Y3Y2Y3Y4 = Y2Y4Y3Y2Y3Y4 = , (Y2Y6 1Y3) 2 = 1, (Y2Y6Y3) 2 = 1, -1 -1 2 -1 2 1 Y4 Y1Y4 Y1Y5 = Y4Y1Y4 Y1Y5 = , -1 -1 -1 -1 -1 1 Y1Y1Y4 Y2Y1Y4 = Y5 Y2Y4 Y2Y5Y4 = , (Y1Y5 1Yi 1) 2 = 1, (Y2Y6 1Yi 1)2 = 1,

Rudvalis' group Ru

524

-1 -1 -1 -1 1 Y4 Y1Y4 Y1Y1Y2 = Y6Ys Y4 Y2YsY4 = , -1 -1 -1 -1 1 Y1Y1Y5 Y1Y1Y5 = Y6 Y3Y5 Y3Y6Y5 = , -1 -1 -1 -1 1 Ys Y2Y6 Y5Y8Y2 = Y1Y5 Y6 Y1Y1Y6 = , Yi: 1Y3Y2YlYBU°i 1Ys 1 = Y5 1Y4Y3YlYs 1 Yi: 1Y8 = 1, -1 -1 -1 -1 -1 1 Y1Y5 Y3Y1Y5 Y7Y3 = Y4 Ys Y4 Y1YsY3Y2 = , (Ys 1Y1 ) 5 = 1, (Y1Ys 1Y1Ys) 3 = 1.

(d) There is a central extension H of H1 with center Z(H) of order 2 having a Sylow 2-subgroup S containing a unique maximal special normal subgroup C of order 211 with elementary abelian center Z(C) of order 8 such that NH(C) ~ D. The group H = (hi 11 :::; i :::; 9) has the following set R(H) of defining relations: hi= h~ (h1,h9)

= h~ = h! =ht= ht= h~ = h: = h~ = 1, = (h2,h9) = (h3,h9) = (h4,h9) = (h5,h9) = (h6,h9) = 1, = (hs,h9) = 1, h~h9 1 = 1, (h1h2) 2 = (h1h3) 2 = 1,

(h1,h9) 1 - h- 1h- 1h- 1h h-l - 1 h- 1 hh 25 h -lh25 95 8 6 89-, 1 2 (h1h5 h2) = (h1h4 1h3) 2 = (h2h6h3) 2 = 1,

h2h4 1h3h2h3h4h9 1 = h2h4h3h2h3h4 1 = 1, h2h5 1h3h2h5 1h3h9 1 = h4 1h1h4 1h1h~hg 1 = 1, h4h1h4 1h1h~ = h1h1h4 1h2h1h4 1hg 1 = 1, h"i;1h2h4 1h2h5h4 1 = 1, (h1h5 1h-.; 1)2 = 1, 1 - h- 1h h- 1h h h h- 1 - l h- 1hh 26 h -lh4 1h 26 4 1h94 74 1 7 2 9 - ,

= h1h1hi: 1h1h1h·i;1h9 1 = 1, = hs 1h2h5 1h5hsh2h9 1 = 1, h1h5 1h 61h1 h1h6 = h 5 1h3h2h1hsh4 1h81 = 1, h5 1h4h3h1hs 1h5 1hshg 1 = h1h5 1h3h1h5 1h1h3 = h 4 1h 81h 4 1h1hsh3h2 = 1, (h"i 1h1 ) 5 = 1, (h1h"i 1h1hs) 3 = l.

h6hs 1h-.; 1h2hsh4 h5 1h3h5 1h3h6h5

1,

Furthermore, H has a faithful permutation representation of degree 2048 with stabilizer (h1, h 8 ). (e) A system of representatives of the 49 conjugacy classes of H = (x, y, h) is given in Table 12.5.1. (f) The character table of H is given in Table 12.6.1.

12.1 Construction of the 2-central involution centralizer

525

Proof (a) Using the faithful permutation representation PE of Lemma 12.1.l(e) and MAGMA one verifies that z = (e1e 3 (e 1e 2e 1e 2e1e§)2)8 = x 4 generates the center Z(D) of D = (x, y). (b) D has three normal subgroups of order 211 . This can be checked by application of the MAGMA command Subgroups(PD: Al=Normal, OrderEqual:= 2048);

C of Lemma 12.1.l(g) is one of them. The others have a cyclic center of order 2 generated by z. Another MAGMA calculation shows that only the normal subgroup N with generators given in the statement has an automorphism whose odd order part is larger than the one of IDI. The complement of Nin D has also be found by means of MAGMA. (c) Using the faithful permutation representation PE and the commands AN : = AutomorphisGroup (N) and PAN: =PermutationGroup (AN) MAGMA establishes that PAN is a permutation group acting on a set of cardinality 240 and that IAI = 219 · 3 · 5. Furthermore, it gives 11 generators of the subgroup of inner automorphisms I of N in AN and an isomorphism h : AN ----+ PAN. Let PI = (h(I)). Then PI is a normal subgroup of PA of order 210 . Using the MAGMA command PK: =HasComplement (PAN, PI) one sees that I has a complement K in AN. Furthermore, MAGMA yields that the Fitting subgroup J of K has order 26 , and K is a split extension of a subgroup W c,,, S 4 by J. In particular, IW is a subgroup of AN of order 213 · 3 · 5. Another application of MAGMA yields that the Frattini subgroup of I has order 24 . Hence it is now easy to find six generators for I and two for W such that IW is generated by eight elements. Applying then the MAGMA command FH1: =FPGroup (P [I\r.J]) the author obtained the presentation of the group H 1 given in the statement. (d) H 1 has a faithful permutation representation P H 1 of degree 1024 with stabilizer (y7 ,y8 ) c,,, S5 • Let MGL1_2:=GL(1,2), M1:=MGL1_2! [1] and FEalg:=MatrixAlgebra. Then CM: =GModule (PH_1, FEalg) denotes the cohomology module. Applying the command Cohomologica1Dimension(PH_1, CM, 2) MAGMA establishes that the second cohomological dimension is 4. Thus there are 15 non-split 2-central extensions H of H 1 . The command P : =ExtensionProcess (PH_1, CM,H_1) provides a presentation for each of them. Only the finitely presented group E stated in the assertion has a special normal subgroup C of order 211 with center Z(C) c,,, V such that Ns(C) c,,, D.

526

Rudvalis' group Ru

Furthermore, H = (h; 11 : : ; i ::::;; 9) has a faithful permutation representation PH of degree 2048 with stabilizer (h 7 , h 8 ) ~ 8 5 . (e) The system ofrepresentatives of the 49 conjugacy classes of H has been calculated by means of PH, Kratzer's Algorithm 5.3.18 of [92] and MAGMA. (f) The character table of H has been computed by means of the permutation representation of PH and MAGMA. D Lemma 12.1.3 Let H = (h; I 1 ::::;; i ::::;; 9) be the group constructed in Proposition 12.1.2(d). Let d1 = h7, d2 = (h1hs)4, d3 = (h1hsh1h§)2, h = hs, x1 = (d1d~d2d1d3d2d3d2d1) 5 andy1 = (d3d1d2d1d3d2d3d2d1d3d2

d3) 3. Lets1 =x1, S2 =yr, S3 = (xry1) 2 andt= (xrur) 4. Let C1 = Xf, C2 = Yi, C3 = (XiY1)4, C4 = (x1y1x1)4, C5 = (xfo1)3, C6 = (XiY1X1) 3, C7 = (xrur) 2 and Cs = (xfo1x1) 2· Let E = (x, y, e) be the finitely presented group constructed in Lemma 12.1.l(e). Then the following assertions hold.

(a) H = (x 1 , y1 , h) has a faithful permutation representation of degree 211 with stabilizer (h, (Y1XiYiX1) 4). (b) S = (s1,s2,s3) is a Sylow 2-subgroup of H. (c) C = (e; I 1 ::::;; i ::::;; 8) is the unique maximal special normal subgroup of S and IOI = 211 . (d) z = x 8 is a 2-central involution of E such that DE= CE(z) = (x, y). Furthermore, there is an isomorphism 'ljJ : DE -; DH = NH(C) such that 'l/;(x) = x1 and 'lj;(y) = Yl · (e) The center Z(C) of C is the unique elementary abelian normal subgroup A = ((c1) 2, (c1c2) 2, (c2c3) 2) of order 8 in S. Furthermore, CH(A) = C. (f) V = (z, t) is the unique normal Klein 4-subgroup of S, where t = xHxtyf)2. Furthermore, C2 = CH(V) has order 213 and its center Z(C2) = V. (g) The automorphism group Aut(C2) of 0 2 has order 219 · 3. (h) The derived subgroup of 0 2 has order 29 and its center Z(q) = A. (i) C has a complement K = (k1, k2) ~ 84 in D = NH (C), where k1 = (Y1XiYfX1) 4 and k2 = Y1XfYiXrY1Xi, (j) The Goldschmidt index of the amalgam H +-- D -; E is l.

12.2 Construction of a simple group of Ru-type

527

Proof All statements (a) through (i) are easily checked using MAGMA and the faithful permutation representation PH of H given in Proposition 12.1.2(d). (m) The Goldschmidt index has been calculated by means of Kratzer's Algorithm 7.1.10 of [92]. D Definition 12.1.4 A finite simple group G is called to be of Ru-type if it possesses a 2-central involution z such that Ca(z) ~ H, where His the finitely presented group of Proposition 12.1.2.

12.2 Construction of a simple group of Ru-type

In this section the conditions of Steps 1 and 2 of Algorithm 1.3.15 are verified for the amalgam H - D ---+ E constructed in Section 12.1. Therefore its Step 3 can be applied to give here a self-contained existence proof for a simple group of Ru-type.

Theorem 12.2.1 (Michler) Keep the notation of Lemma 12.1.1 and Proposition 12.1.2. Using the notation of the three character Tables 12.6.1, 12.6.3 and 12.6.2 of the groups H, D and E, respectively, the following statements hold.

(a) There are exactly two dual compatible pairs (x, T) E mf charc(H)x mf chanc(E) of degree 378 of the groups H = (D, h) and E = (D,e):

(x, T) = (x4s

+ X44 + X31 + x10, T24 + Tas)

with common restriction TID = XID = 1Pl9

+ 1P32 + 1P4l + 1P46 + 'lp49 + 1P51 + 'lp52,

and (x, T)

=

(x4s

+ X44 + X31 + x11, T23 + Tas)

with common restriction TID

= XID = 'l/J20 + 'lp32 + 'lp41 + 1P46 + 'lp49 + 'lp51 + 'lp52,

where irreducible characters with bold face indices denote faithful irreducible characters.

Rudvalis' group Ru

528

(b) Let QJ and W be the up to isomorphism uniquely determined faithful semi-simple multiplicity-free 378-dimensional modules of H and E over F = GF(l 7) corresponding to the first compatible pair X, T, respectively. Let Km : H -----t GL37s (17) and "'® : E -----t GL37s (17) be the representations of H and E afforded by the modules QJ and W, respectively. Let b = Km(h), ~ = Km(x) and tJ = Km(Y) in Km(H) s:; GL37s (17). Then the following assertions hold. (1) Q'.JID ~ WID, and there is a transformation matrix T E GL37s (17) such that

Let e = 7-i"'®(e)T E GL37s(l7). QJ = (b, ~. tJ, e) has a faithful permutation representation of degree 424125 with stabilizer Q: = (~, tJ, e). (3) The four generating matrices of Q5 are stored in MAGMA format on the accompanying DVD. (4) The character table of Q5 is stated as Table 12.6.8. It coincides with that of Ru in the Atlas /19}, p. 127. (5) Qj has 36 conjugacy classes. A system of their representatives and their centralizer orders is given in Table 12. 5. 5.

(2)

(c)

is a finite simple group with 2-central involution Km(z) such that

Qj

= (~) 8

Proof (a) The character tables of the groups H = (x1,Y1,h), DH= (x 1, y1), DE = (x, y) and E = (x, y, e) are Tables 12.5.1, 12.5.2, and 12.5.1, respectively. In the following we use their notations. Using MAGMA and these character tables and the fusion of the classes of DH in Hand DE in Ethe two compatible pairs stated in assertion (a) have been calculated by means of MAGMA and Kratzer's Algorithm 7.3.10 of [92]. Since they are conjugate complex it suffices to construct the semisimple representations QJ of x E Irrc(H) and W of TE Irrc(H) of the first compatible pair(x,T) over their common splitting field GF(17). (b) We first construct the semi-simple faithful representation W corresponding to the character x = T24 +T48 . We use the faithful permutation

12.2 Construction of a simple group of Ru-type

529

representation PE of E of degree 2048 constructed in Lemma 12.1.l(e). Using PE and the MAGMA command LowindE2 := LowindexSubgroups(PE, ) the author determined all conjugacy classes of subgroups U of E having index 100 :s; k s; 200. For each representative U MAGMA knows generators of each of these permutation subgroups U of E. Furthermore, MAGMA calculates the character table and the fusion of the conjugacy classes of U in E. Thus one can calculate all the inner products ( (lu )E, r 24). If k is the smallest index for which this inner product is nonzero then we apply the program GetShortGens (E, U). For T24 we found that k = 168, and U = U24 = ((xe)7, (ex)7, (xexe 2)2, (xexexe 2x) 2). Applying then Kim's implementation of Algorithm 1.5.1 to the permutation representation (lu2 4)E over the splitting field GF(l 7) the author obtained the three generating matrices M E42x, M E42y, M E42e of the corresponding irreducible representation of E = (x, y, e) in GL42 (17). Using PE and the MAGMA command LowindE3 := LowindexSubgroups(PE, ) the author determined all conjugacy classes of subgroups W of E having index 600 :s; k s; 700. We calculated again the inner products ((lw )E, r 35 ) for representatives W of conjugacy classes of subgroups of E with index k in the given interval. For k = 672 MAGMA established that the permutation character of the subgroup W35 = ((ex 2e2x ) 3 , e2x 3 e2x 2,x2e2xex 3 e2) has a non-trivial inner product with T3 5 . Applying now Algorithm 1.5.1 to the permutation representation (lw35 )E over the splitting field GF(l 7) the author obtained the three generating matrices M E336x, M E336y, M E336e of the corresponding irreducible representation of E = (x, y, e) in GL 33 6 (17). These matrices can be reconstructed from the four generating matrices of the simple group Q5 stored on the accompanying DVD and the results described in the following. The semi-simple representation fill corresponding to the semi-simple character T of E in GL 378 (17) is described by the following three matrices:

M Ex

:= DiagonalJ oin (ME42_x, ME336_x),

M Ey := Diagona1Join(ME42_y, ME336_y), MEe := Diagona1Join(ME42_e, ME336_e).

530

Rudvalis' group Ru

Similarly one shows that the four irreducible characters X10, X31, X44, of H = (x1, Y1, h) with respective degrees 6, 60, 120, 192 are irreducible constituents of the permutation characters (lv, 0 )H, (lv31 )H, (lViJH, (lVis )H of the following subgroups:

x48

Vio = (xf, x1 h, hx1) of index 24, V:n = ((y1h) 5 , (xr yi)2, hx1y1x1h, (hx1y1hy1)3) of index 120, ¼4 = ((hy1x1) 4 , (h 2y1) 3 , (h 2y1x1) 5 , (x1h 2 xf ) 2) of index 240, ¼s = (Yi Xi hy1, (X1 Yr hxi ) 3) of index 2048 in the group H, respectively. Applying Algorithm 1.5.1 to the permutation representations (lv, 0 )H, (lVa, )H, (lv4 JH, (lVis )H over the splitting field GF(l 7) the author obtained the following four of triples-generating matrices: MH6x1, MH6y1, MEH6h in GL5(17), MH60x1, MH60y1, MH60h in GL50(17), MH120x1, MH120y1, MH120h in GL120(17), MH192x1, MH192y1, MH192h in GL192(17),

of the irreducible representations of H = (x 1, y1, h) corresponding to the irreducible characters X10, X31, X44, X4B, respectively. These matrices can be reconstructed from the four generating matrices of the simple group Q5 stored on the accompanying DVD and the results described in the following. The semi-simple representation QJ corresponding to the semi-simple character x of H in GL37 8 (17) is described by the following three matrices: M Hx1 := Diagona1Join(MH6_x1, MH60_x1, MH120_x1, MH192_x1), MHy 1 :=Diagona1Join(MH6_y1, MH60_y1, MH120_y1, MH192_y1), MHh :=Diagona1Join(MH6_h, MH60_h, MH120_h, MH192_h).

The matrix groups 'JJH = (QJ(x 1),QJ(yi)) and 'JJE = (filJ(x),filJ(y)) are isomorphic by Proposition 12.1.2(d). So are the semi-simple representations ml:DH and filJl:DE by the definition of the compatible pair (X,T) E mfcharc(H) x mfchanc(E) and Maschke's Theorem, which is applicable because the orders of H and E are coprime to 17. Let Y = GL(278, 17) = GL 378 (17). By (a) both FD-modules QJ and filJ are a direct sum of pairwise non-isomorphic irreducible FD-modules which can be ordered simultaneously such that the dimensions of the components are 6, 12, 24, 48, 64, 96 and 128. In the following we assume that

12.3 Fusion

531

we have chosen bases in ~ and W such that the above matrix generators M Hxi, M Hy1 and M Ex, M Ey are built with respect to these bases. Applying then Parker's isomorphism test of Proposition 6.1.6 of [92] by means of the MAGMA command Isisomorphic(GModule(sub),GModule(sub))

we obtain the transformation matrix Ti satisfying ~(x) = (W(xi) )'Ti and ~(Y1) = (W(y))'Ti. Let e1 = (W(e))'Ti. Then le = (i:, IJ, e1 ) ~ E. By Corollary 7.2.4 of [92] the transformation matrix T of the statement is the product matrix V'Ti_, where

V = diag(l 6 ,1 12 ,1 24 ,1 48 , 166 4, 196 ,1 2 4, 2128 ) E GL 378 (17) and en denotes the n x n diagonal matrix with unique diagonal entry c E

GF(l 7).

Let e = (e 1 )v. Let 18 = (fJ, i:, IJ, e). Using the MAGMA commands RandomSchreier(G) and FPGroupStrong(G) the author obtained a finitely presented group G with nine generators r; and a very long list of relations (not restated in this book). The first four generators r; correspond to i:, IJ, fJ and e. Therefore we applied the program

MyCosetAction(G, E:maxsize:=100000000), where E = (r1 ,r2 ,r4 ). Thus we obtained a faithful permutation representation PG of 18 of degree 424125 with stabilizer le. In particular (8 = 210 · 37 · 53 · 7 · 11 · 23. The character table of (8 has been calculated by means of the above permutation representation and MAGMA; see Table 12.6.8. It coincides with the one of Ru in [19], p. 127. The system of representatives of the 36 conjugacy classes of G has been calculated by means of PG, Kratzer's Algorithm 5.3.18 of [92] and MAGMA; see Table 12.5.5. (c) Let 3 = (i:) 4 . Then C® (3) contains Sj = (fJ, i;, IJ). Table 12.5.5 implies that (8 is a simple group and that C® (3) ~ Sj. This completes the proof.

D 12.3 Fusion

Throughout this section G denotes a finite simple group of Ru-type, and H = (x 1 , y1 , h) denotes the group constructed in Proposition 12.1.2 by generators and relations. In this section it is proved that G has exactly

Rudvalis' group Ru

532

two conjugacy classes of involutions, and that they are represented by z = xf and s = hx1 h 2. Furthermore, it is shown that Ca (s) ~ 22 x Sz(8) and that IGI = 214 · 33 · 53 · 7 · 13 · 29. By means of these results it is easy to determine representatives for all conjugacy classes of G consisting of elements of even order. Let X be a group. Then Z(X) denotes its center, and its second center Z 2(X) is defined by Z 2 (X)/Z(X) = Z(X/Z(X)); f2 1(X) is the subgroup of X generated by all elements of prime order in X.

Xi

Lemma 12.3.1 Let z be a 2-central involution of the simple group G of Ru-type with Ca(z) = H = (x1,Y1,h) constructed in Lemma 12.1.3. Let Z = xL t = xf(xtyr)2, V = (yr(xry1) 4 ) 2 in H. Let V = (z,t) and A= (z,t,v). Let E = ( e1 , e2, e3, e4, e5, e6) = (x, y, e) be the finitely presented group constructed in Lemma 12.1.1 ( e). Then the fallowing statements hold.

(a) (b) (c) (d)

Na(V)/Ca(V) = Na(V)/CH(V) ~ GL2(2) ~ 83. t EH is conjugate to z in G. Na(A)/Ca(A) = Na(A)/C ~ GL3(2). The inverse µ of the isomorphism 'ljJ : DE = (x, y) -----+ DH = (x1, Y1) constructed in Lemma 12.1.3{d) extends to an isomorphism v: Na (A) -----+ E.

Proof (a) and (b) By Table 12.5.1 tis a representative of a conjugacy class of involutions of H such that ICH(t)I = 213 . By Lemma 12.l.3(f) V = (z, t) is the unique normal Klein 4-subgroup of the fixed Sylow 2-subgroup S of H, and V = Z(CH(V)). Furthermore, ISi = 214 and IS: Cs(t)I = 2. Since G is a simple group, Glauberman's Theorem 4.7.3 of [92] asserts that that z 9 = t for some g E G - H. Thus S 9 is a Sylow 2-subgroup of K = Ca(t). Since X = Cs(t) has order 213 Sylow's Theorem implies that X ~ S 9 k for some k E K. Hence X = Cs (t) is properly contained in a Sylow 2-subgroup of Ca(t). Therefore both (a) and (b) hold by Lemma 1.4.4 of [92]. (c) The subgroup A= (z, t, v) = Z(q) is a characteristic subgroup of 0 2 = CH(V) by Lemma 12.l.3(h). Hence A is normal in Na(V) because Ca(V) = CH(V). By (b) there is an element d of order 3 in Na(V) which is not in NH(V). In particular, d E Na(A) - NH(A). Lemma 12.1.3(i) asserts that C = Ca(A) = CH(A) has a complement K ~ 84 in NH(A). As INa(A) : NH(A)I is odd, Theorem 1.4.15 of [92] implies

12.3 Fusion

533

that Ne(A) splits over Caswell. Thus Ne(A) is a semidirect product of C and K1 = (K, r) ~ GL3 (2) because 84 is a maximal subgroup of GL3(2). (d) Let E1 = Ne(A). Then CE 1 (z) = Ns(A) = (x1,Y1) =DH.Let DE = (x, y). Then µ(x 1) = x and µ(yi) define an isomorphism between DH and DE by Lemma 12.1.3(d). Furthermore, C = Ce(A) = Cs(A) = 'lj;(Ci), where C1 = CE(A) = (e;, 11::::::; i::::::; 8) and where thee;, are defined in Lemma 12.1.l(g). Using the faithful permutation representation PH of H defined in Proposition 12.l.2(d) it has been checked that K1 = (k1, k2) ~ 84 is a complement of C in DH, where k1 = (Y1XiYiX1) 4 and k2 = Y1X 4 yrxfy1xr. Hence µ(K1) = ((yx 2y2x)4, yx 4 y 2 x 3 yx 2 ) is a complement of 0 1 in DE. It is a subgroup of the complement L = (µ(K1), g) ~ GL3(2) of C1 in E = (x, y, e), where g = (x 4 e2x 6)6 E E has order 7, as has been checked by means of the faithful permutation representation PE of E defined in Lemma 12.1.l(e) and MAGMA. In particular, Ei/01 ~ GL3(2) operates irreducibly on its 8-dimensional module Ci/A~ W over GF(2) by (c) and Lemma 12.1.1. Now another application of Lemma 12.1.1 yields that E 1 = Ne (A) ~ E because up to isomorphism E is the unique iterated extension of GL 3(2) by its 8- and 3-dimensional modules such that DH~ DE. This completes the proof. D Proposition 12.3.2 Let G be a finite simple group of Ru-type with a 2-central involution z such that H = Ce(z). Keep the notation of Table 12. 5.1. Then the following statements hold.

(a) G has two conjugacy classes of involutions represented by z = x~ and s

= Xi hx1 h 2

or 2A and 2B, respectively.

(b) G has 19 z-special conjugacy classes represented by the conjugacy classes 21, 41, 42, 43, 46, 61, 81, 82, 86, 10, 121, 122, 161, 162, 201, 202, 203, 241, 242 of H. Proof (a) By Lemma 12.3.l(d) each simple group G has an elementary abelian subgroup A of order 8 in H = (x1, Y1, h) such that E = Ne (A) = (x,y,e) contains a 2-central involution z such that DE = CE(z) = Ns(A) and IH: DI is odd. By Lemma 12.l.3(d) there is an isomorphism 'lj; : DE----+ DH= Ns(A) such that 'lj;(x) = X1 and 'lj;(y) = Yl· It has been used to identify the conjugacy classes of DE and DH and their inclusions into E and H, respectively.

Rudvalis' group Ru

534

Tables 12.5.1, 12.5.2 and 12.5.3 provide systems of representatives of the conjugacy classes of H, D and E, respectively. Let d = x 6 y 2 xy 2 xy 2 E fl, S = xr hx1 h 2 E H and S1 = ( xe) 7 E E. Using the faithful permutation representations PE and PH of E and H defined in Lemma 12.1.l(e) and Proposition 12.1.2(d), respectively, and the MAGMA commands IsConjugate(PE,d,s_1) and IsConjugate(PH, d,s) the reader can check that d and s 1 are E-conjugate and d ands are H-conjugate. Hence sand s 1 are G-conjugate. By Table 12.5.3 [CE(s 1 )[ = 28 · 7. As [H = Ca (z) I = 2 14 · 3 · 7 the involutions z = x 8 and s are not conjugate in G. The same method can be applied to the representatives of the other seven conjuagcy classes of involutions of D listed in Table 12.5.2. Thus one obtains the following fusion pattern using the notation of Tables 12.5.1 and 12.5.3, respectively. Here the classes of H and E are denoted by the numbers of the representatives of the involutions. The classes of H are stated before the ones of D. The last entries denote the classes of E. 2A := ([1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 6, 8], [1, 2, 4]),

2B

:=

([6], [7], [3]).

In particular, z ands represent all conjugacy classes of involutions of G. (b) This statement follows immediately from the power map information given in Table 12.5.1. This completes the proof. D

Lemma 12.3.3 Let G be a finite simple group of Ru-type with a 2central involution z such that H = Ca ( z) = (x 1, Y1, h). Then the following assertions hold.

(a) n1 = X1Y1 generates a Sylow 3-subgroup T of H. (b) CH(T) = (n1,n2,n3) = T x S, where S = (n2,n3) is semidihedral of order 16 and n2 = (h 2 y1x1) 5 and n3 = (xrhxryr) 3 have respective orders 4 and 2. (c) NH(T) = (CH(T), n4), where n4 = (x 1hx 1hy 1hx1) 3 has order 8. Furthermore, Q = (n 2 , n~) is a quaternion subgroup of S. (d) n~ = nj = px~ = z and n 2 and n~ are not conjugate in G. (e) Ca(T)/T ~ PGL2(9). (f) Na(T)/T ~ Aut(A5). (g) [Ca(s)/ is divisible by 5, wheres= xrhx 1h 2 is a representative of the unique class of non-2-central involutions of G. Proof (a) holds by Table 12.5.1. (b) Let PH be the faithful permutation representation of H defined in Proposition 12.1.2(d). Using it and MAGMA the reader can check

12.3 Fusion

535

that CH(T) = (n1,n2,n3) = T x S, where S = (n2,n3) = (v,n3), where v = n2n3 has order 8. As vn 3 = v 3 the Sylow 2-subgroup S of CH (T) is semi-dihedral of order 16. (c) and (d) These assertions can easily be checked by means of PH and MAGMA. In particular, Q = (n 2 ,n~) is a quaternion subgroup of S with center z = n~ = n!. The z-special elements n 2 and n~ of order 4 are not H-conjugate. Thus they are also not G-conjugate by Lemma 2.2.6 of [92]. (e) The semi-dihedral group S is also a Sylow 2-subgroup of R = Cc (n1) = Cc (T) because z generates the center of S. Clearly, Vi = (z, n3) is a Klein 4-group of S. Moreover, CR(Vi) = CH(Vi) nR = T x Vi and CR (Q)Q = T x Q by (b) and Lemma 1.3.9 of [92]. Since R has two conjugacy classes of elements of order 4 by (d), Proposition 1.6.5 of [92] implies that R = Cc (T) has a normal subgroup K of index 2 with a dihedral Sylow 2-subgroup S1 such that K does not have a normal subgroup of index 2. Furthermore, R has a unique conjugacy class of involutions. Hence Theorem 1.6.4 of [92] due to Gorenstein and Walter implies that K/T ~ PSL(2, 9) ~ A 6 . Therefore R/T = Cc(T)/T ~ PGL2(9). (f) By (b) and (c) we know that INH(T)/CH(T)I = 2. Let R 1 = Nc(T). Then R is normal in R1 and IR1: RI= 2. Hence R1 = RNH(T). Thus NH (T) /T is a non-trivial subgroup of Ri/T ::; Aut( Cc (T) /T) ~ Aut(A6 ). As IAut(A 6 )1 = 1440 it follows that Ri/T = Nc(T)/T ~ Aut(A5). (g) Now (f) implies that Ne (T) contains an element f of order 5 which is centralized by n 1. But f is also centralized by an involution s 1 E Ne (T) which inverts n 1; see (f) and the Atlas [19], p. 5. By Table 12.5.1 the element Ji = (y1h) 4 of order 5 has a centralizer CH (Ji) of order 23 · 5. Using PH and MAGMA the reader can check that a Sylow 2-subgroup S2 of CH (!1) is a quaternion group of order 8. Now Theorem 4.6.5 of [92] asserts that L = Cc(f1 ) = OCL(z), where O is the largest normal subgroup of L = Cc (f1) of odd order. Operating with the Klein 4-group W generated by z and w = (x 1h) 2 on O an application of Theorem 4.2.2 of [92] yields that IOI divides 53 because z, w and zw are conjugate in G by Proposition 12.3.2 and Table 12.5.l. Thus Cc (!1) is a {2, 5}-group. In particular, f and Ji are not G-conjugate. Since H Cc ( z) has a unique conjugacy class of elements of order 5 the involution s 1 E Ne (T) is not G-conjugate to z. Thus s 1 is conjugate to s = xrhx1h 2 by Proposition 12.3.2 and 5 divides ICc(s)I. This completes the ~~

D

Rudvalis' group Ru

536

Proposition 12.3.4 Let G be a finite simple group of Ru-type with a 2-central involution z such that Cc(z) = H = (x1,Y1,h). Keep the notation of Proposition 12.3.2. Then the following assertions hold.

(a) s = Xihx 1h 2 of G represents the only non-2-central conjugacy class of involutions of G. It is G-conjugate to b = (xe) 7 E E = (x,y,e). (b) P = CE(b) = (P1,P2), where Pl = xe and P2 = (ex 2ex) 2. The center R c

= Z(P) of P is a Klein 4-group generated by b and

= (P1P2)7 ·

(c) P =Rx J, where J of order 26 · 7 is generated by j1 = Pi and j2 = (P1P2) 2· The Fitting subgroup F J of J has order 26 • Its center Z(F J) and F J /Z(F J) are both elementary abelian of order 8.

(d) N = NE(R) = (Y1, Y2), where Y1 = Pl and Y2 = (x 2e2x 3 e) 2 have orders 14 and 6, respectively. Furthermore, /N/ = 26 • 3 · 7. (e) U = Cc(s) "" Rx Y, where R = (b, c) is a Klein 4-group and Y"" Sz(8), the simple Suzuki group of order /Y/ = 26 · 5 · 7 · 13. (f) M = Ne (R) is isomorphic to the finitely presented group M = (d1,d2) with the following set R(M) of defining relations: di

= d3 = l,

(d~d1) 12 = 1,

(d2d1d~d1d~d1) 6

= 1,

d2d1d~d1d2d1d2d1d2d1d~d1d2d1d2d1d~d1d~d1d~d1d~d1d2d1 ·d~d1d2d1d2d1d2d1d2d1d~d1d~d1d2d1d~d1d~d1d~d1

= 1,

d~d1d~d1d2d1d~d1d2d1d2d1d~d1d~d1d2d1d~d1d~d1d~d1d~(d1d2) 2 ·d1d~d1d2d1d~d1d~d1d2d1d2d1d2d1d2d1d~d1d2d1d2d1

= 1,

d2d1d2d1d~d1d2d1d~d1d2d1d~d1d2d1d~d1d2d1d~d1d~d1d~(d1d2) 2 ·d1d~d1d2d1d~d1d~d1d2d1d2d1d~d1d2d1d~d1d~d1d2d1 = 1, (d1d~) 2d1d2d1d~d1d2d1d~d1d~d1d2d1d~d1d~d1d~d1d~d1d§d1d2d1 ·d~d1d~d1d2d1d~d1d2d1d~d1d2d1d~d1d~d1d2d1d~d1d2d1d~

=

1.

Furthermore, M has a faithful permutation representation PM of degree 260 with stabilizer (d2, (d1d2d1d~(d1d2)5)2). (g) A system of representatives of the 28 conjugacy classes of M = (d1,d2) is given in Table 12.5.4.

(h) The character table of M is given in Table 12.6.5. (i) G has eight classes of s-special conjugacy classes represented by the following classes of M: 2b, lOa, l4a, l4b, 14c, 26a, 26b, 26c.

12.3 Fusion

537

Proof (a) holds by Proposition 12.3.2(a). In particular, the involutions s = xrhx 1 h 2 of Hand b = (xe) 7 of E are conjugate in G. (b), (c) and (d) The reader can check all three statements by means of MAGMA and the faithful permutation representation PE of E = (x,y,e) given in Lemma 12.1.l(e). (e) Since s and b are G-conjugate Cc (s) is conjugate to Cc (b) in G. Hence we calculate N = NE(R) containing CE(b). The stated generators Y1 and Y2 of N have been found by means of the program

GetShortGens(E,N). (g) and (f) By (c) Cc (b) = RxY, where Y?: J has a metabelian Sylow 2-subgroup S of order 26 with an elementary abelian center of order 23 . In particular, IY : JI is odd. Now Lemma 12.3.3 implies that 5 divides IYI- Hence Ne (S)/ Risa strongly embedded subgroup of Ne (R)j R by (c) and Theorem 4.8.4 of [92] because all involutions of W = Nc(R)/R are conjugate in W by (a) and Proposition 12.3.2(b). Now Theorem 1.7.3 of [92] asserts that W possesses a normal series WC> W1 C> W2 l::': 1 such that IW/W1I, IW2I are odd, and Wi/W2 ~ Sz(8). Here Sz(8) denotes the Suzuki group of order 26 · 5 · 7 · 13. Since the outer automorphism group Out(Sz(8)) has order 3, and IN/ JI = 3 by (c) and (d), it follows that IW/W1 = 3. Since Cg(s) has order 28 by Table 12.6.1, and all involutions of J are conjugate to z, an application of the Brauer-Wielandt Theorem 4.2.2 of [92] yields that W2 = l. Therefore U1 = Cc(b) = (b,c) x Y and Y ~ Sz(8). Thus (e) holds. Now Theorem 1.4.15 of [92] and (c) imply that the derived subgroup M' of M = Ne (R) = (N, Y) is a direct product of R and Y ~ Sz(8) such that there is an element r E M - M' of order 3 which operates on Y as the generator of order 3 of Out(Sz(8)) and (R, r) ~ A4 . Therefore it is easy to establish a presentation of M. The one of (g) has been found by means of MAGMA. (d) The system of representatives of the 49 conjugacy classes of M has been calculated by means of PM, Kratzer's Algorithm 5.3.18 of [92] and MAGMA. (e) The character table of M has been computed by means of the permutation representation of PM and MAGMA. This completes the proof. D 1

Proposition 12.3.5 Let G be a finite simple group of Ru-type having a 2-centml involution z with Cc(z) = H. Let M = (d 1 ,d2 ) be the finitely

Rudvalis' group Ru

538

presented group of Proposition 12.3.4(f). Then the following assertions hold.

(a) G has two conjugacy classes of involutions represented by z and s = xfhx 1h 2 , and Cc(s) ~Rx Sz, where R = (r 1 ,r2 ) is a Klein 4-group generated by the involutions r 1 = (d 1d 2 d1d§) 7, r 2 = (d1d~d1d2) 7 and Sz = ((d1d2)3, (d1 (d1d2)3)2) ~ Sz(8). (b)

IGI = 214 · 33 · 53 · 7 · 13 · 29. Proof. (a) By Propositions 12.3.2 and 12.3.4 each simple group G of Ru-type has two conjugacy classes of involutions zG and sG such that

IHI= ICc(z)I = 214 · 3 · 5 and

IUI = ICc(s)I = 28 · 5 · 7 · 13.

Furthermore, Cc(s) ~Rx Sz(8) by Proposition 12.3.4(e). Using the faithful permutation representation PM of M defined in Proposition 12.3.4(f) and MAGMA the reader can check that the involution u = r 1 = (d 1 d 2 ) 3 of M has a centralizer CM(u) =Rx Sz generated by the four elements given in the assertion such that R = (r 1 , r 2 ) is a Klein 4-group and Sz ~ Sz(8). (b) Using the faithful permutation representation PH of H described in Proposition 12.1.2(d) and the fusion pattern of the conjugacy classes of involutions of H in G given in Proposition 12.3.2 an application of MAGMA yields r(z, s, z) = l{(x, y) E (zG

n H) x (sG n H)lz

E

(xy)}I = 26 · 3 · 5 · 856.

By (a) we may assume that U = Cc(s) = CM(u) = Rx Sz, s = u = r 1 , and that z represents the unique conjugacy class of involutions of Sz ~ Sz(8). From Table 12.6.5 and Proposition 12.3.2 it follows that the six conjugacy classes (r 1 f, (r 2 )u, (r 1 r 2 )u, (r 1 zf, (r 2 z)u and (r1 r 2 z)u fuse to sG, and zu is the only conjugacy class of involutions of U which extends to zG. Using the faithful permutation representation PM of M = Nc(R) with R = (r1 ,r2 ) given in Proposition 12.3.4(f) it has been checked that r(z, s, s)

= l{(x, y)

E (zG

n U) x (sG n U)ls

E

(xy) }I= 5 · 7 · 13 · 449.

12.3 Fusion

539

Now Thompson's group order formula stated as Theorem 4.2.1 in [92] implies that IGI

= r(z, s, z)IUI + r(z, s, s)IHI = 214 · 33 · 53 · 7 · 13 · 29.

This completes the proof. Proposition 12.3.6 Let G be a finite simple group of Ru-type having a 2-central involution z with Cc (z) = H (x 1 , y 1 , h). Then the following assertions hold. (a) A Sylow 3-subgroup P3 of G is extra-special of exponent 3 and has order 33 • Its center Z (P 3 ) has normalizer N c (Z (P 3 )) ~ 3Aut(A6 ). Its character table is given in Table 12.6.4(b) G has one conjugacy class of elements of order 3. It is represented by p = X1 Y1 E fl.

Proof (a) A Sylow 3-subgroup P3 of G has order IP3 = 33 by Proposition 12.3.5. If it were abelian, then Nc(P3 ) ~ T x Aut(A 6 ) by Lemma 12.3.3(f), where T is generated by p = x 1 y1 E P3. By the proof of Lemma 12.3.3(g) Aut(A6 ) has two non-conjugate involutions z 1 and z2 which are G-conjugate to z and s, respectively. Hence x = ps is an element of order 6 which powers onto s. But G does not have such elements by Proposition 12.3.2(b) and Table 12.6.5. Therefore P3 is extra-special, and Nc(T) is a uniquely determined non-split extension of Aut(A6 ) by T. Its character table is given in Table 12.6.4. It asserts that P3 has exponent 3. (b) This statement follows now immediately from Tables 12.6.4 and 12.6.1. This completes the proof. D 1

Proposition 12.3. 7 Let G be a finite simple group of Ru-type having a 2-central involution z with Cc(z) = H = (x1,Y1,h). Let M = (d1,d2) be the finitely presented group of Proposition 12. 3.4 (f). Let R = (r1, r2) be the elementary abelian Fitting subgroup of M of order 4 defined in Proposition 12.3.5(a). Then the fallowing assertions hold.

(a) There is a simple subgroup Sz ~ Sz(8) of M such that U = CM(r 1 ) =Rx Sz is G-conjugate to Cc(s), wheres is a representative of the unique class of non-2-central involutions of G.

Rudvalis' group Ru

540

(b) G has two conjugacy classes 5A and 5B of elements of order 5 represented by Ji = (y1 h )4 E H and f = (d1 d2 )4d1 d§ ) 2 E M, respectively. A Sylow 5-subgroup P5 of G is extra-special of order 125 and exponent 5.

= Nc((fi)) ~ P5 : ((y) x Qs), where y has order 4 and Q 8 denotes the quaternion group of order 8. The character table of N is given in Table 12. 6. 6.

(c) N

(d) Ne((!))~ (A5 x F20), where F20 denotes the F'robenius group of 20. Its character table is given in Table 12.6. 7. (e) G has one conjugacy class of elements of order 7. It is represented by a= (d1d2d1d~) 2 EM, and P1 = (a) is a Sylow 7-subgroup of G. Furthermore, there is an element b of order 6 in M such that Nc((a)) =[((a): (b3 )) x (r1,r2)]: (b 2). (f) G has one conjugacy class of elements of order 13. It is represented by k = ((d1d2)2(d1d~)2) 2 EM, and A3 = (k) is a Sylow 13-subgroup of G. Furthermore, there is an element c of order 12 in M such that Nc((k)) =[((a): (c3 )) x (r 1 ,r2 )]: (c4 ). (g) A Sylow 29-normalizer Ne (29i) is a F'robenius group of order 14 · 29, i = 1, 2. Proof (a) By Proposition 12.3.4 we know that M has an involution r 1 , a Klein 4-subgroup Rand a simple subgroup Sz such CM(r 1) =Rx Sz and Sz ~ Sz(8). Furthermore, Proposition 12.3.4(e) states that CM(r1) is G-conjugate to Cc ( s), where s = Xi hx 1 h 2represents the second class of involutions of G. (b) through (f) Using the faithful permutation representation PM of M defined in Proposition 12.3.4(g) and MAGMA the reader can check that the given elements f, a and k of Sz have respective orders 5, 7 and 13. Furthermore, the automizers NM ( (x) )/CM (x) of the cyclic subgroups generated by them have respective orders 4, 6 and 12. Hence (e) and (f) follow at once from Proposition 12.3.5 and Sylow's Theorem. By Table 12.5.1 Ji = (y1 h)4 represents the unique conjugacy class of elements of order 5 in H = Cc ( z). Since all involutions of R are Gconjugate to s, Table 12.5.4 implies that Ji and f are not conjugate in G. By means of PM and MAGMA it has been checked that NM((!)) is isomorphic to a direct product of an alternating group A 4 and a Frobenius group F2o of order 20. Thus (d) holds by Theorem 1.6.2 of [92] and Proposition 12.3.5.

12.4 Uniqueness proof

541

A Sylow 5-subgroup A of G has order 53 by Proposition 12.3.5. Assertion (d) implies that a generator h of a Sylow 5-subgroup of A 5 in Ne ( (!)) is centralized by an involution in zG n F 20 . Hence Ji and h are G-conjugate. By the proof of Lemma 12.3.3(g) we know that Cc (Ji) is a {2, 5}-group with a normal subgroup O of 5-power order. In particular, Q 8 operates on 0. It has an irreducible GF(5)-module of dimension 2. As f is not centralized by Q 8 it follows that O = A. Furthermore, A is extra-special of exponent 5. Thus N c (f1) / A is a direct product of Q8 and the cyclic group F 20 / (!) of order 4 because Cc (Ji) = A : Q 8 by Table 12.5.1. Therefore (b) holds. (g) This assertion follows from Propositions 12.3.4, 12.3.5, 12.3.6, Burnside's Theorem 1.4.17 of [92], Sylow's Theorem and the other asD sertions of this proposition. This completes the proof.

12.4 Uniqueness proof

In this section we prove the following result due to D. Parrott [106]. Theorem 12.4.1 (D. Parrott) Let G be a finite simple group which possesses an involution z such that H = Cc(z) satisfies the following properties.

(a) J = 0 2 (H) has order 211 and class at least 3. (b) H/J ~ S 5 , the symmetric group of degree 5. (c) If Q is a Sylow 3-subgroup of H then CJ (Q) ~ Qs, the quaternion group of order 8. Then G

~

Ru, the Rudvalis simple group.

In [106] Parrott does not prove that a finite group H with the above three properties is uniquely determined up to isomorphism. In fact, that need not be true. Therefore we prove here that any simple group G of Rutype is isomorphic to the finite simple group Q5 constructed in Theorem 12.2.1. By means of Lemma 12.1.3 the reader can check that any simple group of Ru-type satisfies the three conditions of Theorem 12.4.1. In [106] Parrott shows that any simple group G meeting his three conditions is a rank 3 permutation group with stabilizer isomorphic to the finite group of Lie type F 4 (2) having index 4060 in G. Thus he is able to identify G with the original Rudvalis group Ru discovered by Rudvalis [115].

542

Rudvalis' group Ru

This last step can be adapted for Q5 = (F, IJ, fJ, e) as follows. Let PQ5 be its faithful permutation representation of degree 424125 constructed in the proof of Theorem 12.2.1. In Sj = (F, IJ, fJ) find a subgroup Sj 1 of order 211 · 5 which is isomorphic to the group H of Proposition 10.1.5. In particular, S'J1 has a uniquely determined elementary abelian normal subgroup Qt which is not normal in Sj 1 . Let ::D = NYJ 1 (sit). Then calculate ~ 1 = N 6 (sit). In ~ 1 find a subgroup ~ of order 2 11 · 3 which is isomorphic to the finitely presented group E of Lemma 10.1.2. Using MAGMA and PQ5 again show that '.r = (S'J 1, ~) is isomorphic to the simple group of order 211 · 33 · 52 -13 constructed in Theorem 10.3.1. Now let J = N 6 ('.£'). Then I® : JI = 4060, and the permutation character (1'.r) 6 is a sum of the irreducible characters x1 , x 5 and x6 of Table 12.6.8. Because of lack of space these calculations are not documented here. Lemma 12.4.2 Let G be a finite simple group of Ru-type with a 2central involution z such that Cc (z) = H described in Proposition 12.1. 2. Then H is a maximal subgroup of G.

Proof Lemma 12.4.2 must be true, otherwise there is a proper subgroup X of G containing H = Cc (z) properly such that Xis of minimal order with respect to these two properties. Hence Cx (z) = H. Suppose that X has a normal subgroup Y of odd order such that X = YH. Then YnH = 1 because H does not have any normal subgroups of odd order. Therefore Lemma 1.5.9 of [92] and Proposition 12.3.5 assert that Y is an abelian group of order dividing q = 33 · 53 · 7 · 13 · 19. By Table 12.5.1 and Lemma 12.1.3(f) V = (z, t) with t = x~(xtyi) 2 is a Klein 4-subgroup of H such that ICs (v) I = 213 . Lemma 12.3.l(b) asserts that z and t are G-conjugate. Thus H = Cc ( z) ~ Cc (t). By Theorem 4.2.2 of [92] we know that IYIICy(V)l 2 = ICy(z)IICy(t)l 2 because t and tz are conjugate in H. But H does not have any normal subgroups of odd order. Hence ICy (z) I = ICy (v) I = 1. Thus Y = 1 and X = H, a contradiction. Let S be a Sylow 2-subgroup of H. It is a Sylow 2-subgroup of X also. Now Theorem 4.7.3 of [92] due to G. Glauberman implies that there is an x EX such that zx ES- {z}. By Lemma 12.1.3(f) we may assume that V = (z, t) is the unique normal Klein 4-group of S. Another application of Lemma 12.3.1 asserts that z and t are X-conjugate and Nx (A)/Cx (A) ~ GL 3 (2), where A is the unique elementary abelian normal subgroup of order 8 in S. Now all the arguments of the proof of Proposition 12.3.2 can be applied. Thus Nx (A) ~ E, where E =

543

12.4 Uniqueness proof

(x, y, e) is the group constructed in Lemma 12.1.1. Furthermore, X has two conjugacy classes of involutions represented by z and s = Xi hx1 h 2 E H. Therefore Propositions 12.3.4 and 12.3.5 yield that IXI = 214 · 33 · 53 • 7 · 13 · 29 = IGI. This contradiction completes the proof. D Proposition 12.4.3 Each finite simple group G of Ru-type has a unique pair of complex conjugate characters x: G---+ 20 26 • 5 101 xm2xm2y 2·3·7 2b 2li (xm2xm2y ,z 212 2·3·7 21i 221 xm2 2 · 11 111 222 (xm2 ' 2 · 11 lb 241 xym2 26 • 3 122 242 (xym2 1 26 • 3 122 yzm2 243 2" · 3 121 28 x"m2xm2 2" · 7 14 x:,m2xy 301 2 · 3" · 5 151 x"m2y"m2 2 · 3" · 5 152 302 x"m2 401 2° · 5 20 (x"m2 ' 2° · 5 20 402 xy,m2xm2 421 2·3·7 2b 422 ( xy" m2 xm2) 11 2·3·7 21i

3P 1 21 22 1 1 1 41 42 51 52 21 21 22 21 22 22 22 7 81 82 83 84 33 101 102 111 lb 41 41 42 14 51 51 64 20 7 7 221 222 82 82 81 28 101 101 401 402 14 14

5P 7P llP 1 1 1 21 21 21 22 22 22 31 31 31 32 32 32 33 33 33 41 41 41 42 42 42 1 51 51 1 52 52 61 61 61 62 62 62 63 63 63 64 64 64 65 65 65 65 65 65 61 61 61 1 7 7 81 81 81 82 82 82 83 83 83 84 84 84 9

9

9

21 21 111 lb 121 122 123 14 32 31 18 41 21i 2b 221 222 241 242 243 28 62 61 81 81 421 422

101 102 112 111 121 122 123 21 151 152 18 20 31 31 222 221 242 241 243 41 301 302 402 401 61 61

101 102 1 1

121 122 123 14 151 152 18 20 2b 211 21 21 242 241 243 28 301 302 402 401 422 421

Lyons' group Ly

592

13.6.2 Conjugacy classes of D

= Ng(A) = (x,y)

Class Representative I Centmlizerl 1 21 22 23 24 25 25 31 32 33 41 42 43 44 45 61 62 63 64 65 65 67 6s 69 610 611 81 82 83 8, 85 85 121 122 123 124 125 125 24

1 (x3y)12 (x)4 (y)3 (xy)6 ( x 3 y 2 x 2 11 ) 3 xv 2 (xy)4 ( y )' ( xyxy 3 ) 2 (x3y)6 ( X )2 (x4y)3 (x3yx2y2)3 (xy)3 (x3y)4 ( x 3 yxyxyxy ) 2 (xy)2 (xy)10 (x4y)2 x 3 u 2 x'u X4112 X1JX1J 3 x 3 uxu 3 x 2 uxvxv 2 V

(x3y)3 x 2 1JXU 3 x 2 v 3 xv X x 2 11 3 x 4 ux 3 V (x3y)' x 3 vx 2v 2 X 3 'UX1/X'UX'U xv (xy)5 X41J x 3v

28 . 32 28 · 3 2 27 . 32 25 · 3 25 · 3 25 · 3 24 27 . 32 23 · 3 2 22 · 32 26 · 3 25 · 3 25 · 3 25 · 3 23 · 3 27 · 3 2 26 . 32 25 · 3 25 · 3 23 · 32 24 · 3 22 · 32 22 · 32 22 · 32 22 · 32 22 · 3 24 · 3 25 25 24 24 24 25 · 3 24 · 3 24 · 3 23 · 3 23 · 3 22 · 3 23 · 3

2P 1 1 1 1 1 1 1 31 32 33 21 22 21 21 24 31 31 31 31 32 31 32 33 33 33 32 41 41 41 42 41 42 61 61 62 6, 64 65 121

3P 1 21 22 23 24 25 25 1 1 1 41 42 43 44 45 21 22 24 24 21 25 22 22 21 22 23 81 82 83 8, 85 85 41 44 42 45 45 43 81

593

13.6 Representatives of conjugacy classes 13.6.3 Conjugacy classes of N = Ne (A)= (x, y, c2)

xyxc2xc2 2 X2C2 (x 2C2 ) 5

1 (x)4

28 . 33

6e

28 . 32

61

25 . 32 25 . 3

69 6h 6;

2e

(xy)B (y)3 xy 3c2 (xc2 )3

3a 3b

(xy)4 (y)2

3c

C2 (xc2) 2

22 . 32

8a 8b

2. 32

8c

(x)2 (yc2 ) 3

26 • 3 25 . 3

8d

4c (xc22y)3 (xy)3 4d

25 . 3 23 . 3

12c

(xyc2) 4 (xy)2 (xy)lO (yc2 ) 2

27 . 32

12d

xy (xy)5

25 . 32

12e

yc2

23 . 3 22 . 3

25 . 32

24a

xyc2

23 . 3

la 2a 2b 2c 2d

3d 4a 4b

6a 6b 6c 6d

25 . 3 24 • 3 27 . 33 23 . 32

23 . 32

24 . 3 22 . 32 22 . 32

xyxyc2 x4y2

22 . 32

6j

y

22 . 3

6k

XC2 (xyc2 )3

2·3 24 . 3 25

12a 12b

X

xyc2yc2 yc2 2 (xyc2 ) 2 xc2 2y

22 . 32

25 24 25 . 3 24 • 3 23 . 3

Lyons' group Ly

594

13.6.4 Conjugacy classes of E

ICE(e;)I

e, L 2a 2b 3a 3b 3, 3d 4a 4b 5a 5b 6a 6, 6, 6d 6. 61 7n 8n 8b 8, 9n 10a 10, 11a 11b 12n 12b 12, 12. 12 14. 15. 15, 15, 15. 15 20n 20b 21. 21b 22n 22b 24 24b 24, 30n 30b 30, 30, 33n 33b 42 42b

1

3 7 • 5 3 · 7 · 11 28 • 3 3 · 5 · 7 2 5 · 3 2 · 5 · 11 27 • 3 7 • 53 · 7 · 11 24 • 37 • 5 23 · 37 • 5 23 . 35 25 · 3 2 · 5 26 . 32 22 . 32 . 53 2 · 3 · 52 27 • 33 · 5 · 7 24 • 3 3 · 5 23 · 3 3 · 5 23 . 33 22 . 33 22 . 32

28

(x)4

(y)3 (xy)4 (v)2 (vc2 ) 4 (v)2 (xv) 5 (x)2 (xv) 4 (yvc2 ) 2 (x11)2 V (vc2 ) 2

(11c2 ) 2 X 2 C?

y x 2v xv 2 (xyc2 ) 3 X v 2 c2VC2

(xv) 2 VVC2 (xvc2 ) 2 (xvc2 ) 4 xv

v 2 c~ vc2 VCo x 4 vv ( xv 2 11 ) 3 x 2 vxv 2 xvv (xyv) 2 x2

v2

X 2 C?.VC?.V

xv (xv) 11

vc~ (vd) 2 xvc, (xvc2

= Ne (3A) = (x, y, v, c2)

)7

XVC2 x 2 vv 2

(x2 11 ,,2 113 xvv (xvy) 7 x 3 vxv

x 2 VC?. VC?. x 2 v 2 c2 (x2v2c2)2 xv 2 v (xv2y)11

•

2·3·7 25 · 3 24 • 3 25 33 22 · 3 2 · 5 2·5 2 · 3 · 11 2 · 3 · 11 25 . 32 23 . 32 22 . 32 22 . 32 22 . 32 2.3.7 2 · 32 · 53 2 · 32 · 5 2 · 32 · 5 2 · 32 · 5 3 · 52 22 · 5 22 · 5 2.3.7 2·3·7 2 · 11 2 · 11 23 · 3 23 · 3 23 · 3 2 · 32 · 5 2 · 32 · 5 2 · 32 · 5 2 · 32 · 5 3 · 11 3 · 11 2·3·7 2·3·7

I

2P L la la 3a 3b 3, 3d 2a 2. 5a 5b 3a 3, 3, 3d 3. 3d 7n 4, 4b 4 9. 5a 5b 11b 11a 6n 6b 6. 6. 6b 7. 15n 15, 15b 15. 15 10. 10. 21b

21. 11n 11b 12n 12b 12b 15. 15b 15, 15n 33, 33a 21a 21b

I

3P L 2. 2b 1. la la la 4a 4b 5. 5b 2a 2n 2a 2a 2n 2b 7n 8n 8b 8. 3. 10a 10b 11. 11b 4, 4b 4n 4, 4a 14a 5n 5a 5a 5n. 5, 20. 20b 7a 7a 22n. 22b 8, 8a 8a 10n 10. 10n 10. 11. llb 14a 14a

I

5P L 2. 2b 3a 3b 3, 3d 4a 4b la la 6a 6, 6, 6d 6. 61 7n 8n 8b 8. 9 2. 2b 11a 11b 12. 12b 12. 12. 12e 14a 3n 3, 3, 3, 3n 4n 4a

21. 21b 22. 22b 24n 24, 24, 6. 6, 6, 6n 33 33b 42a 42b

I

~ 3Mcl: 2 7P 1 2. 2b 3a 3b 3, 3d 4a 4b 5. 5b 6a 6'

6, 6d 6. 6t

L 8n 8b 8 9. 10. 10b 11b 11a 12n 12b 12. 12. 12. 2. 15n 15, 15b 15d 15, 20. 20b 3a 3a 22, 22. 24n 24, 24, 30, 30a 30, 3033 33. 6a 6a

I

llP L 2. 2b 3a 3b 3, 3d 4a 4b 5a 5b 6a 6, 6, 6d 6 6t 7n 8n 8b 8. 9.

10. 10b 1 1 12. 12b 12. 12. 12e 14a 15n 15b 15, 15. 15 20, 20. 21b 21. 2, 2b 24n 24, 24, 30n 30b 30. 30. 3. 3a 42, 42a

I

13. 6 Representatives of conjugacy classes

13,6.5 Conjugacy classes of R

=

Ne (f)

= (x, y, h)

Class Representative I Centralizer! 1 21 22 31 32 41 42 43 44 4.s 51 52 53 54 55 55 61 62 81 82 83 84 85 8, 101 102 103 104 105 105 121 122 123 124 151 152 153 201 202 203 204 20.s 241 242 243 244 245 245 247 24s 25 301 302 401 40,

1 (:u)l2 (x)lO (x2 h )s (y)8 (y)6 ( y )18 (xh 110 (x )' h (xh ) 8 (x)4 (xhyh ) 3 (xy)4 (x3y)2 (xh 2 y) 2

(y)4 (x 2 h)4 (y )3 (y)g (x 2 h) 3 (x 2 h ) 9 (xh) 5 x3h (xh ) 4

(x)2 (x-ul2 x"y xh 2 y xy 3 h 2 (y )2 (x 2 h ) 2

( xy212 (yh2)2 ( xyxy 2 h ) 2 (x 2 y 2 hy) 2 xhvh (xh) 2 X XY

x

2 hy

x4h 7J

x 2h x-u 2 x-uh -uh' x 3 hv x 2 hyh x11 2 h11 yh x 2 1ih-u x-uxv 2 h xh x 2 h2

2 6 · 3 2 · 56 26 . 32 . 52 25 · 52 23 . 32 . 53 23 · 32 · 5 26 · 32 · 5 26 · 32 · 5 25 · 5 24 · 5 24 · 5 24 . 32 . 56 22 . 55 2 · 3 · 54 2 2 · 53 2 · 53 2 · 53 23 . 32 . 5 23 . 32 . 5 25 · 3 25 · 3 25 · 3 25 · 3 24 · 5 24 24 . 32 . 52 22 · 52 22 . 52 2 2 2 23 23 23 23

· 52 · 52 · 52 . 32 · 32 . 32 . 32

2 · 3 2 · 53 2 · 32 · 5 3 · 52 23 . 5 22 · 5 22 · 5 22 · 5 22 · 5 23 . 3 23 · 3 23 . 3 23 . 3 23 . 3 23 · 3 23 . 3 23 . 3 52 2 · 32 · 5 2 · 32 · 5 23 . 5 23 . 5

2P 1 1 1 31 32 21 21 21 22 22 51 52 53 54 55 56 32 31 41 42 42 41 43 4s 51 52 54 55 55 53 61 62 61 62 151 152 153 101 102 103 103 10? 121 122 123 121 124 124 122 123 25 152 151 201 201

3P 1 21 22 1 1 42 41 43 45 44 51 52 53 54 55 55 21 21 82 81 84 83 85 Se 101 102 103 104 105 105 41 42 42 41 51 51 53 201 20.s 204 203 20, 81 8, 82 81 84 84 83 82 25 101 101 401 402

5P 1 21 22 31 32 41 42 43 44 4, 1 1 1 1 1 1

61 62 81 82 83 84 85

Se 21 22 21 21 21 22 121 122 123 124 31 32 31 43 44 42 41 4, 241 242 243 244 245 246 247 24 8 51 61 62 8s 8,

595

Lyons' group Ly

596

13.6.6 Conjugacy classes of L

= N R (V) = (x, y)

Class Representative I Centralizer! 1 21 22 3 41 42 43 44 4, 45 47 51 52 53 54 55 56 57 5s 6 81 82 101 102 103 104 105 105 107 10s 121 122 15 201 202 203 204 205 20, 207 241 242 243 244 25 30

1 (y)" (x)10 (y)8 (y)6 (y)JS (x)5 X2'U2

( xvxy" ) 5 x 3 VX 2 1J (x 3 1JX1/ 2 ) 5 ( xy' ) 4

(x)4 (xyxy 2 ) 4 (x7y3)2 (x 4 y 2 x 3 y)2 (xy)4 (.1"3 y 12 (x7 y)2

(y)4 (y)3 (y)g (xy3 )2 (x )2 (xy)2 (xyxy 2 ) 2 x 3y x7 v x7 v3 x 4v 2 x 3v (y)'

25 · 3 · 25 · 3 · 24 . 23 · 3 25 · 3 25 · 3 24 24 24 24 24 23 · 3 · 22 . 22 .

56 52 52

2 · 2 ·

·5 ·5 ·5 ·5 ·5 ·5 ·5 ·5 56 55 54 54 54 53 53 53

23 · 3 23 23 23 · 3 · 22 . 22 .

·5 ·3 ·3 52 52 52

22 · 2 · 2 · 2 · 2 · 23 23

52 52 52 52 52

2 · 2 · 22 .

1J x112 x31J4 x 3 vx 2 v 3 x 3 vx 2 v 2

·3 ·3 2·3·5 22 · 5 22 .5 22 .5 22 · 5 22 · 5 22 · 5 22 · 5 23 · 3 23 · 3 23 .3 23 .3 52

X 3 1JX1JX1J

2 ·3 ·5

I x11' ) 2 (x 3 yxyxy) 2 X X1J X1J3 xvxv 2 X31JX1J2 X21J2X1J2 X 5 'UX'UX'U

2P 1 1 1 3 21 21 22 21 22 22 22 51 52 53 54 55 55 57 5s 3 41 42 51 52 56 53 57 5s 54 55 6 6 15 102 103 101 104 102 104 103 121 122 122 121 25 15

3P 1 21 22 1 42 41 47 44 46 45 43 51 52 53 54 55 55 57 5s 21 82 81 101 102 103 104 105 10, 107 10s 41 42 51 20, 207 203 205 201 204 202 81 82 82 81 25 101

5P 1 21 22 3 41 42 43 44 45 45 47 1 1 1 1 1 1 1 1 6 81 82 21 22 21 22 21 21 22 22 121 122 3 4, 42 44 45 47 4, 41 241 242 243 244 51 6

13.6 Representatives of conjugacy classes 13.6. 7 Conjugacy classes of M = Ne (V) = (x, y, e) CJlass

1 2 3 41 42 43 51 52 53 54 6 81 82 101 102 10, 104 121 122 15 20, 202 20, 241 242 24, 244 25 30 31, 312 31 314 31, 316 3h 31s 3lo 3110

I Centralizer! 2P 1 2 5 . 3. 5 6 . 31 1 (x)lO 25 . 3 . 52 1 (v)B 2 3 . 3. 5 3 (y)6 2 5 . 3. 5 2 (v)lB 2 5 . 3. 5 2 (x)5 2 24 · 5 (x)4 2 3 . 3 . 56 51 22 . 54 (xy)4 52 (x3y)2 2 · 5 4 53 x 2 yey 2 · 5 4 54 (v)4 2 3 . 3. 5 3 (y)3 2 3 . 3 41 (v)g 2 3 · 3 42 (x)2 2 3 . 3 . 5 2 51 22 . 52 (xv) 2 52 x3y 2 · 5 2 53 yeye 2 2 · 5 2 54 (11)2 23 . 3 6 (xy2 )2 23 . 3 6 x 2 vxe 2-3·5 15 2 2 . 5 10, X 2 2 . 5 10, xv 2 2 . 5 102 xvexe 2 3 . 3 12, V X'l12 2 3 . 3 122 2 3 . 3 121 xeye 2 3 . 3 122 xu 2 exe 52 25 xu 2 e 2.3.5 15 xvxvev 31 314 ve 2 31 31, X'UXe xeu 2 31 3lg xuxey 31 312 xy 2 xe 31 3110 xyeye 31 31s xue 2 y 31 311 11 3 e2 31 31, xvxeve 31 315 31 31s xvxe 2 v

Representative

3P 1 2 1 42 41 43 51 52 53 54 2 82 81 101 102 103 104 41 42 5, 20, 203 202 81 82 81 82 25 10, 3110 3Is 311 31s 312 31, 315 310 314 31,

5P 31P 1 1 2 2 3 3 41 42 41 42 43 43 1 51 1 52 1 53 1 54 6 6 81 82 81 82 2 101 2 102 2 103 2 104 121 122 122 121 15 3 4, 20, 20, 42 202 41 24, 242 242 241 24, 24, 244 243 5, 25 6 30 31, 1 1 312 31, 1 1 314 31 1 1 316 31, 1 1 31s 1 310 1 3110

597

Lyons' group Ly

598

13.7 Character tables of local subgroups 13.7.1 Character table of H= (x, y, m2) c,,c 2A11 2 3 5 7 11 la

2a

2b

3a

3b

3c

4a

4b

5a

5b

6a

6b

6 C 6d

6e 6f

6g 7a

8a

Sb

Sc 8d 9a

2P 3P 5P 7P llP

la la la la la

la 2a 2a 2a 2a

la 2b 2b 2b 2b

3a la 3a 3a 3a

3b la 3b 3b 3b

3c la 3C 3C 3c

2a 4a

2b 4b 4b 46 4b

5a 5a la 5a 5a

5b

3a 2a 6a 6a 6a

3b 2a 6b 6b 6b

3a 2b 6c 6C 6C

3b 3c 3c 7a 2 b 2 b 2b 7a 6e 6f 6g 7a 6c 6f 6g la 6e 6f 6g 7a

4a Sa Sa 8a Ba

4a Sb 8b Sb 8b

4b 4b Sc 8d 8 C Sd Be 8d Sc Sd

X X X X X.5 X.6 X.7 X.8 X.9 X X. X X X.

1 10 16

1

4 4 4 5

1 1 -2 -2 -1

10 -16

16

44 45 110 120 126 126 132 144

594 594 616 616 660 672 693 8 25

880 880 924 X .34 990 X 35 990 X .36 1100 X.37 1155 X.38 1200 X .39 123 2 X.40 12 32 X .41 12 3 2 X .42 13 2 0 1440 1540 1584

1584 1584 X .48 1584 X .49 2310

4

45 110 120 126 126 132 -144 165 210 2 31 330 -528 550 -560 5 94 594

-3 6 -8 6 6 4 5 2 -1 2 g -2 -2 -6 -6 -6

-616

660

-4

-672

693 82 5 -880 -880 924 990 990 1100

-3 9 4 6 6 12 -5

1155 -1200 123 2 -16 -1232 -1232 132 0 8 -1440 1540 -4 -1584 -1584 -1584 -1584 2310 -2

1 7 -8 -8 20 21 2g 36 21 21 6 -48 21 42 63 -6 49 -42 48 70 -112 90 -45 -45 -56 -56 -36

6

4a

4a 4a

1 5 -4 -4 16 13 1 26 -2

2 6 -2 6 -14 -2 6 20 4 6 -3 12 3 29 1 9 3 -14 -2 3 -3 35 -1 g -3 22 2 -8 -2 21 1 2 3 3 14 6 -5 1 50 -2 2 8

-8 -8 -3 12 6

21 -15 4 64 4 64 9 84 -42 6 21 -12 .so -10 -21 9 -120 5 6 -4 56 -4 56 -4 -84 6 -48 12 28 1 -24 -12

-24 -] 2 48 6 48 6 -63 -6

16 49 6 -15 -2 -2 -3 -56

5b la 5b 5b

1 1

9 -1

10 5 10 1 1 1 1 -8 2 -16 -1 -5 5 16 -10 5 7 2 -12 -2 -20 9 9 9 -4 -4 5 12 -17

-1 -1 -1 1

1 2 -2

-10 -10 -1 -1 34 -2 -5 34 -2 -5 2 -20 4 7 -1 -10 3 -6 -1 -8 -1 -8 -1 -8 -3 10 20 -.S 1 -56 4 -1 4 -1 4 -1 4 -1 -3 -14 -2 5

3c 2a 6d 6d 6d

1 1 1 1 1 7 4 -1 8 -4 2 2 8 -4 4 -1 20 5 21 6 -3 2 -3 29 4 36 6 6 -3 21 21 6 -3 6 -2 -3 6 48 -12 21 3 3 42 2 3 9 3 -1 -3 63 -6 2 -3 9 49 -8 9 -2 -42 3 -2 3 -48 -6 70 -5 -2 1 112 -8 -2 -6 90 3 3 -4 56 8 -4 56 8 -36 -3 -4 3 -12 -6 21 6 -3 -15 9 6 -64 -4 2 -64 -4 2 -3 84 9 -42 6 21 -12 -3 2 50 -10 -6 -21 9 -5 3 120 6 8 -1 56 -4 -56 1 4 -56 4 J -84 6 -4 -3 48 -12 28 1 1 -4 24 12 12 -6 -48 -6 -63 -6 1 -3

1 1 -1 -1

-2

1 3 2 2 2 3 -2 1

2 -3 -4 -4

-2 -1 -1 -1 -1

-1 -1 -1 -1

1 -1 4 -1 -3 1 1 -1 -6 2 -1 5 1 -4 -1 -1 3 -6 2 -4 1 -3 -4 -1 -4 -1 -4 -1 -4

-1 -1 -1

9a 3C 9a

5b lOb 2a 1 Ob I Ob

llb l la l la llb la

1 5 4 -1 4 -1 9 -1 -1 -1 10 -J 5 10 1 1 -8 16 -1 -1 -5 5 16 -10 - I -1 5 7 12 1 -1 20 9 -1 g -1 9 -1 -2 2 4 -1 2 -2 4 -1

1 -1

9a

9a

1 1 1 1 -1

-1

A

A

-1

.s

2 -6 -2 -1 -2 -2

A

A

-12 -2 -17 -2

-1 3 5 -3

10 10 -1 - ] -5 -5

-6 -2 -4 -4 -1 1 -1 -1

-10

-4 2 -1 -1 -3 3 3 -3

-1 -1 -1

2 -1 -1 -3

-2 -1 -1 -1

-2

-8 2 8 -2 8 -2 10 -20 -1 1 -5 -4 -4 -4 -4 5

13.7 Character tables of local subgroups

= (x, y, m2)

Character table of H

599

{continued)

2 3 5 7 11 2P 3P 5P 7P llP X.l X.2 X.3 X.4 X.5 X.6 X.7 X.8 X.9 X.10 X.11 X.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19 X.20 X.21 X.22 X.23 X.24 X.25 X.26 X.27 X.28 X.29 X.30 X.31 X.32 X.33 X.34 X.35 X.36 X.37 X.38 X.39 X.40 X.41 X .42 X.43 X.44 X.45 X.46 X.47 X.48 X.49

llb 11a 11b llb lla la 1 -1

A A

12a 6a 4a 12a 12a 12a 1 3

12b 6b 4a 12b 12b 12b 1

.j i 1 -2 5 2 -i -4 2 A 1 -2 A 1 -2 2 2

i

5 -i

-2 1 -1 -1 -2 1 -3 2 -i

-i

-i -6

3 3

12c 14a 15a 6c 7a 15a 4b 14a 5a 12c 14a 3b 12c 2a 15a 12c 14a 15a 1 1 1 -1 3 -1 -2 -1 -2 -1 2 3 i -2 2 1 1 1 1 1 1 -2 -i 1 -4 2 i -3 -2 -2 -1 -1 -2 i -1 2 1 2 2 -2 .j -2 -3

2 -i -1

-1 -1

i -2

-3

4 i

-2 -2 -2 1 4 1 -2 -2 -2 -5 1 -1

-i

-4

-1

-i2

i

-1 1 1 1

18a 9a 6d !Ba 18a 18a 1 1 -1 -1 -1

-i

2

i

-2 1 -2

2 1 2 -2 -1 -1 -2 3

-i

-i -i -i

1

-A -A -A -A

-i -i

-i -i -i1 -i

1

A A

A A

-i -i i

i

B C

1

-i -1

i

-1

-i

1

2 -i -1 -1

-1

i -i

-i -i -i -i -2 -i

-i -i

i

2 1 1 1 1 -2 -2 -1 -2 -1 2 .-2

d

B -1 -1

i

1

-i -i Ei -Ei

: -E

E

42b 21a 14a 42b 6a 42a

1 1 1 1 1 -1 -1

-i -i -1 -1

-1

-i -1

C

i

C B

1 2 -1 3 1 i

i

-i -i -i -i -i -i -i -i i v -v -1 -1

1 -D 1 -1 -1 -1 1 -2 1 -2 1 1

D

-1 1 1

-1

-1

-i -i i i 1

C -1 -1

i -i

42a 21b 14a 42a 6a 42b

-i -i B -2 -2

-1 -1

Ii

i

-1 -i 1 1 1 1 -2 1 i 2 -1 -2 -1 -1 -i -2 -1 -3

40b 20a 40b Ba 40a 40a 1 -1

2 -i -1

1

1

-i i

-1 -1

C B

i -i -i -i -i -i -i -i 1 1 -i -1

2

-1 1 -2 1 -1 -1

i

-1 -1 1 1 -1 -1

-i -i -i -i

-i -i

-i

28a 30a 30b 40a 14a 15a 15b 20a 28a 10a IOa 40a 28a 6b 6a Ba 4a 30a 30b 40b 28a 30a 30b 40b 1 1 1 1 -1 -1 2 -1 1 -2 1 -2

-2 1 2 2 -1 -1 -1

-1 -1 1 -2 -2 -1 -1 2 2 2

i

-i -i

. -Ii -d

. -C -B 1 1 1 1

-i -i

((11)5 - ((11)4 - ((11)3 - ((11) - 1, B 2((21 )3((21 )f + 2((21 )3 ((21 )? + 2((21 )3 ((21 )1 +((21 )3 +((21 )f +((21 )? + ((21) 7 + 1, C = -B + 1, D = 2((40)~((40)~ + 2((40)~((40)g + ((40)~ 2((40) 8((40)~ - 2((40) 8((40)g -((40) 8, E = -2((24)~((24) 3 - ((24)~ where A

= -((11)9

22b 24a 24b 24c llb 12b 12b 12a 22b Sb Sb Ba 22b 24a 24b 24c 22a 24b 24a 24c 2a 24b 24a 24c 1 1 1 1 -1 1

-i -i

2 -1 -1 2 -1 -3 1

-2 -2 -2 1 -2 1 -1

22a lla 22a 22a 22b 2a 1 -1

2

1 -1

2 -i -i

1 1 1 2 -2 1 1 -2 -2 2

21b 21a 7a 21b 3a 21a 1 . -1 -1 -1 -1 -1 -1

1

-2 -2

1 1 1 2

21a 21b 7a 21a 3a 21b 1

1

-i -i -i i i -i

-3

20a IOa 20a 4a 20a 20a 1 1

1

1

2 -1 -1 -1 -1 -1 1 -2 -2 1

-2 -2 i -2

15b 15b 5a 3a 15b 15b 1 2 2 2

2((24) 8 ((24) 3 -((24)s.

Lyons' group Ly

600

= (x, y)

13.7.2 Character table of D !

53. (52

GL2 (5))

la

2a

'b 3a

4a

4b

4c

4d

4e

4f

4g

5a

5b

5c

5d

5e

5f 5g 5h 6a

la I a la la la la la la la la

la la 3a 2a 2b la 2a 2b 3a 2a 2b 3a 2a 2 b 3a 2a 2b 3a 2a 2b 3a 2a 2b 3a 2 a 2 b 3a 2a 2b 3a

2a 4b 4a 4b 4b 4a 4a 4b 4b 4a

2a 4a 4b 4a 4a 4b 4b 4a 4a 4b

2b 4g 4c 4g 4g 4c 4c 4g 4g 4c

2b 4f 4d 4f 4f 4d 4d 4f 4f 4d

2a 4e 4e 4e 4e 4e 4e 4e 4e 4e

2b 4d 4f 4d 4d 4f 4f 4d 4d 4f

2b 4c 4g 4c 4c 4g 4g 4c 4c 4g

5a 5a I a 5a 5a 5a 5a 5a 5a 5a

5b 5b la 5b 5b 5b 5b 5b 5b 5b

5C 5c la 5C 5C 5c 5c 5C 5c 5C

5d 5d I a 5d 5d 5d 5d 5d 5d 5d

5e 5e la 5e 5e 5e 5e 5e 5e 5e

5f 5f la 5f 5f 5f 5f 5f 5./ 5f

5g 5g la 5g 5g 5g 5g 5g 5g 5g

1 l 1

1

-1

1 1 I 1 4 4 4 4 4 4 4 4 4 4 5 5 5 5

1 1 1

1 1 1 1

1 1 1 1

4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 6

-1

1 l 1 -[ 1 1 1 -[

1 1

l

-C

-C

1

-[

-[

C

C

-[

4

1

4 4 4 -4 -4 -4 -4 -4

1

4 4 -4 -4

-4

5 5 5 5 6 6

1 1 1 1 -2 1 1 -2 1 -1 1 -1 -[ -I -1 - [ 2 -2

A -A A -A A -A -A A -A A -A A 5 5 -5 -5 -6 6

5 5 -5 -5 -6 6

-1 1

C -C

-1

1

C -C

B -B D -D B -B -D D -B B -/D /D -B B /D -/D

-6 -6 -6 -6

-1

1

I

-C C

-/D /D /D -/D D -D -D D

4

4 -4

-4 -4

-4

A -A

-A A

8

-8 8

-8

-4 -4 4 4

-4

-C C

6 6

-A A

4 -4

-8

1

-2 -2

-1

4

4

-4

1 1 l

-4

4 4 -4 -4

-8

4 4 -4 -4

1

-[

1 -1

5 5 6 6

5 5 6 6

6

6

6

6

6

6 6 -4 24 4 24 A 24 24 -A 96 -25 120 120 120 120 -50 -50 240 240 240 240 -75 -75 -100 -100 -125 -150

6 6

6 6 6

24 24 24 24 96

-5 -5 -5 -5 -10 -10 -10 -10

-[ -[ -[

1 1 1 1 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6

6 -1

1

-]

-4

-1 -1 -4

-1 -1 -1 -4

20 20 20 20

-5 -5 -5 -5

-5 -5 -5 -5

-I

-[

Sa

Sb lOa 10b 1 Oc

4a 8b 8a 8b 8b 8a Ba Sb 8b 8a

4b 8a Sb Sa Sa Sb Sb

5a lOa 2a 10a 10a lOa IOa

1 1 1 -C C 1 - [ -1 1 C -C 1 2 2 1 -2 -2 1 E -E

1

5h 3a 5h 2a la 6a 5h 6a 5h 6a 5h 6a 5h 6a 5h 6a 5h 6a 5h 6a 1 1 1 l

-[ -[

-1 - [ -1 -1 -1 - [

-1 -1 -1

1

1 1 1

1 1 4 4 4 4

E

1

-[

-]

-1

-C C

1 I 1 1 1 1 4 4

1 1 1 -1 -1 1 1 1 l -1 - [ 4 4 4 4

-4 -4 -4 -4

-4

2 -1 -[

l 1 1 1 l 1 4 4 4 4

10b 10c 1 Ob 1 Oc IO b I Oc 8a IOa lOb 10c Sa IOa IOb 10c Sb lOa lOb 10c

1

-1 -1 -1 1 -E -1 -1 -1 - [ -1 -1 -1 - [ 2 -1 -1 - [ -1 -1 -1 - [ -1 - [ -1 - [

5 b 5c l Ob 10 c 2 b 2b lOb 10c

-4 5 1 1 5 1 l 5 -1 - [ 5 -1 - [ 2 2 6 6 -2 -2

-6 -6

-6 -6 4 -[ 4 -1 -4 1 -4 1

4

4

-4 -4 -4

-1

5 -5

-[

4 -1 4 1 -4 1 -4

10 -5 -10

-10 15 -10 15 -10 -10 -10 -10 10 -10

-5 5

-5 -5

5 -5

-4 4 -4

-3 -3 4 -4 -5 6

-2 -2 2 2 -2 -2 2 2

Lyons' group Ly

610

Character table of L = NR(V)

2P 3P 5P 7P !IP 13P 17P 19P 23P 29P X I X 2 X .3 X 4

X.5 X 6 X 7 X .8 X.9 X 10 X 11 X 12 X .13 X 14 X 15 X. 16 X I7 X 18 X .I 9 X 20 X . 21 X 22 X 23 X . 24 X 25 X .26 X .27 X 28 X 29 X .30

10d lOe lOJ 10g lOh 12a 12b 15a 20a

2 Ob 2 De

20d

20e

20/ 20g

24a

24b

2 4e

24d 25a 30a

5g 5d 5 e 5h 6a 6a 15a 10a !Oh 4b 4a 5a 20a lOe lOJ 2a 2b 2a 12a 12b 3a 4e lOe 10g lOh 12b 12a 15a 20a 10g lOh 12b 12a 15a 20a lOe l Og I Oh 12a 12b 15a 20a lOe lOe 10g 10h 12a 12b 15a 20a 10e l Og lOh 12b 12a 15a 20a lOe 10g IOh 12b 12a 15a 20a lOe 1 Og lOh 12a 12b 15a 20a

I Ob !Od

!De 20e

I De 20d 4e 20d 20d 20e 20e 20d 20d 20e

10b 20b 4d 20b 20b 20/ 20/ 20b 20b 20/

20g 20g 20c 20c 20g

12a Sb 24a 24c 24e 24d 24d 24b 2 4b 24a

12 b Sa 24b 24d 24d 24c 24e 24a 24a 24b

12 b Sa 24c 24a 24a 24b 24b 24d 24d 24e

l2a 25a 8b 25a 24d 5 a 24b 25a 24b 25a 24a 25a

I

I

C -1

-C

C

C

-C

-1

-1

-1

-1

-C -1

C

-C

-C

C

-1 -1

-1 I

-1 I

-1 I

-] I

-C C C C -C -C F /F -/F -F -/F /F

-C C -F F

2a 10d 10d 10d 10d 10d 10d 10d I

I I I I I I -1 -1 -1 -1 -1 -1 -1 -1 I

-1

-1

I

I -1 -1 -1 -1 -1 -1 -1 -1

I -1

I -1

20b 20c 2 Og 20g 2 Oc

-1

-1

I -1 -1 -1 -1 I

-1 -1 2 -2

-1

-1 -1

-1

-1 2 -2

I

-C C C -C E -E C -C C -C -E E

-C -C -C C C C

-1 -1 I I

-2

-2 -1 -1 -1 -1

C

-C

I

I

-1

I

C C C -C -/F -C /F -C

1

-F F

24c 25a 24c 25a 24d 25a

F /F -F -/F

-1 l

-1 l

-1 I

-1

I -1

-1

-1

C -C

C -C

C -C

-C C

-C C

C -C

-1

-1

I

I

-1

-1

I -1

C -C

-C C

-1

-2 2

-1 -1 -1 -1 -1 -1 -1 -1

-1 -1 I

-1 I

-C C

C -C -1 -1

-3 -3

6a 30a 30a 30a 30a 30a 30a 30a

-1 -1 -1 -1

2 3

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

15a 10a

D -D C -/D -C /D D C /D -C -/D -D D C -D -/D /D -C -D C D /D -/D -C

-1 -1

2 -3

-1

24a 25a

-C C

-2

-1 -1 -1 I

C

-]

20c

-1 I

-1 -1

4 -1

I

-C

lOd 20 C 4a 20c

-C C -2

-1 -1 -1 -1 -1 -1

-1 -1 -1

-1 -1

-1 -1 I I

-1

C

-1 I

20e 20d 20d 20e 20 e 20d

1

I

-C -1

I I

X .31 X .32 X .33 X.34 X .35 X .36 X.37 X .38 X.39 X .40 X .41 X.42 X .43 X.44 X .45 X .46

53 .(5 2 :GL2(5)) (continued)

S"

-2 2 3 -3

-3 2 -3 4 -1

-4

-1

-1 -1 I

-1 -1

-4

-4i, B = 6i, C= -i, D = - i - 1, E where A and~ is a primitive 24th root of unity.

2i, F

2e

3

+ ~5

14 Suzuki's group Suz

In 1969 M. Suzuki discovered his sporadic simple group Suz. It is a primitive rank 3 permutation group [124]. Soicher's nice presentation of Suz is stated in [110], p. 98. It also provides generators for the stabilizer T Sc' G 2 ( 4) of the rank 3 permutation representation of degree 1782 of Suz. In 1974 Wright determined the character table of Suz [139]. In particular, he showed that Suz has two conjugacy classes of involutions. Furthermore, he determined the abstract group structure of the centralizer H = Csuz(z) of a 2-central involution z and of some other local subgroups of Suz. In [108] Patterson and Wong showed that each finite simple group G with a centralizer Cc (z) Sc' H of a 2-central involution z E G is isomorphic to Suzuki's simple group Suz. Their uniqueness proof is not self-contained. This chapter provides another existence and uniqueness proof for Suzuki's sporadic group. It is self-contained except for a quotation of Phan's uniqueness theorem of U4 (3) of [109] which can also be proved completely by means of the methods and results of [92]. The sporadic Suzuki group can be constructed by means of Algorithm 1.3.8 from an iterated extension E of an irreducible subgroup L Sc' 3A6 of G L5 ( 2). Let E 1 be the split extension of L by the natural 6-dimensional vector space V1 over GF(2). Then E 1 has an irreducible 4-dimensional representation½ over GF(2) on which the center of L acts trivially. The group E is a non-split extension of E 1 by ½; see Proposition 14.2.2. It turns out that ½ is the unique maximal elementary abelian normal subgroup A of any Sylow 2-subgroup of E. However, the group E is not uniquely determined by the irreducible subgroup L of GL 6 (2). In view of the limited space we do not dismiss here all the extensions which do not lead to a simple group. Instead we start our construction of a simple group G of Suz-type from a presentation 611

612

Suzuki's group Suz

of a finite group H which is isomorphic to the centralizer Ca (z) of a 2-central involution z of Suzuki's sporadic group Suz. It is stated in Lemma 14.1.1 of Section 14.l. A classification of the conjugacy classes and the character table of H are derived from this presentation also. The presentation of His due to A. Stein (University of Kiel). Using Algorithm 7.4.8 of [92] we construct in Section 14.2 the correct iterated extension E which is isomorphic to Na (A). In order to prove that Na (A) is uniquely determined by the given presentation of H we also show in Section 14.2 that any simple group G with such a 2-central involution centralizer has two conjugacy classes of involutions represented by z and u. Thus we are able to determine the fusion of the conjugacy classes of H and E = Na (A) in G; see Proposition 14.2.2. Furthermore, we derive in Proposition 14.2.3 a presentation of U = Ca (u), its character table and the fusion of the conjugacy classes of U in G. Thus Theorem 1.6.4 allows us to show that all simple groups G of Suz-type have order jGj = 213 · 37 · 52 · 7 · 11 · 13; see Proposition 14.2.4. In Section 14.4 we apply Algorithm 7.4.8 of [92] to H and construct a simple subgroup \8 of GL 143 (13) having a 2-central involution J such that CG (3) ~ H; see Theorem 14.3.l. Using the algorithm described in Theorem 6.2.l of [92] we constructed a faithful permutation representation P\8 of \8 of degree 135135. The character table of \8 has been calculated by means of P\8 and MAGMA. It is equivalent to that of Suz given in the Atlas [19], p. 128. In Section 14.5 it is shown that any finite simple group G having a 2-central involution z such that Ca (z) ~ H is isomorphic to \8; see Theorem 14.4.10. In particular, \8 ~ Suz. Hence Patterson's and Wong's Uniqueness Theorem is a consequence of Theorem 14.4.10. Sections 14.6 and 14. 7 contain the systems of representatives of the conjugacy classes and the character tables of the local subgroups dealt with in this chapter. They are essential tools for the given existence and uniqueness proofs of Suz. The four generating matrices of \8 are documented on the accompanying DVD.

14.1 The centralizer of a 2-central involution

The presentation of the following group His due to A. Stein (University of Kiel). So is the faithful permutation representation PH of H. Several properties of the abstract group H were observed first by Patterson and Wong [108], Suzuki [124] and Wright [139].

14.1 The centralizer of a 2-central involution

613

Lemma 14.1.1 Let H = (h; I 1 :S i :S 16) be the finite group with the following set R(H) of defining relations:

= h~ = h§ = h~ = h~ = hio = hi1 = hi2 = hr3 = hi4 = hr 5 = 1, h~ = h~ = h15, h~ = h~ = h1, hi 6 = 1 and (h15,h;) = 1 for all 1 :Si :S 15, (h1,h2) = (h1,h3) = (h1,h4) = (h1,h5) = (h1,h1) = 1, (h1,h6) = h15, (h2, h3) = (h2, h4) = (h2, h6) = (h2, h1) = 1, (h2, h5) = (h3, h4) = h15, (~,~)=(~,~)=(~,~)=(~,~)=(~,~)=(~,~)=1, (h5,h6) = (h1,hs) = (h2,hs) = (h3,hs) = (h5,hs) = 1, (h1,hg) = (h2,hg) = (h4,hg) = (h5,hg) = 1, (h1,hs) = (h1,hg) = (hs,h10) = (hg,h10) = (hn,h12) = (h1,hu) = 1, (h1, h12) = (h1, h13) = (h10, h13) = (h13, h15) = 1, (h1,h10) = (h3,h10) = (h4,h10) = (h5,h10) = (h1,hu) = (h2,hu) = 1, (h6, hn = (h1, h12) = (h2, h12) = (h6, h12) = (h1, hrn) = (h2, h13) = 1, (h5,h13) = (h5,h13) = (h1,h14) = (h6,h14) = (h3,h15) = (h4,h15) = 1, (h1, h10) = (hs, hg) = (hg, hu) = (hs, h12) = h1, h 1}hshuh1hs = h 1}h10huhgh1oh1 = h 91h 1}hgh12h1 = 1, h 1}h1}h1oh12h1hs = h 1}hgh13hs = h 1}huh13h12 = 1, h 1} h 1} h12h13hu = h 1_l h1h14h10 = h 1 } h1oh14h7 = 1, hs 1h1}hsh13h9 = h-;_lhsh14h9 = h-;_lhgh14h3 = h1, (huh14) 5 = (h12h14) 5 = 1, (h13h14) 2 = 1, (h10h15) 2 = 1, h1 }h1 }h1}huh12h14 = h 1a1h 1lhnh14h13h14huh12h14 = 1, (h1,h15) = h1h16, hs 1h1lhsh15h1h12 = h4, h 91h 1lhgh15h1hn = h2h3h4, h 1lhnh15h9 = h3h4, h1lh12h15h3 = h1h4, (h15,h14,h15) = h1h16, h~ 7 = h1h5h16, h~ 7 = h2h6, h} = h1h4h16, h~B = h1h3h6, h~g = h1h3h16, h~g = h1h4h6, h~IO = h1h2h15, h~IO = h5h5, h~ll = h2h3h15, h} 11 = h4h16, h~ll = h2h4h5, h~ 12 = h3h15, h} 12 = h2h4, h~ 12 = h2h3h5h16, h~ 13 = h3h4h16, h} 13 = h3h15, h1h = h1h2, h2h 15 = h1h16, h5h 15 = h5h6, (h5, h13) = (h6, h13) = (h1, h14) = (h6, h14) = (h3, h15) = (h4, h15) = 1, hi

8

15

h14 -_ h 5, h2

hh14 3 -_ h4 ,

hh14 4 -_ h 3,

hh14 5 -_ h 2,

hh15 _ h 5· 6 -

614

Suzuki's group Suz

Then the fallowing statements hold.

(a) H has a faithful permutation representation of degree 1728 with stabilizer Ho = (h1, h1, hs, hg, h11, h12, h13, h14) of order 27 · 3 · 5. (b) The subgroup S = (h; I 1 s:; i s; 12) is a Sylow 2-subgroup of H with center Z(S) = (z), where z = h 16 . Furthermore, Z(H) = Z(S) is the center of H.

(c) V = (z, t) is the unique Klein 4-subgroup of S, where t = h1. (d) CH(V) has center Z[CH(V)] order 2 in NH(V).

=

V and it has a complement of

(e) A= (z,t,h 2 ,h7 ) is the unique maximal elementary abelian normal subgroup of S, and its order is 16.

(f) C = Cs(A) = (z,h1,h2,h1,h3,h4,h7,hs,h11,h12) is a special 2group of order 210 with center Z( C) normal Klein 4-group of C.

=

A, and V is the unique

IDI =

213 ·3 2, C is normal in D and has a complement K = (h 5, h4, h 13 , h 15 ) in D; its center Z(K) = (h 13 ) has order 3 and K/Z(K) ~ D/C ~ S4 •

(g) D = NH(A) = (C,h5,h5,h10,h15) has order

(h) H is a perfect non-split extension of U4 (2) by its extra-special normal subgroup P = 02 (H) = (z, h1, ... , h5) of order IPI = 128. Furthermore, P and Z(H) = Z(P) are the only non-trivial proper normal subgroups of H.

(i) D = (x, y), where x = hsh13h5h10h15 and y = h5h12h11h15 have orders 24 and 12, respectively.

(j) H = (x, y, h), where h = h14 has order 2. (k) A system of representatives h; and the corresponding centralizer orders ICH (h;)I of the 57 conjugacy classes hf of H = (x, y, h) are given in Table 14.5.1.

(l) A system of representatives d; and the corresponding centralizer orders ICD (d;)I of the 77 conjugacy classes df of D = (x, y) are given in Table 14.5.3

(m) The character table of His given in Table 14- 6.1. (n) The character table of D is given in Table 14.6.3. Proof (a) The permutation representation of H given in (i) is due to A. Stein. Using MAGMA it has been checked that the permutation representation PH of H is faithful.

14.2 Even conjugacy classes and group order

615

(c) and (e) The reader can verify these two statements using PH and the MAGMA command

Subgroups(X : Al:=Normal,IsElementaryAbelian :=true), where XE {S,C}. (k) and (1) The author used PH and Kratzer's Algorithm 5.3.18 of [92] to calculate the tables of the representatives of the conjugacy classes of Hand D. (m) and (n) Both character tables have been computed by means of MAGMA and the given permutation representations. All other statements have been established by using standard MAGMA commands and the faithful permutation representation PH of H. D Definition 14.1.2 A finite simple group G is said to be of Suz-type if it has a 2-central involution z such that Ca ( z) is isomorphic to the finitely presented group H stated in Lemma 14.1.1.

14.2 Even conjugacy classes and group order

In this section we determine the conjugacy classes of even order and the order of a finite simple group G of Suz-type. Proposition 14.2.1 Let G be a finite simple group of Suz-type. Let A = (z, h1, h 2 , h 7 ) be the unique maximal elementary abelian normal subgroup of the Sylow 2-subgroup S of H = Ca(z) defined in Lemma 14.1.1. Then the following statements hold.

(a) H has four conjugacy classes of involutions. They are represented by the generators z, t = h1, h1 and h14 of H defined in Lemma 14.1.1. (b) T = CH(h14) is a self-normalizing subgroup of the centralizer U = Ca(h14), and ITI = 29 • (c) T is a Sylow 2-subgroup of U.

(d) z, t, h1 r/. hf4 and z 0 = t 0 or z 0 = M/. (e) If zg = t for some g E G then g E Na(V) - NH(V) and Na(V)/Ca(V) = Na(V)/NH(V) ~ GL2(2). (f) If zg = t for some g E G then Ne (V) is isomorphic to the finitely presented group M = (mi 11 :Si :S 8) with the following set R(M)

Suzuki's group Suz

616

of defining relations:

m 31 = m 22 = m 33 = m 43 = m 35 = m 72 = m 82=1 ,

m 62m7 = 1,

(m1,m3) = (m1,m4) = (m1,m5) = (m2,m3) = (m2,m4) = 1, (m2, m7) = (m3, m1) = (m4, m1) = (m5, m7) = (m6, m7) = 1, (m3,ms) = (m4,m) = (m5,ms) = (m6,ms) = (m1,ms) = 1, (m2m6) 4 = (m 41m6m31m6) 2 = (m11m2) 2 = 1, (m31m 41)2 = 1, (m41m 51)2 = 1, (m6m11)3 = 1, m 1-1 m7m1m8-1 = m 1-1 msm1m 7-1 ms-1 = 1 , m 2-1 msm2m 7-1 ms-1 = 1, m2m 5-1 m2m5 = m3m5-1 m 3-1 m5m4-1 = 1, (m4m5m5 )2 = (m6m 1-1)3 = m2m5m5-1 m 4-1 m2m5m4m5m7-1 = 1, 1 m 4-1 m6m2m6m4m5m2m5=m3m6m3-1 m5m3m5m3-1 m5ms-1 =, m 1-1 m5m 3-1 m5m2m 1-1 m6m3m5m2m7-1 ms-1 =1, m6m2m1m5m3-1 m 1-1 m5m 3-1 m5m2m 3-1 m 8-1 =1.

(g) M has a faithful permutation representation PM of degree 1024 (h)

and stabilizer (m1, m2, m3, m4, m5). Let x1 = (m~m5m3m5)4, x2 = (m3m5m~m5)4, X3 = (m2m~m5 m6m3) 3, and r1 = (m2m6m3m5m3mg) 2, r2 = m3m5m5m3m§m5 m2, r3 = m3m5m5m6m~m§m3. Let W1 = xrx2X3 and W2 = (r2r~) 3 . Then z = is a 2-central involution of M with centralizer DM = CM(z) = (w1,w2) and M = (w1,w2,m), where m=m1. A system of representatives mi and the corresponding centralizer orders ICM(mi)I of the 49 conjugacy classes ofmf1 of M = (w1,w2,m) are given in Table 14.5.5. If z 9 = t for some g E G then there is an isomorphism T : NH(V) = (v1,v2) --. DM such that w1 = T(vi) and w2 = T(v2), where v1 = d1d2d3d5 and V2 = dtd3(d1d2d3) 2d5d4d5, and d1 = hs, d2 = h14, d3 = (h14h15) 6 , d4 = h15hsht 5, d5 = (h15hsh14ht 5)2. If z 9 = t for some g E G then G has two conjugacy classes of involutions. They are represented by z and h = h 14 . If z 9 = h1 for some g E G then there is a 2-element y E Na (A) NH(A). In any case, Na(A) > NH(A).

Wi

(i)

(j)

(k) (1)

(m)

14.2 Even conjugacy classes and group order

617

Proof (a) Let w = h 14 . Using MAGMA and the faithful permutation representation PH of H given in Lemma 14.1.l(a) one sees that w ES, \CH (w)\ = 29, \CH (h1)I = 212 .3.5 and \CH (h1)I = 210 ·3 2. Furthermore,

{z} U {hf} U {hf} U {wH} is the set of all involutions of H. (b) Let U = Cc(w) and T = Cc(w). Using MAGMA again we checked that Z(T) n T" = (z). Therefore each c E Nu(T) belongs to Cu(w) n Cu(z)

= CH(w) = T.

(c) This assertion follows immediately from (b). (d) Let u E {z,h 1 ,h7}. Suppose that w E uc. Then \Cc(w)\ = \Cc(u)\. By Table 14.5.1 a Sylow 2-subgroup of Cc(u) has order at least 210 because h = h 14 . Now (c) provides a contradiction. In particular, G has at least two conjugacy classes of involutions represented by z and h. Now Theorem 4.7.3 of [92] due to Glauberman implies that that zg = t or zt = h 7 for some g E G. (e) Suppose that xg = t and let Y = Cc (t). Then Cy (t) is properly contained in a Sylow 2-subgroup T of Y. By Lemma 14.1.1 V is the unique normal Klein 4-subgroup of S and it is the center Z (Y) of Y. Hence (e) holds by Lemma 1.4.4 of [92]. (f) By Lemma 14.1.l(d) Vis the center of W = Cc(V) = CH(V). Using the faithful permutation representation PH and MAGMA the author checked that X = (h 2, h3, h4, h5 , h 7, h8 , h9 , h10) is a unique normal subgroup of order 210 of W. Its derived subgroup X' is the center of V and equals W. Hence Xis normal in N1 = NH(V) = (h5,h7,h11,h12,h14). It has a complement Q = (q1, q2) with center Z( Q) = ((q1q2 )5) oforder 3 and Q/Z(Q) ~ A5, where q1 = h5h14h12h14 and q2 = (h5h7h11h12h14h7 ) 4. Now Theorem 1.4.15 of [92] and (e) imply that Nc(V) is isomorphic to the split extension M of the direct product of the symmetric group S 3 of order 6 and the alternating group A 5 by X. In order to construct M we calculated the automorphism group AW of W = Cc(V) using MAGMA and PH. It has order 212 -3 2 ·5. MAGMA provides a faithful permutation representation PAW of degree 320. Using it and MAGMA the author observed that the Frattini subgroup F of the group I of inner automorphisms has a complement K in AW of order 24 · 32 · 5. Furthermore, \K'\ = 22 · 32 · 5. In particular, K' is isomorphic to a direct product of a cyclic group (c) of order 3 and L ~ A 5 • MAGMA also provided three generators aj of order 3 for L. Using PAW

618

Suzuki's group Suz

and the command

exists(m){x : x in Kl Drder(x) eq 2 and c-x eq c-2} we find an involution m of K such that K = (c, m, L). Clearly, F ~ X/V is elementary abelian. Let B = {f;ll::::; i::::; 8} be a basis of F. The conjugation actions of the generators of K 1 induce endomorphisms of M having the following matrices w.r.t. B: 11001000 01000000] 10101000 01001111 Mc= ( 01100000 , 11101101 01011110 01001100

M

a2

11000000 (10000000] 10100000 00001000 Mm= 01101000 , 10000100 01010110 00100101

00101000 11100000] 10101000 11110000 = ( 11 OOOOOO , 10001100 11100010 11100001

M

a3

00001010 (00110100] 10111110 00010000 Ma,= 11111110 , 01001110 11010110 00100011

00101101 (01000100] 11001101 01010100 = 11 0 O1 OOO · 11101001 11101111 10000001

The subgroup MK= (Mc, Mm, Ma 1 , Ma 2 , Ma 3 ) of GL 8 (2) is transformed into a finitely presented group K = (k; 11 : : ; i ::::; 5) by means of the MAG MA command FPGroup (MK). Its set R( K) of defining relations consists of the following relations:

= k~ = kl = kj = kg = 1, (k1,k3) = (k1,k4) = (k1,k5) = (k2,k3) = (k2,k4) = (k1 1 k2) 2 = 1, (ki 1 ki 1)2 = 1, (ki 1k5 1 )2 = 1,

kf

(k2,k5)

= 1,

1 k- 1 - 1 k 35 k -lk3 k5 4-.

Then we construct its faithful permutation representation PK of degree 30 with stabilizer (k3 , k4 ). Thus we can apply Holt's Algorithm to calculate the second cohomological dimension of K with coefficients in F; it is 0. Since K acts irreducibly on F we use the MAGMA commands Q: =Getvecs (MK) and Semidir (MK, Q) to construct the uniquely (up to isomorphism) determined split extension K 1 of K by F. Its set R(Ki) of defining relations consists of R(K) and the following relations: kl

= 1,

(k4k6k5) 2

= 1,

k2k6k5 1 ki 1 k2k6k4k5 (k3k6k3 1 k6) 2

= 1,

=

(k6k1 1 )3

= 1,

ki 1k6k2k6k4k6k2k6

(ki 1 k6k3 1 k5) 2

(k2k6) 4

= 1,

= 1,

= 1,

k1 1 k0k3 1k6k2k1 1 k5k3k6k2k0k2k1k6k3 1k1 1k6k3 1k6k2k3 1

= 1.

14.2 Even conjugacy classes and group order

619

Let P K1 be its natural permutation representation of degree 256 with stabilizer K. In order to construct the extensions of K 1 by V we now use the matrices belonging to the conjugate actions of the generators m = k2 and c = k1 of K on V because all other generators of K 1 act trivially on V:

Let Nki

=

I be the identity matrix in GL 2 (2) for i

= 2, 3, ... , 6 and

FEalg :=MatrixAlgebra, Vi := GModule(PK_i, FEalg). Application of the MAGMA command CohomologicalDimension(PK_i, Vi, 2) yields that this dimension is 1. Since the Sylow 2-subgroup S of H does not split over its normal subgroup V by Lemma 14.1.l(c) we know that Nc(V) is the non-split extension M = (mJll :S: j :S: 8) of K 1 by V. Its presentation is given in the statement. (g) This statement can easily be verified by means of MAGMA. (h) Using the faithful permutation representations PH of H given in Lemma 14.1.l(a) and PM of Mall assertions of this statement can be checked computationally by the reader. (i) The author used PM and Kratzer's Algorithm 5.3.18 of [92] to calculate the table of the representatives of the conjugacy classes of M. (j) Since z 9 = t for some g E G not in H assertion (f) implies that Ne (V) ~ M. Using PH and MAGMA the author verified that D 1 = NH (V) = (dk I1 :S: k :s; 5) for the elements dk given in the statement. Furthermore, D 1 = (v 1 ,v2 ). Using PH and PM and MAGMA we found an isomorphism T: D 1 --+ DM satisfying T(v 1 ) = w 1 and T(v 2) = w 2. (k) An application of Kratzer's Algorithm 5.3.18 of [92] provides a natural bijection between the conjugacy classes of D 1 and DM. In particular, we determined the fusion of the D 1 -classes in H and also the fusion of the DM-classes in M. For the eight conjugacy classes of involutions of D 1 = DM we obtained the following fusion patterns into the four classes of involutions of H and the six classes of involutions of M using the notation of Tables 14.5.1 and 14.5.5, respectively: ([1, 2, 3], [1, 2, 3, 4, 5], [1, 2, 3]),

([4], [6, 7, 8], [4, 5, 6]),

Suzuki's group Suz

620

where the classes of H and M are denoted by the indices of the involutions of the respective tables, and where the classes of H are given before the ones of D. The final triples denote the classes of M. In particular, (d) and Tables 14.5.1 and 14.5.5 imply that z E 21 , t E 22 and h7 E 23 of H are conjugate in G, because the involutions of M fuse only to z and h in G. (1) If z 9 = h 7 for some g E G then g ¢:. H. By means of another MAGMA calculation with PH the author observed that the derived subgroup S7 of the centralizer C 7 = CH (h 7) has order 210 . Therefore it is a Sylow 2-subgroup of C7 by Table 14.5.1. It also has been verified computationally that its center Z(S7 ) = A. Hence 8 7 is properly contained in a Sylow 2-subgroup T of Cc (h 7 ) and T contains a subgroup T1 such that IT1 : 8 7 = 2. In particular, the characteristic subgroup A of 8 7 is normal in T1 . Hence there is a y E T1 such that y E Nc(A) - NH(A). (m) By (1) we assume that z 9 = t for some g E G - H. Then G has exactly two conjugacy classes of involutions by (k). Hence z and h7 fuse in G and Ne (A) > NH (A) by (1). This completes the proof. D 1

Proposition 14.2.2 Let G be a finite simple group of Suz-type having a 2-central involution z with centralizer H = Cc ( z) defined in Lemma 14.1.1. Let A= (z, h 1 , h 2 , h 1 ) be the unique normal subgroup of order 16 of the Sylow 2-subgroup 8 and let D = NH(A) = (x, y) be its normalizer, where the generators x and y defined in Lemma L4,.1.1(i). Then the fallowing statements hold.

(a) Nc(A) is uniquely determined up to isomorphism and it is isomorphic to the finitely presented group E = (e;ll :S: i :S: 13) with the following set R(E) of defining relations:

(es, eg) = (e2, e11) = (e4, e10) = (e4, e11) = (e4, e12)

= (e4, e13) = 1, (e5, e10) = (e5, e11) = (e5, e12) = (e5, e13) = (e6, e10)

= (e5, eu) = 1, (e5, e12) = (e5, e13) = (e7, e10) = (e7, eu) = (e7, e12) = (e7, e1 3) = 1, (es, e10) = (es, eu) = (es, e12) = (es, e13) = 1,

621

14.2 Even conjugacy classes and group order (eg, e10) = (eg, en)= (eg, e12) = (eg, e13) =

1,

1,

(e10,e11) = (e10,e12) = (e10,e13) = (e11,e12) = (e11,e13) = -1 -1 -1 ( e12,e13 ) = e 1-1 e10e1e -1 ell = e 1 elle1e 10 = 10

1,

1,

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 e 1 e12e1e 10 e 13 = e 1 e13e1e 11 e 12 e 13 = e 2 e1oe2e 10 e 11 = -1 -1 -1 -1 -1 -1 -1 e 2 e12e2e 10 e 12 = e 2 e13e2e 10 e 12 e 13 -1 -1 e3-1 e11e3e 10 e 11 -1

=

-1 e3-1 e12e3e 11

-1 -1 -1

-1 -1 1 = e3-1 e10e3e12 e 13 = ,

-1 -1 -1 = e 4-1 e6-1 e4e5e10 ell e 12

-1 -1

-1

e3 e13e3e 10 e 11 e 12 = e 4 e 5 e4e5e 11 =

1

=

1,

,

-1 -1 -1 -1 -1 -1 -1 -1 1, e 5 e 6 e5e5e 11 e 4 e7 e4e7e 10 e 11 e 12 = -1 -1 -1 -1 -1 1 e 5 e7 e5e7e 10 e 11 e 12 -1 -1 -1 -1 -1 -1 -1 -1 e6-1 e7-1 e5e7e 11 e 4 e 8 e4ese 11 e 5 e 8 e5ese 10 e 11

=

= ,

=

-1 -1 e6-1 e 8-1 e5ese 11 e 13 -1 -1 -1 e 4 e 9 e4ege 10

= =

=

-1 -1 -1 e7-1 e 8-1 e7ese 10 e 12 e 13 -1 -1 -1 1 =

= 1,

= 1,

= ,

e 5 e 9 e5e9ell -1 -1 -1 -1 1 e6 e9 e5e9e 12 e7-1 e9-1 e7ege 13 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 e 1 e4e1e4 e 8 e 9 e 12 e 13 e 1 e5e1e 5 e 8 e 11 = -1 -1 -1 -1 -1 -1 -1 -1 1 e 1 e 6 e 1 e 4 e 5 e6 e7 e 10 e 11 e 12 -1 -1 -1 -1 -1 -1 -1 -1 1, e 1 e7e1 e6 e 8 ei 3 = e 1 ese1 e 9 e 11 e 12 =

= ,

=

1

,

= ,

-1 -1 -1 -1 -1 -1 e 1 ege1e 8 e9 e 11 e 12 e 13

-1 = e 2-1 e4e2e4-1 e 10 = 1,

-1 -1 -1 e 2 e5e2e 5 e 12

-1 -1 = e 2-1 e5e2e4-1 e 5-1 e6-1 e9-1 e 10 e 13 = -1 -1 -1 -1 -1 -1 -1 -1 1 e 2 e7e2e 4 e7 e 8 e 9 e 11 e 12 e 13 = , -1 -1 -1 -1 -1 -1 -1 -1 -1 1 e 2 ese2e4 e 8 e 11 = e 2 e9e2e 5 e 9 e 11 e 13 = ,

e3-1 e4e3e 4-1 e 5-1 e 8-1

=

1,

-1 -1 -1 1 e3-1 e5e3e4-1 e 9-1 e 10 e 11 e 13 = ,

-1 e3-1 e7e3e 5-1 e7-1 e 8-1 e 13 = 1, -1 -1 -1 -1 -1 -1 -1 -1 1 e 3 e 8 e3e4 e6 e7 e 8 e 10 e 11 e 12 = , -1 -1 -1 -1 -1 -1 -1 1 e3-1 ege3e4-1 e6-1 e9-1 e 10 e 13 = e 2 e 3 e2e 1 e3 e1e10 = , -1 -1 e3-1 e5e3e4-1 e 5-1 e6-1 e 9-1 e 10 e 11

=

-1 -1 -1 -1 -1 -1 -1 -1 -2 -1 -2 -1 -1 -1 e3 e1 e3 e1 e3 e1 e10 e13 = e3 e1 e2 e1 e3 e2 e10 e12 e13 -1 -1 -1 -1 -1 -1 -1 1. e2e 3 e 2 e 1 e3 e2e1e 10 ell e 13 =

1

= '

(b) E has a faithful permutation representation of degree 1536 with stabilizer (e3, (e1e4e3e1e4) 2 , (ere3e1eD 5 ). ( C) Let d1 = ( e1 e4 )3, d2 = e3 e1, d3 = (e1 e~ )2, d4 = e~ ere~ . Let x1 = (d4d1d3) 4 (d1d3d2d1d2d4) 23

Yl

and

= (d1 d3)2 (d2d1 d4d2)2 ( d4d2d4C4C5) (d1 d3) (d4d2d4) (d1 d3 ).

622

Suzuki's group Suz Then z = xP is a 2-central involution of E with centralizer DE = CE (z) = (x 1, y1) and there is an isomorphism a- : D ---+ DE such that x1 = a-(x) and YI = a-(y). Furthermore, E = (x1, Y1, e1).

(d) A system of representatives (e;) and the corresponding centralizer orders /CE (e;) I of the 47 conjugacy classes of E = (x1, Y1, e1) are given in Table 14- 5.2.

(e) The character table of E is given in Table 14.6.2. (f) Each finite simple group G of Suz-type has 2 conjugacy classes of involutions represented by z and u = h of H = (x, y, h). The classes 21, 22 and 23 of H belong to z 0 and 24 belongs to u 0

(g) The Goldschmidt index of the amalgam H

+--

D

---+

.

E is 1.

Proof (a) As in Lemma 14.1.1 we let H = (x, y, h). Then D = NH (A)= (x,y), where A= z,t,h 2 ,h7 is the unique maximal elementary abelian normal subgroup of the fixed Sylow 2-subgroup S of H. Furthermore, C = C8 (A) has a complement Kin D. Since D contains S we applied the program GetShortGens (D, S) and obtained the four generators of s 1 = x 6 , s 2 = (yx) 3 , s 3 = (x 2y) 3 and s4 = (yx 2)3 of S. Similarly, we observed that z = sr, t = s§, a 1 = S§ and a 2 = (s 1s 3s 1) 2 generate A. The normal subgroup C of D is generated by b1 = S3, b2 = S4, b3 = (s1s3)2, b4 = (s1s4)2, b5 = (s2s3) 2 and b5 = (s2s4) 2 . With respect to these six generators the center Z (C) = A is generated by z1 = (b1)2, z2 = (b2)2, Z3 = (b3) 2 and Z4 = (b1b2) 2. The elements k1 = (x 4y 2) 4 and k2 = (x 3 yx 2 yx 2 ) 3 generate the complement K of C in D. Hence D = (b;, kj I1 :S: i :S: 6, 1 :S: j :S; 2). Using the permutation representation PH and the command FPGroup(D) MAGMA established the following set R(D) of defining relations of D:

14.2 Even conjugacy classes and group order

623

1 2 111l 1-li4 11 4 11-l 2 -- 11 115 112 111 1! 2f 5- -1, 1-l1-l1-l1-l 1-l -- 1-11-11-11 1l-l J6 2 1 2 J6 2 7 2 1f 7I1 -- 1 , f 8-li1 113 111f 81-l 3 -- f 712 113 11217 114--1, 1J2 1 1- 1 - 1- 11-21 1- 11-l - 1 .f:- 3 f8 f -l123 3-53151-, 1 1 ls 1513 lifsfi = lsl613-l 12-l ls- 112-l = 1, 1-11-11-lf !:8f 2-11-lf 4 2f-11-l 8 4 -- f-l 8 J6 4 2 81-l 2 -- 1, 1-l J5 1-l11-l 1-l -- i-lf-21-112 f 7J6 1 7 J6 5 8 J5 8 -- 1 , I - 1-11 1-11-11-11-lf - 1 !:8-11-11-21: J5 1 81-l 4 J5 - J6 4 2 1 7 5 7- , 1 f f f-l -- 1· f 7-lf-lf-lf8 7 8 ! 7878 Therefore a set R( D 1) of defining relations of D 1 = D / A consists of R(D) and the relations ff = 1, 1, fr = 1, (!1!2) 2 = 1.

Ji=

Let

=

Ii A

E D 1. Then the elementary abelian normal subgroup B = C/A = (qill S i S 6) has a complement K = (k1, k2) in D1, where k1 = q7 and k2 = Qs. Thus D1 has a faithful permutation representation P D 1 of degree 64. The conjugate actions of k1 and k2 induce endomorphisms given by the following matrices w.r.t. the basis B := {q1,Q2,q3,q4,q5,q6}: Qi

100011) 010010 111100 M k1 = ( 0 0 1 0 1 0 , 000001 000011

100000) 010000 111001 ( Mk 2 = 1 OO111 · 100010 010001

Let MK be the subgroup of MG= GL 6 (2) generated by the matrices Mk 1 and Mk 2 . Its center Z(MK) is generated by Mc = (Mk1Mk2) 4 of order 3. Using MAGMA it has been checked that MR= CMc(Mc) has order IM RI = 26 · 34 · 5 · 7. It has a simple composition factor PSL3(4). By Proposition 14.2.1 Nc(A) > NH(A) in any simple target group G of Suz-type. Thus INc(A)/A: D 1 1 is a positive odd number. It is not divisible by 7 because z and h1 in A are G-conjugate by Proposition 14.2.1. Furthermore, A contains eight elements which are H-conjugate to h7 . Therefore all non-trivial a E A are G-conjugate. Hence an element y E Nc(A) of order 7 must have a non-trivial centralizer CA(y). But IH = Cc(z)I = 213 · 34 • 5.

Suzuki's group Suz

624

Using the standard faithful permutation representation PG63 of MG= GL 6 (2) of degree 26 - 1 and the MAGMA command

Subgroups(MR: OrderEqual := 2-3*3-b*5-c), where b E {2,3,4} and c E {0,1}, the author has checked that up to conjugacy there is exactly one subgroup of MR which has a dihedral Sylow 2-subgroup as MK and has an order of the above form. It follows that b = 3 and c = 1. In particular, Nc(A)/A has a non-trivial Sylow 5-subgroup. Hence it is of order 5. Using the command

exists(q){x: x \in MRI Order(x) eq 5 and Drder(sub) eq 1080} MAGMA established a q E PG63 corresponding to the matrix

110010) 100001

Mk 3

111001 = ( 0 10 110 101110 101001

·

The subgroup MX = (Mk 1,Mk2,Mk3 ) has order 23 · 33 · 5. As before we transform M X into a finitely presented group F X by means of the command FPGroup (MX) . It has the following set R( K) of defining relations:

= ki = k~ = 1, 1 k:; ki:1k2k1 1ki: 1k1 = (k;;:1k-:; 1) 3 = 1, k2k3 1k2 1k1 1k3 1k2k1 = k3 2k1k2 1k1k3 2k2 kr

= 1.

Since X is given as a matrix group and as a finitely presented group we can apply Holt's Algorithm stated as Algorithm 7.4 5 in [92] to get the second cohomological dimension of X with coefficients in the GF(2)Xmodule B. It turns out to be zero. Let Eo = (k1, k2, k3, k4, k5, k5, k7, ks, kg) be the split extension of X by B, where the first three generators correspond to the generators k1, k2 and k3 of X. Then a set R( Eo) of defining relations of E 0 consists of R( K) and the following relations:

ka

= kl

= kl = k? = k~ = ki = 1,

(ki,kj) = 1,

for all

4 ~ i,j ~ 9,

(k2,k4) = (k2,k5) = 1, k k- 1kk- 1k515 k k- 1k1 - 1 - , k 1-lk414 8 1k91 s1 -1 - , k k- 1kk- 1k 716 k k- 1k1 k 1-lk614 5 1k6 1k71 s1 -

14- 2 Even conjugacy classes and group order

625

k1 1ksk1k9 1 = k1 1k9k1k3 1kg 1 = 1, 1k- 1k- 1k- 1 - k- 1 k k k- 1k- 1k- 1k- 1 - 1 k 2-lk6k2k4 5 6 9 2 724 7 8 9 , 1k1 - k- 1 k 9k2k1k1- 1 k 2-lksk2k4 s-2 5 9-, 1k- 1k- 1 - k- 1 k k k- 1k- 1 - 1 k 3-lk4k3k4 5 s-3 534 9-, 1k- 1k- 1k- 1 - k- 1 k k k- 1k- 1k- 1 - 1 k 3-lk6k3k4 5 6 9 3 735 7 8 , k -lk k k- 1k- 1k- 1k- 1 - k- 1 k k k- 1k- 1 k- 1 - 1 3

834

6

7

8

-

3

934

6

9

-

·

In order to determine all the possible extensions E of E 0 by A we first determine the 4-dimensional irreducible representations of E 0 over GF(2) because Nc(A) acts irreducibly on A. In fact, the exterior square of B contains two irreducible constituents of dimension 4, one of which is described by the following three matrices: Mvi

1100) = ( 11 00 00 0l 0111

,

Mv2 =

1100) 100 ( 01 O 1O 1011

,

Mv 3

= (

0011) 1100 0 1 OO 1110

·

Let Mv; = Jd(GL4(2)) for 4 =:; i =:; 9 and let N be the GF(2)E0 module described by these nine matrices. The author checked with MAGMA that it is irreducible. Let FEalg:=MatrixAlgebra and V : = GModule (PE_O, FEalg). Then, using the MAGMA command CohomologicalDimension(PE_O, V, 2), MAGMA asserts that this dimension is 1. Since S does not split over its normal subgroup A Ne (A) also does not split over A by Theorem 1.4.15 of [92]. Hence Nc(A) is isomorphic to the uniquely determined non-split extension E of E 0 by N. The MAGMA commands P:=ExtensionProcess(PE_O,V,E_O) andE:=Extension(P, [1]) yield the extension group E = (e;ll =:;is; 13) with the set R(E) of defining relations given in statement (a). (b) The permutation representation PE has been constructed in two steps. First we took only the cyclic group (e 3 ) of order 5 as a stabilizer and obtained a permutation representation P El of degree 221184. Then we used the MAGMA command DegreeReduction(PE1) and obtained a faithful permutation representation PE of degree 1536. Its stabilizer is given in the statement. (c) Using the faithful permutation representations PH of H given in Lemma 14.1.1 (a) and PE of E all assertions of this statement can easily be checked by the reader computationally.

626

Suzuki's group Suz

(d) The author used PE and Kratzer's Algorithm 5.3.18 of [92] to calculate the table of the representatives of the conjugacy classes of E. (e) The character table of E was computed with MAGMA and PE. (f) DH= (x,y) and DE= (x1,Y1) are subgroups of Hand E, respectively. The constructed isomorphism er : DH -+ DE identifies also the classes of DH and DE. We worked out the fusion of the DH-classes in H and the fusion of the DE-classes in E. For the eight conjugacy classes of involutions of DH = DE we obtained the following fusion patterns into the four classes of involutions of H and the four classes of involutions of E using the notation of Tables 14.5.1 and 14.5.2, respectively: ([1, 2, 3], [1, 2, 3, 4, 5], [1, 3]),

([4], [6, 7, 8], [2, 4]), where the classes of H and E are denoted by the indices of the involutions of the respective tables, and where the classes of H are stated before the ones of D. The last entries denote the classes of E. In particular, (d) and Tables 14.5.1 and 14.5.2 imply that z E 21 , t E 22 and h 7 E 23 of H are conjugate in G, because the involutions of E fuse only to z and h in

G. (g) The Goldschmidt index has been calculated by means of Kratzer's Algorithm 7.1.10 of [92]. D Proposition 14.2.3 Let G be a finite simple group of Suz-type having a 2-central involution z with centralizer H = Cc (z) defined in Lemma 14.1.1. Let A= (z,h1,h 2 ,h1) be the unique normal subgroup of order 16 of the Sylow 2-subgroup S and let E = Nc(A) = (x,y,e) be its normalizer whose generators x, y and e are defined in Lemma 14.1.1(i) and Proposition 14.2.2(c). Then the following assertions hold. (a) u = (ye) 2 is a representative of the conjugacy class of non-2central involutions of G and U1 = CE(u) = (s1,s 2,s3,s4) has order 29 , where s1 = Y1 e1, s2 = (yfe1 ) 2, s3 = (Yi e1Y1 ery1 e1 ) 5 and 84 = (yre1yre1y1ei) 2 .

(b) The involutions q1 = sr, q2 = (s1s2)4, q3 = s1s2s3s1s2s4 and q4 = s~ generate the unique maximal elementary abelian normal subgroup L of U1. (c) The normal subgroup J = Cu1 (L) of U1 has a complement of order 2 generated by j = S4s3s1. Furthermore, J is the direct product of the Klein 4-group Q = (q1 , q2q3q4) and the normal

14.2 Even conjugacy classes and group order

627

subgroup T = (t1, t2, t3, t4) of order 26 of U1, where t1 = s1s2s1, t2 = Si S2 S4, t3 = Si S2 and t4 = S1 S2 S4 S1 . (d) T has two maximal elementary abelian normal subgroups B 1 , B 2 of order 16. Furthermore, B{ = B2, and B1 nB2 is the non-cyclic center of T of order 4. (e) Cc (u) is isomorphic to the finitely presented group U = (r 1 , r 2 , r 3, r4,r 5) with the following set of defining relations: r 12 =r32 =r24 =r42 =r52 = 1, r 5r3 =r4r5, (r4, ri) = (r4, r2) = (r4, r3) = (r4, r5) = (r5, r1) = (r5, r2) = 1, r 2-1 r1r2r1r 2-1 r1r2r3r2r1r 2-1 r3 = 1 , (r21r1r3r1r3r1) 2 = 1, (r1r 21)7 = 1, r1r2r1r2r1r3r22 r3r1r 2-1 r3r1r3 = 1, r 2-2 r1r 2-2 r1r 2-2 r1r22r1r 22r1 = 1, r 2-2 r3r1r 2-1 r1r2-2 r3r1r3r1r2-1 r3r1 = 1, r2r3r2r3r1r 2-1 r1r 2-1 r3r2r3r2r3r2-1 r3 = 1.

(f) U has a faithful permutation representation PU of degree 480 with stabilizer (r1, r5, (r2r1 r2r1 r~ )2, (r1 r2r1 r~r1 r2r1 r~)2).

(g) A system of representatives (Ui) and the corresponding centralizer orders ICu(ui)I of the 38 conjugacy classes of U = hll : 3 is an odd prime power or C 2 ~ A 7 . All involutions of C 2 are G-conjugate to u by Table 14.5.6 and (c). Furthermore, U = Cc(u) does not have any elements of order 6 whose centralizer has a Sylow 3-subgroup of order divisible by 27. Hence C 2 cannot have elements of order 6. Therefore C 2 =f. A 7 • Now Proposition 14.2.4 and statements (a) and (d) complete the proof because J is isomorphic to a maximal subgroup of A 6 . D Proposition 14.4.6 Let G be a finite simple group of Suz-type with a 2central involution z such that Cc (z) = H = (x, y, h) described in Lemma 14.1.1. Let U = (r1, r 2 , r 3 , r4, r 5) ~ Cc (u) be the finitely presented group constructed in Proposition 14. 2. 3. Let N1 = (g1, gz, g3, g4) be the finitely presented group constructed in Proposition 14.4.S(c). Then the following statements hold.

646

Suzuki's group Suz

(a) e = (x 3 hx 2 hx 3 hx) 4 (hx 3 hxhx 2 hxh) 4 E H is a representative of the conjugacy class 5A of elements of order 5 of G and Cc (e) ~ (e) x A5. (b) Nc((e))/Cc(e) is a cyclic group of order 4. (c) The element f = r 1 r~ E U is a representative of the conjugacy class 5B of G and Cc(!) ~ (!) x A5. (d) Nc((f))/Cc(f) is a cyclic group of order 4. (e) The element s = r1r2 of U = Ca(u) = (rill ::S i ::S 5) is a representative of the unique conjugacy class of elements of order 7 of G and Ca(s) ~ (s) x A4. (f) The normalizer Ne ( (s)) is a split extension of a cyclic group of order 6 by Cc((s)). Proof (a) and (b) Using the faithful permutation representation PH of Lemma 14.1.l(a) and MAGMA the reader can verify that the elements k1 = (x 3 hx 2 hx3 hx) 4 and k2 = (hx 3 hxhx 2 hxh) 4 of H both have order 3, K = (k 1, k2) ~ A5 and that e = k1k2 has order 5. Furthermore, k1 is fl-conjugate to d = x 8 and CH (K) is a dihedral group D 8 of order 8 generated by the involutions s 1 = (yxy 2 ) 6 and s2 = (x 4 yx 3 yx)3. Hence CH(e) = (e) x D 8 . Moreover, the element w = (xhx 2 hx 5 ) 4 (hx 5 hx3 hx) 3 has order 4 and satisfies the relation ew = e2 and normalizes both subgroups Kand D 8 of H. Hence FA = (e, w) is a Frobenius group of order 20. In particular, (b) holds. Another application of MAGMA yields that K has a unique conjugacy class of involutions uf and that u1 is fl-conjugate to the representative u = h of the second conjugacy class u 0 of G. Now Proposition 14.2.4 and Table 14.5.6 imply that Cc(e) = K x L, where a Sylow 5-subgroup of L is cyclic of order 5. Since L is not a 2-group Theorem 1.6.4 of [92] due to Gorenstein and Walter implies that L is isomorphic to some PSL 2(q), where q > 3 is an odd prime power or L ~ A7. By Table 14.5.7 Ne ( (d)) does not have any elements of order 18. Hence L does not have any elements of order 6. Thus L ~ A 6 . (c) and (d) By Tables 14.5.1 and 14.5.6 the elements e of Hand f = r 1r~ of U represent two distinct conjugacy classes 5A and 5B of elements of order 5 of G because the involutions z and u are not conjugate in G by Proposition 14.2.2(i). Using the faithful permutation representation PU of U given in Proposition 14.2.3(f) and MAGMA it has been checked that T = Nu ( (!)) has order 24 • 5 and Cu(!) has a cyclic complement (j) of order 4. Hence (d) holds.

14.4 Uniqueness proof

647

As Ca (e) is a subgroup of Ca (L) ~ A 5 x L and f is G-conjugate to an element Ji of order 5 of L it follows that Ca (f) ~ (f) x A 5 . (e) and (f) By Proposition 14.2.4 the Sylow ?-subgroups of G are cyclic of order 7. By Table 14.5.6 the elements= r 1 r 2 of U = Ca(u) =(rill:::; i :::; 5) is a representative of a 7-class of G. Using the faithful permutation representation PU of U given in Proposition 14.2.3(f) and MAGMA it can be verified that w = r4(r2r3rir3r2r3) 4 (r3r2r3r~r3r2r3r2r3) 3 E U has order 6 and satisfies the relation sw = s 3 . Hence G has a unique conjugacy class of elements of order 7 and (f) holds. Another application of MAGMA shows that the Klein 4-group V = (r4,r 5 ) centralizes s. But (r 5 ,w) = r 4 . Now Table 14.5.7 implies thats is centralized by an alternating group A 4 and that its Sylow 3-subgroup centralizes w. Furthermore, V is not centralized by w because (r 5 , w) = r4. Thus Na((s)) is a split extension of (w) by Ca(s). D Proposition 14.4. 7 Let G be a finite simple group of Suz-type with a 2central involution z such that Ca (z) = H = (x, y, h) described in Lemma 14.1.1. Let N1 = (g1,g2,g3,g4) and N2 = (t1,t2,t3,t4) be the finitely presented groups constructed in Proposition 14-4.S(c) and Proposition 14.4.4, respectively. Then the following statements hold. (a) G has three conjugacy classes 3A, 3 B and 3c of elements of order 3 represented by the classes 3 1 , 33 and 35 of N 1 given in Table 14.5.7. (b) G has two conjugacy classes 9A and 9 B of elements of order 9 represented by the classes 91 and 9 2 of N 2 given in Table 14.5.8. (c) G has two conjugacy classes 5A and 5B of elements of order 5 represented by the classes 5 of H = Ca ( z) and 5 of U = Ca (u) given in Tables 14- 5.1 and 14- 5. 6, respectively. (d) G has one conjugacy class 7A of elements of order 7 represented by the class 7 of U given in Table 14.5.6. (e) G has one conjugacy class llA of elements of order 11. (f) G has two conjugacy classes 13A and 13B of elements of order 13. (g) G has three conjugacy classes 15A, 15B and 15c of elements of order 15. (h) G has two conjugacy classes 21A and 21B of elements of order 21 represented by the conjugacy classes 21 1 and 21 2 of N 1 given in Table 14- 5. 7.

648

Suzuki's group Suz

Proof (a) This is a restatement of Proposition 14.4.3(i). (b) Applying the same argument to the groups N 1 and N 2 we verified that the classes 3 1 , 33 and 35 of N 1 and the classes 3 1 , 32 and 3s of N 2 fuse in G, respectively. Hence the power map information of Table 14.5.8 implies that G has two conjugacy classes of elements of order 9. (c) and (d) These statements hold by Proposition 14.4.6. (e) and (f) These two statements are now obvious by Sylow's Theorem and Proposition 14.2.4. (g) By Propositions 14.4.3(i), 14.4.4 and 14.4.5 the 3-part x 3 of an element x of order 15 is G-conjugate either to an element of 3A or to an element of 3c. By Table 14.5.7 there is a unique conjugacy class 15A of elements x with x 3 E 3A. Since A 6 has two conjugacy classes of elements of order 5 Proposition 14.4.5 implies that there are two conjugacy classes 15B and 15c of elements x of order 15 with x3 E 3c. (h) This assertion holds by the power map information given in Table 14.5.7. Finally, Corollary 14.2.5, Proposition 14.2.4, the power map information of the tables in Section 14.5 and the class equation of G yield that G has exactly 43 conjugacy classes. In particular, G has 16 conjugacy classes of elements of odd order with representatives given in the statement. D Lemma 14.4.8 Let G be a finite simple group of Suz-type with a 2central involution z such that Cc (z) = H = (x, y, h) described in Lemma 14.1.1. Then His a maximal subgroup of G. Proof Lemma 14.4.8 must be true, otherwise there is a proper subgroup X of G containing H = Cc (z) properly such that X is of minimal order with respect to these two properties. Hence Cx (z) = H. Therefore Proposition 14.2.4 implies that X has a proper minimal normal subgroup Y =/- 1. Thus H n Y < H. By Lemma 14.1.1 the center Z(H) = (z) of order 2 and the Fitting subgroup P of order 27 are the only nontrivial proper normal subgroups of H and H/P ~ U4 (2). Therefore IY n HI E {1, 2, 27 }. Suppose that IYI is odd. Then Lemma 1.5.9 of [92] and Proposition 14.2.4 assert that Y is an abelian group of order dividing q = 33 · 5 · 7 · 11 · 13. Operating with the Klein 4-group V = (z, v) defined in Proposition 14.2.1 on Y yields by Theorem 4.2.2 of [92] and Table 14.5.1 that IYIICy(V)l 2 = ICy(z)IICy(t)l 2 because v and tz are conjugate in H. Now Cy(z) = 1 as H does not have any normal subgroup of

14.4 Uniqueness proof

649

odd order. Furthermore, ICy(V))I :S: 3 · 5 and ICy(t)I :S: 34 · 5 by Table 14.5.1. Thus IYI divides 32 as IYI divides q. However, the perfect group H does not have any irreducible 3-modular representations of degree 2. Suppose that IY n HI = 2. Then Corollary 1.4.18 of [92] implies that the cyclic Sylow 2-subgroup of Y has a normal complement Q. As a characteristic subgroup of Y it is normal in X. Hence Q = 1 by the previous argument. Furthermore, Y = Z(H) and X = Cx (z) = H, a contradiction. Hence Y n H = P and P is a Sylow 2-subgroup of Y. Since Y is a minimal normal subgroup of X its commutator subgroup Y' = Y because P is not abelian. By the previous argument Y does not have any normal subgroups of odd order. Hence any normal subgroup N # 1 of Y has even order. Thus P n N = N 1 # 1 is a normal subgroup of P. Therefore z E N 1 by Theorem 1.3.4 of [92]. Hence M = (zy) is a normal subgroup of X contained in N. Thus M = N = Y, and Y is a simple group. By Lemma 14.1.l(h) Pis extra-special of order 27 • In particular, P is not a dihedral 2-group of order 8. Therefore Y is not simple by Lemma 4.8.2 of [92]. This contradiction completes the proof. D Proposition 14.4.9 Let G be a finite simple group of Suz-type. Then G has exactly one complex character x E Irrc( G) of degree 143. Proof Let H = Ca (z) be the centralizer of a 2-central involution of G. Let S be the Sylow 2-subgroup of H defined in Lemma 14.1.1. Let A be its unique maximal elementary abelian normal subgroup. Then D = NH(A) is a proper subgroup of E = Na(A) by Proposition 14.2.2. Hence G = (H, E) by Lemma 14.4.8. Proposition 14.2.2(g) states that the amalgam H +--- D --, E has Goldschmidt index 1. Therefore the free product P = H *D E of H and E with amalgamated subgroup Dis uniquely determined up to isomorphism. By Theorem 14.3.1 there is exactly one compatible pair (X,T) E mfcharrc(H) x mfcharrc(E) of degree 143. As G is a simple epimorphic image of P it follows that G has at most one character x E Irrrc( G) with minimal degree 143. Using Brauer's characterization of characters we now show that any finite simple group G of Suz-type has a complex irreducible character x E Irrc( G) of degree 143. The simple group of Suz-type ® constructed in Theorem 14.3.1 has 43 conjugacy classes. Also G has 43 conjugacy classes by Corollary 14.2.5 and Proposition 14.4. 7; ® has an irreducible character of degree 143 by Theorem 14.3.l(b)(4). Inspired by its values

650

Suzuki's group Suz

we define a class function

X :

x(g,)

G

-+ (C

by

lA

2A

2s

3A

3s

3c

4A

4s

4c

4n

5A

5s

143

15

-1

35

8

-1

15

-1

-1

-1

8

3

6A 3

x(gi)

6s 6

lOA

lOs

llA

0

-1

0

x(gi)

x(gi)

15A -1

7A

8A

8s

8c

gA

9s

3

-1

3

-1

2

2

12c -1

12n 2

l2E

l3A

13s

0

-1

0

0

15c

l8A

18s

20A

21A

21s

24A

0

0

0

0

0

0

-1

6c 6

15s -1

6n 0

12A 3

6E -1

12s

14A -1

Let p be any prime divisor of jGj. If Y is an over-group of a pelementary subgroup X, and the restriction XIY of X to Y is a generalized character, then xrx is also a generalized character of X. By Proposition 14.4.7 the Sylow 13-normalizer N13 of G and Sylow 11-normalizer Na(ll) of Gare Frobenius groups of orders 13 · 6 and 11 · 12, respectively. As x(x) = 0 for all 13-singular and 11-singular elements and x(l) = 143, its restrictions X[N 13 and X[Nu are projective characters of N 13 over GF(13) and N 11 over GF(ll), respectively. Therefore by Proposition 14.4.7, and Corollary 2.8.10 and Theorem 2.8.9, both of [92], it remains to show that Xry is a generalized character of Y whenever Y belongs to the following set of subgroups:

of G. By means of the results of this section the fusion of the conjugacy classes of the groups Y E ZJ in G can be determined easily. Therefore the inner products (Ai, X)Y can be calculated for each Ai E Irrc(Y) and each Y E { H, U, N 1 } by means of the character tables given in Section 14.6. It turns out that X[H X[U

X[N 1

= X1 + X7 + X11 + X15 + X16 + X17,

where

Xk E

Jmc(H),

= X3 + X5 + X15 + X27, where Xk E Jmc(U), = X2 + X5 + X6, where Xk E Irrc(N1).

Now N 1 n N 2 is a normal subgroup of N 2 of index 2 by Proposition 14.4.4. Hence X[N2 is a positive sum of irreducible characters of Irrrc(N2 ) by Clifford's Theorem.

14.4 Uniqueness proof

651

The character tables of the four remaining local subgroups and the fusion of their conjugacy classes in G can be calculated from the results in Propositions 14.4.5, 14.4.6 and 14.4.7. Therefore the restrictions of x to these subgroups can be determined similarly. The details are left to the reader. It has been checked that the inner product (x, x)c = 1. Therefore x is an irreducible character of G with x(l) = 143 by Corollary 2.8.10 of [92]. This completes the proof. D Theorem 14.4.10 Each finite simple group G ofSuz-type is isomorphic to the finite simple group Q5 ::::; GL 23 (23) of order [\!5[ = 210 · 37 · 53 · · 11 · 23 constructed in Theorem 14.3.1.

Proof By Theorem 14.3.1 there exists a finite simple group Q5 of order [\!5[ = 211 . 33 . 52 . 7 · 11 · 13 which is of Suz-type. Let G be any finite simple group of Suz-type. By Proposition 14.2.4 G has exactly one 2-central conjugacy class zG of involutions and the same order as Q5. Proposition 14.2.2 asserts that the Sylow 2-subgroup S of H contains a maximal elementary abelian normal subgroup A such that D = NH (A) is a proper subgroup of E = Ne (A). Furthermore, E is uniquely determined by H up to isomorphism. Lemma 14.4.8 implies that G = (H, E). The amalgam H +-- D --. E has Goldschmidt index 1 by Proposition 14.2.2(g). Proposition 14.4.9 states that G has a unique irreducible complex character of degree 143. It is of defect zero for the prime 13. Therefore each finite group G of Suz-type has a unique irreducible 13-modular representation of degree 143 by Theorem 3.12.2 of [92]. Hence G ~ Q5 by Theorem 7.5.1 of [92]. This completes the ~~

D

Suzuki's group Suz

652

14.5 Representatives of conjugacy classes 14.5.1 Conjugacy classes of H = (x, y, h) Class Representative I Centralizer! 1 21 22 23 24 31 32 33 34 41 42 43 44 45 46 47 4s 49 5 61 62 63 64 65 65 67 6s 69 610 611 612 613 81 82 83 84 85 91 92 101 102 103 121 122 123 124 125 126 127 12s 129 1210 1211 181 182 20 24

1 (x)12 (xv) 6

(x2v)6 h (x)B (xh) 3 (xh) 6 (v)4

(x)6 (y)3 (xv) 3

(x3y)2 (xv3 )3 (y3 h)2

(x2y)3 (x 2 hxh) 3 xvxh yh (x)4 (xhy2)3 (xhv2)15

(y)2 (xv) 2 (x2y)2

(x2y)10 X2Y2

(x2y2)5 xyxhy (xyxhv) 5 x 4 1/h (x4y2h)5

(x)3 x 2 v 2 hv x 3y v 3h x3y2 xh (xh) 2 (x2 h)2 xy 2 h (xy2 h)3

(x)2 xy 2 hxhy xy X'l/3 (xy3 )5 x 3 vxh y X2'U (x2y)5 x 2 hxh (x 2 hxh) 5 xhy 2 (xhy2)5 x 2h X

213.34.5 213 . 34 . 5 2 12 · 3 · 5 210 . 32 29 27 . 33 24 . 34 24 . 34 24 . 33 210 . 32 . 5 210 . 3 29 · 3 29 · 3 28

•3 29

27 · 3 27 · 3 25 23 · 5 27 . 33 24 . 34 24 . 34 24 . 33 26 · 3 24 . 32 24 . 32 24 . 32 24 . 32 24 . 32 24 . 32 24 . 32 24 . 32 26 · 3 26 25 25 25 2 · 32 2 · 32 23 · 5 22 · 5 22 · 5 25 . 32 23 . 32 24 · 3 24 . 3 24 · 3 24 · 3 23 · 3 23 · 3 23 · 3 23 · 3 23 · 3 2 · 32 2 · 32 22 · 5 23 · 3

2P 1 1 1 1 1 31 33 32 34 21 21 22 21 22 22 23 23 24 5 31 33 32 34 31 32 33 33 32 31 31 34 34 41 42 44 45 43 92 91 5 5 5 61 64 65 65 65 61 64 65 67 65 67 92 91 101 121

3P 1 21 22 23 24 1 1 1 1 41 42 43 44 45 46 47 4s 49 5 21 21 21 21 22 23 23 23 23 23 23 23 23 81 82 83 84 85 32 33 101 103 102 41 41 43 45 45 44 42 47 47 4s 4s 62 63 20 81

5P 1 21 22 23 24 31 33 32 34 41 42 43 44 45 46 47 4s 49 1 61 63 62 64 65 67 65 69 6s 611 610 613 612 81 82 83 84 85 92 91 21 22 22 121 122 123 125 124 125 127 129 12s 1211 1210 182 181 41 24

14.5 Representatives of conjugacy classes 14.5.2 Conjugacy classes of E = Na (A) = (x1, Yl, e1) Class Representative I Centralizer! 1 21 22 23 24 31 32 33 34 41 42 43 44 45 45 47 43 51 52 61 62 63 64 65 65 67 63 81 82 83 84 101 102 121 122 123 124 125 125 127 123 151 152 153 154 241 242

1 (xi )12 (xi 'l/1 e1 ) 5 (x1 e 2 ) 6 (y1 e1 ) 2 (xi ) 8 (xi )16 (y1 )4 e1 (y1 )3 (x~yi)3 (xi ) 6 (x2y3)3 (X 3 Y1 ) 2 Xl e1

(xi e2 )3

u1e1 (xi u1e1 ) 2 (x1u1e1 ) 4 (xi ) 4 (xi )20 (vi ) 2 x2 y2 (x2yf )5 (xi ef ) 2 (x1en 10 x2 e2

(xi ) 3 xf yf X1 y 3 e1 x 3 y1

Xl 'l/1e1 (xi Yi e1 ) 3 (xi ) 2 (xi )10 x2y3 (x~yn 5 Yi x 2 y1 xi e~ (x1e~)5 y 2 e1 (y2 e1 )2 (y2 e1 )7 (yfe1) 11

X1 (xi ) 5

213 . 33 . 5 2 13 . 3 2 2 8 . 3. 5 29 · 3 29 27 · 33 · 5 2 7 · 33 · 5 24 . 32 22 . 32 210 . 3 210 . 3 29 · 3 28 · 3 29 28 25 · 3 25 22 · 3 · 5 22 · 3 · 5 27 . 32 27 . 32 24 . 32 24 . 32 24 . 32 25 · 3 25 · 3 22 . 3 25 · 3 26 26 25 22 · 5 22 · 5 25 · 3 25 · 3 24 · 3 24 · 3 23 . 3 23 · 3 23 · 3 23 . 3 3.5 3.5 3.5 3.5 23 · 3 23 · 3

2P 1 1 1 1 1 32 31 33 34 21 21 21 21 21 21 23 24 52 51 31 32 33 33 33 31 32 34 43 42 41 45 51 52 61 62 61 62 63 63 65 67 152 151 154 153 121 122

3P 1 21 22 23 24 1 1 1 1 41 42 43 44 45 46 47 43 52 51 21 21 21 21 21 23 23 22 81 82 83 84 102 101 43 43 44 44 41 42 47 47 51 52 52 51 81 81

5P 1 21 22 23 24 32 31 33 34 41 42 43 44 45 45 47 43 1 1 62 61 63 65 64 67 65 63 81 82 83 84 22 22 122 121 124 123 125 125 123 127 32 31 32 31 242 241

653

Suzuki's group Suz

654

14.5.3 Conjugacy classes of D Class 1 21 22 23 24 25 25 27 2s 31 32 33 34 35 41 42

= (x1, Y1) I Centralizer! 2P 3P

Representative 1 (x1 )12

( x1 Y1 ) 6 (x2y1)6 (x1 y3 )6 (x1 y;)3 (x;yfx 1y 1 ) 2 x;y 4 x1y1 x 5 y1x 2 y 2 x 2 y1 (x1 )s (x1 )16 (y1 )4 (x;y,)4

(x2y1)8 ( Y1 ) 3 (x~yn3 (x1 )6

43 44 45 4o 47

( x; Y1 )3 (x3y 1 x2y2)3

4s 4g

(x;y;)2

410 411

xfy1x1y1 (x 2 Y1 ) 3

412 413 414

x1y1x1v1x 1 y;

415 415 41 7 41s 61 62 63 64 65 60 67 63 69 610 611 612 613 614 615 615 617 61s 619 81 82

(x1111 ) 3 (x;yf) 3

213 . 213 . 2 12 210 .

x; Yt x1 Yt x1 Yt x1 Yt

·3 32

29 · 3 28 · 3 2' 28 28 27 . 32 27 . 32 24 . 32 24 . 32 24 · 210 2 10

32 .3

29 28 28 28 28

·3 ·3 ·3 ·3 ·3 2'

(x3y,)2

(x2y4)3

32 3'

·3

2' 2' 27 · 3 27 · 3 28 28

4

28 28

x;11fx1y1 (x1 ) 4 ( Xt )20

25 · 3 25 27 . 32 27 . 32

x;y1x;y1

4

x1y1x1y x1y (x1y 3 )3

(x1 Yt ) 2 (x1 Yt ) 19

(y1 )2 (x; YI ) 2

(x 2 Yt ) 10 x2y; (x;y; )5 x4y1

(x4yi)5 x4y3 (x;y;)5 XiY1X7Y1X1Y1

(x 3 y1x 2 y1x1v1) 5 (x1 y 3 ) 2 (xi y3 )10 X1Yi (x1 y;) 5 (x1 ) 3 x;y1x;y;

26 26 24 . 24 .

·3 ·3 32 32

24 24 24 24 24 24 24

32 32 32 32 32 32 32

· . . . . . .

24 · 32 24 . 32 25 · 3 25 · 3 24 · 3 24 · 3 25 · 3 26

1 1 1 1 1 1 1 1 1 32 31 33 35 34 21 21 21 22 22 22 21 21 2, 22 23 23 21 22 22 22 24 2o 31 32 32 31 33 34 35 35 34 34

1 21 2, 23 24 25 25 27 2s

1 1 1 1 1 41 42 43 44 45 45 47 4s 49 410 411 412 413 414 415 415 417 41s 21 21 22 22 21 23 23 23 23 21

35 31 32 33 33 32 31 31

21 23 23 23 23 24 24 25

32 43 41

25 81 82

14.5 Representatives of conjugacy classes Conjugacy classes of D

= (x 1 ,y1 )(continued)

Class Representative I Centralizer! 2P 26 42 x5y2x1y2 83 X3Y! 2' 4s 84 25 4g x{y2 85 25 410 x~yf 85 (x1 )2 121 2 5 · 3 61 122 123 124 125 125 127 12g 129 1210 1211 1212 1213 1214 1215 1215 1217 121s 241 242

(x 1 ) 1o

1 21 22 31 32 33 34 35 41 42 43 51 52 61 62 63 64 121 122 123 124 125 125 151 152 153 154

25 24 24 24 24 24 24 24 24 23 23 23 23 23 23 23 23 23 23

X1Y1 (x1 y,) 5 x 2 uf (x"un' x;y1 (xfy1) 5

x~y1x;y; (x3y1x2y2)5 YI x;y1 (x;yi)5 X1Y3 (x1 y3 )5

x;y; (x;y;)5 x6y; X1 ( X[ 15

14.5.4 Conjugacy classes of W Class

655

·

· · · · · · · · · · · · · · · · · ·

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

62 63 64 64 63 64 63 61 62 65 65 67 615 617 65 67 65 121 122

3P 83 84 85 86 43 43 44 44 45 45 45 45 47 47 41 411 411 417 41 7 412 412 42 81 81

= (m1, m2) I Centralizer!

Representative 1 (m1 m2 ) 6 (m;m,m, m;m'm! ) 3 (m1m2 ) 4 (m1m2) 8 m1 (m1 ,2 m 2 m2m1 m2 (m1m2 ) 3 m~m2m1

m;

(m 2 m2 m 2 m2m1 m~ ) 3 m, (m2 )' (m1m2J2 (m, m, y10 m 2 m2m1m~m'l.m; (m;m2m1m;m;m;) 5 m1 m, (m1m2)' m~m2m~m2m1 m~ (m;m,m 2 m2m1m~)5 m 2 m2m1 m2m1 m;m1 m2 ( m: m2 m1 m2 m 1 m; m1 m2 ) 5 m~m2

(m;m2) 2 (m 2 m2) 7 (m 2 m2)1 1

26 · 32 · 5 26 · 3 24 · 3 26 · 32 · 5 26 · 32 · 5 32 32 32 24 · 3 24 · 3 24 · 3 3.5 3 ·5 26 · 3 26 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 24 · 3 3.5 3.5 3.5 3.5

2P 1 1 1 32 31 34 33 35 21 21 21 52 51 31 32 31 32 61 62 62 61 62 61 152 151 154 153

3P 1 21 22 1 1 1 1 1 41 42 43 52 51 21 21 22 22 41

41 43 43 42 42 52 51 51 52

5P 1 21 22 32 31 34 33 35 41 42 43 1 1 62 61 64 63 122 121 124 123 125 125 31 32 31 32

Suzuki's group Suz

656

14.5.5 Conjugacy classes of M

= Nc(V) = (w 1 ,w 2 ,m)

Class Representative I Centralizer! 1 21 22 23 24 25 25 31 32 33 41 42 43 44 45 45 47

4s 4g 410 51 52 61 62 63 64 65 65 81 82 83 84 85 85 87 101 102 103 104 121 122 123 124 12s 151 152 201 202 24

1 (w2 ) 4 (w1 ) 6 (w2w2)5 (w;m) 3 (wfw2m)3 w~mw';m 2 (w1 ) 4 m (w 2 m) 2

(w1m 2 ) 3 (w,m) 2 (w4w; )3 (w2 ) 2 (w1 ) 3 w 2 w2w1w; w~w2mw1m

w;mw2m 2 4 w mw1w2w1m W 6 W2m W1W2

(w 2 w 2 )2 (w1m2)2 (w1)2 (w1w;) 2

w1w'.;m 2

w~m wfwzm w 3m w, w2m 2 w2m wfwz

w 3 w~ w 5 w2w1m (w 2 w2m) 2 W1Wg W~Wz W 4 W2W1 Wz

w2

w;

w1m 2

w 4 w'; W1 W1W; w1w2m W1WzW1Wzm

w 2 wzm w 4 w2mw1 wz w1w;m

213.32.5 2 13 .3.5 2 11 28 · 3 29 27

·3 ·5 ·3 ·3 28 27 . 32

2P 1 1 1 1 1 1 1

31 32 33 21 21 21 21 2 7 · 3 22 28 22 28 22 28 21 27 22 25 24 2 3 · 3 · 5 52 2 3 · 3 · 5 51 2 7 · 3 31 2 5 · 3 31 2 5 · 3 31 2 4 · 3 31 2 2 · 3 33 2 2 · 3 32 2 6 · 3 42 26 44 26 42 26 44 26 42 26 44 25 43 2 3 · 5 51 2 3 · 5 52 2 2 · 5 52 2 2 · 5 51 2 5 · 3 61 2 4 · 3 61 2 4 · 3 61 2 3 · 3 62 2 3 · 3 63 3.5 152 3.5 151 2 2 · 5 101 2 2 · 5 102 2 3 · 3 121

2' · 3 2 · 5 22 · 32 28 · 3 · 5 210 . 3 29 · 3 210

3P 1

5P 1

21 22 23 24 25 25 1 1 1

21 22 23 24 25 25 31 32 33 41 42 43 44 45 45 47 4s 4g

41 42 43 44 45 45 47 4s 4g 410 52 51 21 22 22 23 24 25 81 82 83 84 85 85 87 102 101 104 103 42 41 43 45 45 52 51 202 201 81

410 1 1 61 63 62 64 65 65 81 82 83 84 85 85 87 21 21 23 23 121 122 123 125 124 32 32 41 41 24

14.5 Representatives of conjugacy classes 14.5.6 Conjugacy classes of Cc(u)

= hll :S k :S 5)

Class Representative I Centralizer! 2P 1 21 22 23 24 25 26 3

1

r4 T5 r1 T1 T4

r3 TJ T5

(r 1 r; r 1 r3 ) 2

41 42 43 44 45 45 47 4s 49 410 411

r1 r2 r1 r2 T3 r2

412

r1 r2 r1 r2 r., r, rs

T3 T5

r2 ( r, r3 ) 2 r2 r4

(r1 r3 r,? ( r1 r3 r5) 2

(r1r3r2r5 ) 2 r2r5

r1 r,r1 r3r5 r1 r3 r; r3 r5

r 1r;

5 61 62 63 7 81 82 83 84 85 86 101 10, 103 12 141 142 143

(r1r;r1r3r5)2 r1 r 2 r1 r3 r1 r3 r1 r3 r2 rs r1 r2

TI T3 T1 T3 Tz

r 1 r3 rs r1r2r1r3

r1 r3 r2 rs

r3 rs r1r;r4

r1 r2 r1

2 9 ,3 2 •5•7 29 . 32 .5.7 28 . 32 · 5 · 7 29 29 25 . 32 28 23 . 32 25 . 32 27 27 27 27 27 27 26 26 26 25 25 2' 23 . 22 · 22 · 22

.5 32 32 32 ·7 24 24 24 24 24 24

r1r2r4

22 .5 2' .5 22 .5 22 · 32 22 .7

r2 rs r1 r2 r 4 rs

2' .7 2' .7

r1

r;

r5

r1r;r4r5 r1r;r1r3r5 r1

1 1 1 1 1 1 1 3 21

23 23 23 23 23 23 23 23 23 23 24 5 3 3 3 7 43 45 46 42 47 44 5 5 5 61 7 7 7

3P 1 21 22 23 24 25 25 1

5P 1 21 22 23 24 25 25 3

7P 1

41 42 43 44 45 45 47 4s 49 410 411 412 5

41 42 43 44 45 45 47 4s 4g

41 42 43 44 45 45 47 43 4g

410 411 412 1

410 411 412 5

21

22 23 24 25 26 3

21

61

61

25 22 7

62 63 7 81 82 83 84 85 85 21 22 22 12

62 63 1 81 82 83 84 85 85 101 103 102 12

141 143 142

21 22 22

81

82 83 84 85 85 101 103 102 41 141 143 142

657

Suzuki's group S uz

658

14.5.7 Conjugacy classes of N1 Glass 1 21 22 31 32 33 34 35 41 42 43 5 61 62 63 64 65 66 67 7 81 82 83 84 91 92 10 121 122 123 124 125 15 211 212 241 242 243

Representative 1 a1 03

Q4 ( 02 03 ) 8 (02 ) 2

(g1 g3 g; g3 ) 2 /g1g3 ) 2 (g2g3 ) 6 ( 01.Q3 02 ) 3 (01g2.a1.a3 ) 2 gig~

Q1 04 (.q2g3) 4 g2

g1 g3g; g3 .Q1.Q2 _q3 g2 g1.Q3

(g1 a2 03 a2 g1 a3) 5 .Q1 .Q3 (a1a2 ) 3 (.02 .Q3 ) 3 01 a2 g3 01 03 a2 g; g3

01 02 01 03 01.a3 .a2 03

( 01 03 .Q2 03 ) 2

= Nc(3A) = I Uentralizerl 28 · 37 · 5 · 7 28 . 33 25 · 3 2 · 5 2 7 .3 7 .5.7 24 . 37 23 . 37 22 . 35 2 · 34 26 . 32 25 · 3 25 · 3

2 ·3 27 . 24 . 23 . 22 . 22 . 22 .

·5 33 33 33 33 33 33 2 · 32 3.7 25 · 3 24 · 3 25 23 33 33

02 g3 (g203 )13

2·5 25 . 32 23 . 32 24 · 3 22 . 32 22 · 3 3.5 3.7 3.7 23 · 3 23 · 3

g1g2grni

2° · 3

g1g3g; (g1g2g1g*) 2 (g2 g3 ) 2 01 02 Yi 02 03 02 03 gig;

Q1 03 .Q2 g1g~g4

01.Q2 ( .Qi .Q2 ) 2

(g1,g2,g3,g4)

2P 1 1 1 31 32 33 34 35 21 21 21 5 31 32 33 34 34 34 35 7 41 41 41 43 92 91 5 61 62 61 63 62 15 212 211 122 122 121

3P 1 21 22 1 1 1 1 1 41 42 43 5 21 21 21 21 21 21 22 7 81 82 83 84 33 33 10 41 41 43 41 42 5 7 7 81 81 82

5P 1 21 22 31 32 33 34 35 41 42 43 1 61 62 63 64 66 65 67 7 81 82 83 84 92 91 22 121 122 123 124 125 31 211 212 241 242 243

7P 1 21 22 31 32 33 34 35 41 42 43 5 61 62 63 64 65 66 67 1 81 82 83 84 91 92 10 121 122 123 124 125 15 31 31 241 242 243

14.5 Representatives of conjugacy classes 14.5.8 Conjugacy classes of N2

= Na(Z) =

(t1,t2,t3,t4)

Ulass Representative I Centralizer! 1 21 22 23 31 32 33 34 35 36 37 3s 41 42 43 44 45 46 61 62 63 64 65 66 67 6s 69 610 611 612 613 614 615 616 81 82 91 92 121 122 123 124 125 126 127 181 182 241 242

1 /t1 ) 2 t3 /t1t3t4) 6 (t1 t2 ) 4 /t2t3) 6 (t2t3t4 )2 (t) t3 ) 2 (t~t3) 4 (t2 ) 2

(t;td /tit3t4) 4 t4 /t1t3t4) 3 (t1 t3 ) 2 (t3t4) 3 t1

t~ t3 t4 /t2t3) 3 (t2t3 ) 15 /t1 t2 ) 2 (t3t4) 2 ( t1 t2 t3 t1 t3 )2 t 2 t3 (t~t3 ) 5 t2 t3 t4 (t2t3t4 ) 5 t2 t4 (t;t2t4 ) 5 t2t3t2t4 (t2t3t2t4 ) 5

t;

t2

t; t3

/t1t3t4) 2 t1 t4 t1 t3 (t2t3) 2 /t2t3) 4 t 2 t4 (t1t2t4) 2 t1 t3 t4 (t1t3t4) 7 t3 t4 t1 t2 t1t2t3t1t3 t2 t3 (t2 t3 ) 5 t1 t2 t4 (t1t2t4) 13

26 26 24 25 24 24 23 22 22 2 2 22 25 24

. . . . . . . . .

37 33 34 32 37 37 35 35 35 35 35 34 32 32

· · . . . 25 · 3 24 · 3 23 · 3 24 24 . 34 24 . 34 24 . 33 24 . 33 23 . 33 22 . 33 22 . 33 22 . 33 22 . 33 22 . 33 22 . 33 22 . 33 22 . 33 2 · 33 2 · 33 22 . 32 23 · 3 23 2 · 33 2 · 33 23 . 32 23 . 32 22 . 32 22 . 32 23 · 3 22 · 3 22 · 3 2 · 32 2 · 32 23 · 3 23 · 3

2P 1 1 1 1 31 32 33 35 34 36 37 3s 21 23 21 21 21 23 32 32 31 32 33 34 35 33 33 34 35 34 35 36 37 3s 41 43 92 91 64 63 616 616 64 63 65 91 92 122 122

3P 1 21 22 23 1 1 1 1 1 1 1 1 41 42 43 44 45 46 22 22 21 21 21 22 22 22 22 21 21 22 22 21 22 23 81 82 32 32 41 41 42 42 44 45 43 61 62 81 81

659

Suzuki's group Suz

660

14.6 Character tables of local subgroups 14.6.1 Character table of H= (x,y,h)

2P 3P 5P X.I X.2 X.3 X.4 X.5 X.6 X.7 X.8 X.9 X.10 X.11 X.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19

X.20 X.21 X.22 X.23 X.24 X 25 X.26 X.27 X.28 X.29 X.30 X.31 X.32 X.33 X.34 X.35 X.36 X.37 X.38 X.39 X.40 X.41 X.42 X.43 X.44 X.45 X.46 X.47 X.48 X.49 X.50 X.51 X.52 X.53 X.54 X.55 X.56 X.57

13 4 1 la Ia la la 1

5 5 6 10 10 15 15 20 24 27 30 30 30 32 32 36 36 40 40 45 45 60 64 81 81 81 108 135 135 135 135 160 160 160 160 160 180 180 180 180 270 270 288 288 324 324 360 360 405 480 480 480 512 540 576 640

13 12 4 I I 1 2a 2b la Ia 2a 2b 2b 2a 1 I 5 5 5 5 6 6 10 10 10 10 15 15 15 15 20 20 24 24 -5 27 30 30 30 30 30 30 -32 -32 36 4 4 36 40 40 40 40 45 45 45 45 60 60 64 64 81 81 81 -15 81 -15 108 -20 135 -25 135 -25 135 -25 135 -25 -160 -160 -160 -160 -160 180 20 180 20 180 20 20 180 270 -50 270 -50 -288 -288 36 324 324 36 40 360 40 360 405 -75 -480 -480 -480 -512 540 -100 64 576 -640

10 2

g

2c la 2c 2c I -3 -3 -2 2 2

2d Ia 2d 2d

2 -2 -2

-1 -1

7 4 8 3 -10 6 6

3 4

9 -:i

1 1 4 3 7 3 3

-12 -8

-36 -4 36 12 4 2 6 6 -6

3b 3c Ia 3c I

1\

j[

A

A

E

E

E

3 -3 -3 8 8 6 6 -2 -2

:i

F

F

E

j[

:i

F F

A

A

j[

H

FI

FI J

J

H

J

i

i

N

N

0 N

N

0

0

0

9

J

. -16

-9

. -12

-8

. :

g

24

40 40 45 45 60 64 81 9

g

9

18 -9 -9 -9 -8 16 -8 16 -8 12 -6 -6 12

:i

F

8

-8

4 -2 4 -2 4 -3 6 6 -3

J J

4c 2b 4c 4c 1 1 1 2 -2 -2 -1 3 4

4, 2b 4e 4, I -3 -3 -2 2 2 -1 7 4 8 -1 -52 -10 2 6 2 6

4d 2a 4d 4d I I I 2 -2 -2 -1 3 4

-4 -4 -4

J 6 6 -:i 4 -8 -8 -2

-:i

F

12 -4

3d 3d la 3d

-3 -3

J

-36 4 36 -8 -24 24 -27

3c 3b la 3b 1

6 6 -3 -3 :i 5 2 2 -1 6 6 3

9

12 -4 -12 -8 -8 -3 -3 -3 -3 -4 4

9 9 12 -9 15 -9 -9

3a 3a la 3a 1 -1 -1 3 1 I 3

10 10 2 I 1 4a 4b 2a 2a 4b 4a 4a 4b 1 I 5 -3 5 -3 6 -2 10 2 10 2 15 -1 15 7 20 4 24 8 3 3 30 -10 6 30 30 6

: . 3 3

:i

12 15 15 15 15

4 -8 -8 -3 -3 -4

-3 -3 4

-3 -3 4

9g -:i -:ig 9 12

-9 15 -9 -9

-3 -3 4 7 11

7 7

9 4 -5 -1 -5 -5

-20 4 12 -20 12 8 4 -20 -12 4 8 -20 -4 12 10 -14 30 6 6 -14 30 10 -36 12 12 -36 -12 12 -40 8 -8 -16 -40 -8 -16 -8 45 -27 -3 9

F

6 8 -4

F

60 . -6 -64 -8 -8

12 -4

-4

4/ 4g 2b 2c 4g 4g 1 1 I 2 2 -2 2 -2 2 -1 3 3 -1 4

H

-i

:i

4h 2c 4h 4h 1 I I 2 2 2

3

-1

4i 2d 4i 4i 1 -1 -1

5a 5a 5a la 1

-i I

:i -i

2 -2 -2 2 2 2 2 2 2 4 -4 4 -4

: -4 -8 -8 -3 -3 -3 -3 -4 4

-i

2

-i

6a 3a 2a 6a 1 -1 -1 3 I I 3

9

4 4

4 4

6b 3c 2a 6c 1

6c 3b 2a 6b 1

A

j[

-3

E

E

6 -3 2 6

6d 3d 2a 6d I j[ 2 A 2 -3

E

E

6 -3 :i 2 -1 6 3

6e 3a 2b 6, 1 -1 -1 3 1 1 3

i

3 :i :i 3 -3 F F -3 -3 F F -3 -8 -A -A -2 -8 -A -A -2 3 -2 6 3 -2 6 -2 H FI I -2 -2 FI H 1 -2 J J J J -:i 6 6 -:i -:i 4 -8 -8 -2 4

9 -:i -:i -:i -i 1 5 -3 -3 -3 1 1 5 -3 -3 -3 I I -2 -4 4 9 i 3 -1 18 2 :i i : -1 -5 -9 3 -1 3 . -1 3 -1 3 3 1 -9 . -1 3 -1 3 3 1 -9 4 -4 8 -i -i -4 4 -4 . -16 -N -N 2 8 -0 -0 -4 4 -4 4 -4 . -16 -N -N 2 4 -4 8 -0 -0 -4 . -2 12 -3 -4 -6 6 2 : -2 -6 6 2 : -4 . -8 12 : -3 -4 -2 -6 -6 -6 9 I g -2 2 6 6 I 4 -4 -2 -J -J 4 -4 . -2 -J -J . -1 : -8 4 2 -1 :i -2 8 3 -2 5 -3 -3 -i . -4 . -4 . -4

-4 -4

4 4 4

: -24

-:i -:i

-F

2

i

16 -9 -12 8

-F

-8

-F -6 -F -6

-8

4

. -i

: -6 8 8

4

661

14. 6 Character tables of local subgroups Character table of H 6f

6g

3b 2c 6g 1

3c 2,

B

13

1 A

A

2 1 -2 2

6h 3c

1

2c 6i 1

B

B

6f

13

1

13

1

A

A

A 2 1 -2

A 2

2

1

-2 2

-i -i -i 13

B

B

B

13

13

B

-}]

B

13 -13

13 I

I

K

K 2

I

I

I

I

6i

Gj 6k

3b 3a 3a 2c 2c 2c 6h 6k 6j l 1 J

B 13

C C

1 1 A -l A -1 2 -1 1 -2 -2 1 2 2 . 3 -1 -1

13 B -13

C C

-}]

= (x, y, h) (continued) 61 6m 3d 2c 6l

C C -'.2 1

8a 4a 8a Ba 1 1

_z J

I

I f

~

8c 4d 8c Sc 1 -1

8d 4f Bd 8d 1 -1

Se 4c Be Be 1 -1

-1 -1 -1

9a 9b 3b 9b 1 D jj

l

I

l l

. -2

1 1

.

K

K

K

-3

K 2

K 2

I~

-i -i -i -l -~ . -3 . -3 -4

. . . .

. -3 -3 MM

C K

.1'1

M

(]

K

G

G

G

G

-1 -.5 -1 -1

J 1

.

-1 -1

D

3 1 1 3

-i -i -i 2

3

~

i

3 -1 -1 -1

1

-i -i -1

-1

2 -i

K

K

I? -K K

-K

-i =~ -:=i

3

13 B

3 -3 -3

B -B -JJ 13 -13 -B

i

4 4

2

-3 -3

2 -2 .

. -3

G

(J

(J

G

-l1

-2

1

-1

i

1 1 -1 -]

-1

-1 -1 -J

-1

2

-2 -i I

I

1 1 1

i -i -1

i -i

=fl

3 3 3

-!5 -D -!5 -D -D -tJ . -D -iJ

-K

3

l 2 -1 -1 (] C

2

i -i -i 1 i

-6

2 -2

i

1 2 -1 -1 C (]

1 1

I

C

6

i

C

=I1

C

1

.

2

-3 -3

-1

2

i

2

-2 -2 -2 . -3 -3 -K

-1 -1

2 -1 -2 -2 1 -2 1

-2 -2

-3 -3

-1 -1

1

-2

-1

1 1 1 I L -L -~ -L L

i

C (] 1

-2 -2

-1

-3

1 (] C 1 -1

3

-3 -3

i -i -i -i

1 -1 -1 1 -1 -1 . . .

-1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 -1 1 -1 -~

1

i5

-!5 -D -D -.!5

: -3 -3 -i -1 -i

-3 -3 (] C K K

9a 5a 5a 3c 10a 10c 9a 2a 2b 1 1

-1 -1 -1 -2 2 . -D -15 -1 -1 -1 -2 2 . . . -D -D -1 2 2 -1 3 -1 -1 -1 -2 1 1 3 -1 1 1 1 1 1 1 4 -i -i 2 3 - l - I -i -i -i -1 -i -i 2 -2 C 2 2 (] 2 2 f5 G (J D (J G D f5 3 3 2 -2

-3 -3

I

8b 4b 8b 8b 1 1

1

-2 2

-i

2 2 -1 -1

-i -2 -1

i -i -i -i -2 i -i -i

3

-4

2

2 -i

-2

-i -i -i -i

l

Suzuki's group Suz

662 Character table of H

= (x, y, h)

(continued}

12h 12i 12j 12k 18a 18b 20a 24a

X.1

1

1

1

1

1

1 1

1

X.2-D-D-D-D D jj X.3-D-D-D-D jj D X .4 -1 -1 -1 -1 . X.5 jj D jj D -D -D X. 6 D jj D jj -D -D

1

-1 1 1

-1

X.7

x.s -i -i -i -i -i -i

X.9 X.10

§:B

i

i

i

i

=~

X.13 jj I! D D jj . X.14 D D D X.15-D-D jj D -D -D X.16-D-D D jj -D -D X.17 X.18 X.19 . -D -D X.20 _. _. . -D -D X.21 -D -D -D -D

-i

-1 -1 -1

X .22 -D -D -D -15

X.23 X.24 X.25 X.26 X.27 X.28 X .29 X.30 X.31 X.32 X.33 X.34 X.35 X.36 X.37 X.38 X.39 X.40 X.41 X.42 X.43 X.44 X.45 X.46

i -i

1

-1

-~ -i 2

1

-1

. 1 1

1

jj jj

1

-1 jj -1

-1

1

-1 jj

D

. -D -D -D -D

D D

D

-i -i -i -i

-D -D -D

-1

-D

D

D

D

D

D D

D

-i -1

jj

-i -1

X.47 X.48

X.49

H~ X.52 X.53 X.54 X.55 X.56 X.57

-i -i jj

D

i

i

D -D -D D-D-D

-i -i

where A = -3((3) - 1, B = -((3) - 2, C = 2((3) + 1, D = ((3) + 1, E = 3((3) - 2, F = -9((3) - 6, G = 4((3) + 2, H = 6((3) - 2, I = -2((3), J = -9((3), K = -3((3) - 3, L = 2((5) 3 + 2((5) 2 + 1, M = 6((3) + 3, N = 6((3) + 1, 0 = -3((3) - 8.

663

14,6 Character tables of local subgroups 14.6.2 Character table of E = Nc(A) = (x1,Y1,el) 13 3

13

9

7 3 1

2P 3P 5P

2a

2b

2c 2d 3a 3b 3c 3d

4a

4b

4c 4d 4c 4f 4g 4h 5a

5b

6a

6b

6c

6d

6c

6f

la 2a 2a

la 2b 2b

la 1 a 3b 3a 3c 3d 2c 2d la la la la 2c 2d 3b 3a 3c 3d

2a 4a 4a

2a 4b 4b

2a 2a 2a 2a 2c 2d 5b 5a 4c 4d 4c 4f 4g 4h 5b 5a 4c 4d 4c 4f 4g 4h 1 a la

3a 2a

3b 3c 2a 2a 6a 6c

3c 2a 6e

3c 2a 6d

3a 2c

3b 3d 4c 2c 2b Sa

Sb

6g

bf 6h Ba

8b

1 C C

1

1

A A A A

A A A A

1

1

1

-1 -1

A A A A

A A A A

-1 -1 -1 -1 -1 -1

1

1 10 15 15 18 45 45 54 54 60 60 60 60 60 60 72 90 90 120 120 120 135 135 180 180 180

10 15 15 18 45 45 54 54 -4 -4 -4 -4 -4 -4 72 90 90

-8 -8 -8 135 135 -12 -12 -12

-12 180 -12

180 180 180 360 540 540

-12 -12 -12 -24 -36 -36

10 15 15

-6 5 5 -18 -18

1

1 5 5 -1 1 5 2 -1 2 G 2 G 8 8 -1 -1 8 8 -1 -1 1 9 9 1 I [ 1 [ I

a

X.22 X.23 X.24 X.25 X .26 X .27 X .28 X.29 X.30 X.31 X.32 X .33 X .34 X .35 X .36 X.37 X .38 X.39 X .40 180 X.41 X .42 X .43 X .44 X.45 X .46 X .47

10

la la la

x.10 x.11

X.21

10 l

la

x.1 x.2 X.3 X.'l X.5 x.s X.7 x.s X.9

x.12 X .13 X.14 X.15 X 16 X 17 X 18 X.19 X.20

7 3 1

a

-2 -2 10 10 -1 -1 J J -1 -1

J

10 15 15 2 -3 -3 6 6 -4 -4 -4 -4 -4 -4

.J

6 -2 g

-3

-6 -6 8 -4 15 15 -3 4 15 15 -3

4 8 -4 4 8 -4 -24 10 -30

J .J J J J J J .[

3 3 3 3 3

6 -2 . -3 8 30 30 3 8 N fi -3 8 fi N -3 15 -9 -1 3 -5 15 . -12 12 -4 45 45 -8 4 45 45 . -4 0 6 . -4 6 0 -8 4 6 0 -8 4 0 6 -3 8 12 -12 -8

8

-3

-6 10 -8 -8

. . . .

-1 -1 -1 -1

B B C C

B B

5

1 1

1 1 C

B C B 1 -1 -1 1 -1 -1 1 -1 -1

1 1 -2 -2 -2 -2

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

6

5 2

9 [

I 10

-1

-1 -1

.]

J

-1 -1

f5

K L

1 1 1 1 2 -1

1 -1 -1

1 H

J1

J1

H

-1 -1

-1 -1 1

. -!5 -D -2 -1 -1 -1 -1 -2 . -1 3 1 -1 -3 -3

L K

8 -4 -4

. -2

-1 -1

-2

D

f5 -1

-1 -1 M

-1 -1

8 -4 -4 -4 4 8 -4 -4 -4 4

A

D -1 D -1 f5 -1 3

D

-2

i.11 M 3

M M

f5

-2

f5 f5

M

M

A D

f5

-2

-3

-3

2

2

H FI

it

D

-3 -3 6 6 6 6 -2 -2 6 8 -8 8 -8

-2

-2 -5

-H -it -it -H

8 -8

-9 -9 -1 -1 -9 3 -5 -5 3 -4 8 -4 12 -4 -12 -4 4 4 12 -8 4 -4 12 12 -8 4 12 -8 4 4 -4 12 4 -4 12 -8 8 -8 -12 -12 -8 4 -12 12 4

3 -1 -3 1 2 -2 -2 2 -2 -2 2 2

M

-1 3

-1 3

-2

A

A

3 -2 -2

-3 1

1 -IV.I 1 -M

H 3 -1 -3 3

-3 -3

-3 -3

-3 -3

-A -A -A -A

-A -A

-A -A

. -3

2 -2

-1 -1 -1 -1

. -D -!5 1 -2 -2 f5 D D f5

[

J

D

1

10

J

D D

-1

a

8 -1 8 -1 9 I

f5

f5 f5

4b

1

3 -1 1 5 -3

-4

4 4 8 -6 10 8 8

a G

5 -1 5 2 G

1 D

6 -2 -2 1 1

-3 -3 -6 -6

s

12 12

1

6

-9

8

1 1

-6 -6

-8 4 4 -12 -12 -12 -12 -12 -12

-1 -1 -1 -1

6

-9

-9 .

10 15 15 2 -3 -3

-1 -1 -1 -1

1 1 1 -1 1 1 -1 1 1 -J 1 J -1 1 1 1 -1 -1 1 -1 -1 2 2

6b

6g 6h Sa 8b

-3 -3 -3 1

-A

-3 1

. -2

-A

-A -A -15 -D -D -15

-2 -2

Suzuki's group Suz

664

Character table of E = Nc(A) (continued) 2

3 5

6

5

8, 2P 4a 4e 5a 5b 3P Be Sd 10b 10a 5P Sc 8d 2b 2b 1 1 X.1 1 X.2 -1 C B X.3 -1 C B X.4 -1 B C X.5 -1 B C X.6 X .7 X .8 X .9

X.10 X.11 X.12 X.13 X .14

1

1 2 2

1 1

C B

-1

-1

1 X.15 -2 X.16 -1 -1

§:H -~ -~

-1

1 1

B C

-1

12a 12b

Ba

1

15 D D

D 15 15

1 1

D 1 1

15

ii H

.

1 E E F

1

H

i'I .

.

.

1 1 1 1 -1 -D -D -D -D -1 -D -D -fJ -D

-2

-i -i

D 15

-2

15 D

-2

D 15

§:§~_§ X.22 -2

-D·

-D

-D·

Ba

24b 24a J 1 E -D -i5 E -i5 -D P -i5 -D

1

F -D -i5

-D

-1 -1

-1 -1

-1

C

i

i

. -D

-D

-i -i

B

-1

D [J

B

C

-1 [J

B

C

-1

-1

C

D

B

.

1

-15

-D

D

15

i5

D

D

15

D

i5

.

1

i5 -fJ -D D -D -fJ D

15

~-Fi-d . -C -B

X .23 X .24

3

-1 [J

X.25 X.26 X.27 X.28 X.29 . X.30 -2 X .31 2 X .32 X .33 X .34

-1

-1

15

X.19-3-1

E

1 P F E E

-15 -D -iJ -D

-i

-2 D

F

1 F F E

A

D A

2

ii

.

.

X.36 -1

1

X .35

3 -1

X.37 X .38 X .39 X.40 X.41 X.42 X.43 X .44

. -2 2

X .47

2

H

i

-3

-D

3 -i -i -i

-1

-1

-1

D A

15 D D

D

-1 -1

jj D

-1 -1 -1

fJ

A

15

15

i

-D

-A -A -D -b -A -15 -D

. -A

15

-i -]

i

1 -D

i -1

. -15

-D -D -D -D

§:!~ . -2

15 -D D

i

-1

D

-D 15 -D

i

-1

i5

-D

D

-i5

-1 -[ -[

H -H -H

1

1 -D

i

2 -2 -2

i

1

D -D

-[

H -H -H

-3 -D

-i -i -15

. -D D

-i

15

i -1

-D

-D

15 D

-i -i 1

1

D 15 15 D -15 -D -D -15

where A= -3((3) - 3, B = ((5) 3 + ((5) 2 + 1, C = -((5) 3 - ((5) 2 , D = ((3) + 1, E = ((15)3((15)~ +((15)3((15)g +((15)3, F = ((15)3((15)~ + ((15)3((15)g + ((15)g + ((15)g, G = 6((3), H = 2((3), I= -9((3) - 9, J = 15((3), K = 3((5) 3 + 3((5) 2 + 3, L = -3((5) 3 - 3((5) 2 , M 4((3) + 1, N = 30((3), 0 = -45((3) - 45.

14,6 Character tables of local subgroups

665

14.6.3 Character table of D = NH(A) = (x,y) 2 13 3 2 la 2P la 3P la 1 1

I 1 1 I 2 2 2 3 3 3 3 3 3 4 4

13 2 2a la 2a I 1 I 1 1 1 2 2 2 3 3 3 3 3 3

8 8

8

4 4 4 6 6 6

8 9 9 9 9 12 12 12 12 12 12 12 12 12 12 18 18 24 24 24 24 24 24 24 24 24 24 24 24 24 24 32 32 32 32 32 32 32 32 32 36 36 36 36 72 72 72 72 96 96 96

9 9 9 9 12 12 12 12 12 12 12 12 12 12 18 18 24 24 24 24 24 24 24 24 24 24 24 24 24 24 -32 -32 -32 -32 -32 -32 -32 -32 -32 36 36 36 36 72 72 72 72 -96 -96

-96

10 2 2c la 2c 1 1

I 1 1 I 2 2 2 3 3 3 3 3 3 -4 -4

4

4 4 4 4 4 6 6 6 8 8

4

12 I 2b la 2b

-4 -4 -4 -4 6 6 6

-8 -8 -8

9 12 12 12 12 12 12 12 12 12 12 18 18

-8 -8 -8

9 9 9 9 -12 12 12 -12 -12 -12 -12 -12 -12 -12 18 18

24 -24

-8 24

24

-8 -8 -8 -8

-8 -8 -8 -8

36 -36

36 -36 36 36 36 36 -24 -24 -24 -24

9 I 2d la 2d J 1 1 l 1 I 2 2 2 -1 -1 -1 -1 -1 -1 4 4 4 4 4 4 6 6 6 8

8 2e 2f 2g 2h la la la la 2e 2f 2g 2h 1 1 1 1 -1 1 1 -1 -] 1 I -] 1 1 1 1 1 1 1 1 -1 1 I -1 2 2 2 2 2 2 -i --1] 3 -i I I 3 -1 -1 3 -1 -1 -1 3 -1 1 -1 3 1 1 -1 3 1 -2 2 -2 2 2 -2 -2 2 2 -2 -2 2 -2 -2 -2 -2 -2 -2

7 7 2 2 3a 3b 3c 3b 3a 3c la la la 1 1 I 1 A A A A A A A A 2 2

B 13 13 B

3 3 3 3 C () () C C () () C 1 1 1 I

A A A A A A A A

2 2 -i

13 B -1 B 13 -1

8 8

9 -3 i -3 -3 3 1 3 -3 -3 1 -3 3 1 3 9 -6 -2 -4 -2 6 -6 2 -4 6 2 -4 -2 2 3 -2 3 -4 2 -4 -2 2 C -4 -2 2 () -4 2 -2 () -4 2 -2 C -6 2 -6 -6 -6 2 -4 2 4 2 -4 -2 4 -2 4 -6 -4 2 8 6 -4 -2 6

-8

4 4 -4 4 -4 -4 4 -4

i

1 1 1 1 1

-6 6 2 -6 2 -2 6 -2

-6

-4 -4 4 -4 4 4 -4 4

-4 6 -4 -6 6 6 4 4 -6 6 -4 -6 -4

2 -2 2 2 2 -2 -2 -2

-2

2 2 4 -2 6 -6 -2 2

1

1

1 A

A

A

3e 3d la 1 1

A

A

A

A A - ] -1 -A -A -A -A

-2 -2

-2 -2

-B -13 -13 -B

-13 -B -B -13

3 () C 2

13 B

3 C ()

2

B 13

4a 2a 4a 1 1 1 1 1 I 2 2 2 3

3 3 3 3 3 -4 -4 -4 -4 -4 -4 6 6 6

-8

-8 -8

9 9 9 9

4

-4 -4 4 -12 -12 -12 -12 -12 -12 18 18

3 3

()

C C ()

6

D D D

fj fj fj

D

fj

fj fj

10 3d 3e la

10 I 4b 2a 4b I 1 I 1 1 I 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 8 8

8

9 9 9 9 -4 -4 -4 -4 12 12 12 12 12 12 18 18

-4 -4 -4 -4 -4 -4 -6

-6

I 4d 4e 4f 4g 2b 2b 2b 2a 4d 4c 4f 4g 1 1 1 1 -1 1 -1 -1 -] I - [ -1 1 l 1 1 1 1 I 1 -] 1 - ] -1 2 2 2 -i -1 -i -i 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 -[ 1 -2 -4 2 -2 2 2 -4 -2 -2 -4 2 -2 2 -4 -2 2 -2 -4 2 -2 2 -4 -2 2 6 6 6 -8

D D

-i -i

E E 2 F Ji E E -4 -A -ii E E 2 I I E E 2 Ji F 2 K 8 8 K 2 I I E E E E -4 -A -A 8 8 2 R K

4

3

C ()

3

() C

12 -12 I 2 -12 -12 -12 -12 -12

7 I 4h 4-i ,1j 4k 2a 2b 2b 2c 4h 4i 4j 4k 1 1 I 1 1 I 1 1 1 1 1 1 I 1 1 1 1 I 2 2 2 2 2 2 2 2 2 -] 3 3 -1 3 3 -1 3 3 -1 3 3 -1 3 3 -1 3 3

4l 4m 4n 2c 2b 41 4n I 1 -1 I -] 1 ] 1 l 1 -1 1 2 2 2

-i

-]

4o 2b 4o 1 -J -1 1 1 -1

-i

1 -1 -1 1 1 2 -2 2 -2 2 -2

-1 1 -1 -1 -1 -1 -1 1 -1 1 2 -2 2 -2 2 -2 -2 -2 -2

i -3 -3 -3

i -3

-2 -2 -2 -2 -2 -2 -2 -2 -2

-8 -8 9 -3 3 -3 3 -3 -3 -3 3 9 3 -6 6 6 6 -6 -6 6 -6 4 -2 2 -2 4 2 -2 2 4 2 4 -2 -2 2 4 -2 4 2 -6 -6

-3

-3

-4 -2 -4 2 4 6 -4 -2 4 6 4 -6 -4 2 4 -6

6 -6 -2 6 -2 2 -6 2

6

-4

2 -6 -4 6 -4 2 6 2 4 8 -6 -2 4

-2 -6

-6 -6 6 -12 -6 -12 6 12 -6 12 6

3 1 -3 -3 3 5 5 -3 1 -3 -3 -3 3 1 3 1 -3 -3 2 -4 2 -2 -2 2 2 -2 -2 -2 -2 2 2 -2 -2 -2 -2 2 2 2 2

3 -3 3 6 6 -6 -6 2 -2 2 2 -2 -2

4 6 4 -6 -4 -2

8 -8 -4 -8 -8

-3 -3

fj D fj D 8 8 -4

8

4c 2a 4c 1 1 1 1 1 I 2 2 2 -1 -1 -1 -1 -1 -1 4 4 4 4 4 4 6 6 6 8 8 8 9 -3 -3 9

-6

6 -6 6 -6 6 -6 6

2 -2 -6 2 -6 6 -2 6

4 4 -4 4 -4 -4 4 -4

6 -4

-6 -6

2

6 6

-6 2 2 -6 -6 4 -4 2 -2 4 -4 -2 2 2 -2 4 -4

4 -4

8 4 4 4 4 4 4 4 4

-2

8 -4 -4 -4 -4 -4 -4 -4 -4

-6 -4 -4 -4 -6 -4 -4 6 -4 6 4 -6 -4 -4 4 4 -4 6 4 -6 4 -4 4

4 4 4 4 4 4 4 4 4

-4

-4 -4 -4

-8

2 -2 2 2 2 -2 -2 -2

-2 2 -2 -2 -2 2 2 2

2 -2 2 -2 -2 2 6 -6

4 2

-4 -4 -4 -4 -4 -4 -4 -4 -4

4 -2 2 -2 2 -2 -6 6

Suzuki's group Suz

666

Character table of D = NH(A) (continued) 8

5 5 6 4 4 4 1 1 2 2 2 4p 4q 4r 6a 6b 6c 6d 6e 6f 6g 6h 6i 6j 6k 2P2b2d2f 3a 3b 3b 3a 3c 3d 3e 3e 3d 3d 3e 3P 4p 4q 4r 2a 2a 2b 2b 2a 2c 2c 2c 2c 2a 2a X.1 1 I I 1 1 1 1 1 1 1 1 1 1 X.2 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 X.3 -1 -1 -1 A A A A AAAAAA X.4 1 1 1 A A AAAAAA A A X.5 1 1 1 A AAAAAA A A A X.6-1-1-1 A A A A AAAAAA 2 -1 -1 -1 -1 -1 -1 -1 X.7 2 2 2 B B B B -1 -A -A -A -A -A -A X.8 X.9 . . . B B B B -1 -A -A -A -A -A -A X.10 -1 1 1 3 3 3 3 X.11 1-1-1 3 3 3 3 (j X.12-1 1 1 (j C C (j (j X.13 -1 1 1 C C (j (j X.14 1-1-1 C C (j (j X.15 1-1-1 C C X.16 -2 2 1 1 1 1 X.17 2 -2 1 1 1 1 X.18-2 2 A A A A X.19 2 -2 A A A A X.20-2 2 A A A A X.21 2-2 A A A A X.22 X.23 X .24 X.25 2 2 2 X.26 B B B X.27 . . . B B B X.28-3-3 1 X .29 3 -3 1 X.30-3 3 -1 X .31 3 3 -1 X.32 -2 2 X .33 -2 X.34 2 . . X.35 2 .-2 X.36 2 2 3 3 3 3 X.37-2-2 3 3 3 3 (j (j X .38 2 2 C C . (j (j X.39 2 2 C C . (j (j X.40 -2 -2 C C . (j (j X.41-2 -2 . C C X.42

§:H-2

-2 -2 -2 -2

2

X.50 -2 X.51 2 X.52 -2 X.53 -2 X.54-2 X.55 2 X.56 2 X .57 2 X.58 X.59 X.60 X.61 X.62 X.63 X.64 X.65

6 -2 -2 -

D D jj

: -3 -3 .

3 3 . 3 3 . -3 -3

-3 -3 -3 -3

D D

D-B-B jj -B -B

.-E-E .-E-E

. -E -E . -E -E -8 -8 . -E -E

. -A -,1

.-E-E

§JL2

-2 -

X.68 2 X .69 2 X. 70 -2 X.71 -6 X.72 6 X .73 2 X .74 -2 X.75 X.76 X.77

2

8 - 8

.

4

i

. -A -A -A -A . -A -A -A -A

2

A

i

A

2

A

2

. -A -A -A . -A -A A . -A -A

.

. -2

2

2 -2 -2

B

-2 -2

G

c l

G

-J J (j

. -2 -J

:i -3 -3

i

G

G -F -F

G -F -F J -K -K

-G -C A A J -J -J - l -[

-J -[ - l

C -C -G

-J

J

A

J -K

A

-K

B

. ii

B

B B B -B -ii . -B -ii . . B ii -ii -B

i

G G -J J J -J

G (j

2

B

H

ti

B

B -ii

H

H

H H

. 2 . -2

.

.

. -2 2 2 -2

a

3 -3 -3 -3 -3

(j

C -C -G -G -C

C

(j -G -C -C -G

B

B

.

B -ii

. -B -B . -ii -B . -B

-2

B -B -B

H H H

2

ii . -ii . B

ti

H H

A

-2 -2

3 3 -2 -2 -3 -3 -2 -2

3

2

B

ii

.. -A=i -A=f -A-f

2

-2 2 2

-ii B ii

. -2 . -2

i

: 2

i -2

-2

B

. -2 . -2

i

A

. -A -A

. -B

. -2 . 4

i

ii B

. -2 2 2 -2

-i -i

. -B -B . -B -B

.i

6s Ba 8b Be Bd Be Bf 12a 12b 1 2 c

. -1 -1 -1 -1

-G -C

-C -G -C -G -G -C

D -B -B D-B-B D -B -B jj -B -B jj -B -B D-B-B

. -8 -8

Br

. -3 1 -3 1 -1 -1 3 1 -3 -1 1 1 3 -3 1 -1 -1 -1 . -2 . .

2

. 3

6q

: -3 -3

2

. . . -3

jj jj jj

6p

2

-2 -2

X.45 2 X.46 -2

i: n X.49

6l 6m 6n 60

B

14.6 Character tables of local subgroups

667

Character table of D = NH (A) (continued) 12d 12e 12f 12g 12h 12i 12j 12k 12l 12m 12n 2P 6d 6d 6c 6d 6c 6a 6b 3P 4d 4e 4e 4f 4f 4g 4g X.l 1 1 1 1 1 1 1 1 1 1 1 X .2 -1 1 1 -1 -1 -1 -1 1 -1 1 1 X.3-A A A-A-A-A-A A -A 1 A X.4 A A A A A A A A A 1 A X.5 A A A A A A A A A 1 A X.6-A A A-A-A-A-A A -A 1 A X. 7 2 2 -1 -1 -1 X.8 B B -1-A -A X.9 . B B . . . . -1 -A -A i

X.10 X.11

-1 -1 1 -1

-1 -1

-1 1

-1 1

-1 1

-1 1

-1

X.12 -A -A -A -A -A -A -A X.13 -A -A -A -A -A -A -A X.14 A -A -A A A A A X.15 A -A -A A A A A X .16 X.17

1 -1 -1 -1

-1 -1

-1 1

-1 1

1 -1

X.26 X.27 X.28 X.29 X.30 X.31 X.32 X.33 X.34

u~

:

. -A -A

1

. -A -A . A A

1 1 1 1

-1

1

1

1

. -A -A -i A A -i -i -A

. -A -A . -A -A

-2

X.47 X.48 . X.49 X .50 -B X.51 B X.52 X .53 -B X.54 X.55 X.56 B X.57 X.58 X.59 X.60 X.61 X.62 X.63 X.64 X.65 X.66 X.67 X.68 X.69 X.70 X.71 X.72 X.73 X.74 X.75 X.76 X.77

A

A

A

i

A

1

-1

A

-A -A A A -A -A

-i

-i

i

i 1 A

1

-1

-1 -1

-1 -1

-i 1 -A

i

-1

4 A . A A . -A -A

-1

.

i

-1

-A -A

.-A

-2 -2

-1 -1 -1

-1

. -B -B , -B -B

X.37 X.38 X.39 -A X .40 A X.41 A X.42 X.43 X.44

u~

-1

. -A -A

1 -i -1 -1

A-A-A-A-A A A -1 X.19-A-A-A A A-A-A -1 X.20 A -A -A -A -A A A,-1 §:H -A -A -A A A -A -A -1 -i X.23

-1

. -A -A

i -A -A

4 A . A A . -A -A

X.18

§Jt

120 12p 12q 12r 24a 24b 6e 12a 12b 4b 8a Ba 1 1 1 1 1 1 1 1 -1 1 -1 -1 -A A A 1 -A -A A A A 1 A A A A A 1 A A -A A A 1 -A -A

i

1 -1

i -i -i

-1

1

1 ,

1

A

-i -i 1

1

. -A -A

A A

A

i

i

-1

-1

A

. -A -A

1 i 1 =i =1 A -A -A A A A -A -A

-1 -1 -l

A

. A A . -A -A . -A -A

A A

-2 -2 2

2

-2 -2 -i B B . -B -B

. -B -B

. -B -B

B

B

B

B

B .

. -B

B .

-B

i

A A

A

A

A A A

-i -i

i

1 A A 1

1

1 A A A

-i -i . -A -A

. -A -A

. . . . .

-1

-1

-A -A -1 -A -A

-A -A -1 -A -A

. -A

-A

i

i

. -A -A

A A A A

where A= -((3)-1, B = 2((3), C = 3((3), D = -6((3)-6, E = 8((3), F = 2((3)+3, G = 2((3)+1, H = 4((3)+2, I= -((3)-3, J = ((3)+2, K = 3((3) + 2.

Suzuki's group Suz

668

= Cc(u) = hll :S k :S 5)

14.6.4 Character table of U 8 2

la

2P 3P 5P 7P

la la la la

2a

1 1 2b 2c 2d 2e 2f 1a la la la 2c 2d 2e 2j 2c 2d 2e 21 2c 2d 2e 2j 1 1 1 1 1 1 -1 1 1 1 -1 -1 1 1 1 -1

] -~ 4

4

3a 3a la 3a 3a

4 -2 -4

3 3 3 3 3 3 3 3 3 3 3

-1 -1 1 1 1 l 1 1 -1 -1 -l

4c 2c 4c 4c 4c

4d 2c 4d 4d 4d

4e 2c 4e 4e 4e

4f 2c 4f 4f 4f

§ -2

1 1 1 1

1 1 1 l

1 1 1 1

1 1 1 1

1

2

2 -2

2

4

-3 -3 3 -3 3 -3 3 3 -3 3

4g 2c 4g 4g 4g

4h 4i 4j 4k 41 5a 6a 6b 6c 7a 8a 8b 8c 8d 8a 2c 2c 2c 2c 2d 5a 3a 3a 3a 7a 4c 4e 4f 4b 4h 4i 4j 4k 41 5a 2a 2e 2b 7a Sa Sb Sc 8d 4g 8C 4h 4i 4j 4k 4l la 6a 6b 6c 7a Sa Sb Be Bd Sa 4h 4i 4j 4k 41 5a 6a 6b 6c la Ba Sb Sc Sd 8, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 - ] -1 -1 1 -1

-2

2 -2 -2

2

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

i

1 -1 1 -1 1 -1 1 -1 1 -1

-1 -1 -1 3 3 -1 -1 -1 -1 3

2

-1

-1 , 3 3 -1 -1 -1 -1 3 3 -1 -1

3 -i -i -i -3

-1 -1 -1 -1 3 3 -1 -1 -l -1 3

3 -1 -1 3 -1 -1 3 -1 -1 3 -1

-1 3 3 -l -1 -1 -1 3 3 -1 -1

3 -1 -1 3 -1 -1 3 -1 -1 3 -1

=i 3 -1

i

-1 -1 J -3 1 -3 -1 -1 -3 1 3 -1 1 1 -1 3 -1 3 1 1 3 -1

i

1 1 3 1 -1 -3 -1 -1 -3 -1

-1 1 1 1 1 1 1 -1 -1 -1

8 -~ 8 -8 . -8 1 -8 1 8 . -8 6 -6 8 : ~ 6 -2 -6 -2 2 2 6 -6 . -2 . -2 -2 2 6 2 -6 . -2 . -2 6 2 -2 -6 2 . . . . 6 -6 -90 -6 -6 . 6 2 2 2 2 2 2 -2 -2 -2 90 -6 -6 . -6 2 2 2 2 2 2 2 2 2 -6 6 2 2 -2 2 -2 -2

-64

64 -64 64

126

-126-2-2 2 . -2 2 . -2

-2

2

1 1

2

-~

-1 1 -1 1 -1 1 -1 1 -1 1 -1

2

2

2 -2

. . . . . . .

-1 . -1

.

-l

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 1 1 1 1 1 1 -1 -1 -1

2 -1

i -i

-1 1 1 -1 1 -1 1 -1 -1 1 -1

-11 -i1

-i

12 - l

1

1

2

1 1 -1

i -i -i -i

-1 1 1 -1 l 1 1 -1 1 -1 -1 -1 1 . -1 1 -1 -1 1 . -1 -1 1 l 1 -1 -1 -1 1 -1 -1 l 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 1

-1

-1

1 1 1

• -i -1 -1 -1

i

2

-1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1

-i

2 -2 -2 -1

2

_§ =§ _§ =~ =§ -2 -2 -2

-2-2-2-2-2-2 -2 -2 2 -2 2 2 -2 -2 2 -2 2 2

1 -1 -1 1 1 -1

-§ 2 -2

-r : -i _f

. -i

-1

=~ -~

2

2 -2

2 -2

-I i j 3 -1 -1 -1

§ § -i _§ 3 3 3 3 3 3 3 3 3 3 3

4b 2c 4b 4b 4b

1 -1 1 1 1 -1

2 -4

4 -2

4a 2a 4a 4a 4a 1 J

1 1 1

.

-2 -2

Character table of U

= Cc(u) (continued) 2P 3P 5P

7a

14b 146 2b

7P

I

-1 1 -1

1 1 1 1

1 1 -1 -1

1 1

-1 -1

- 2

1 -1 1 -1

-2 2 -2

1 1 1 1 1 1 1 -1 -1 1 -1 -1

=t -1

i

i

1

1 -1 1 -1 1 -1 1 -1 1 -1 1

l -1

1

-1 -1

1

l -1

1

-1 -1

-]

-i

-1 -1 -1

1

1

-1 1 -1

-1 1 -1

-i _j

i

1 -] -1

1

-1

1

-i

i

-1

i

-1

-1

I I

A

1

A

1 -1

1

i A A

-1

-1

-1 B -B -1 -B B

2

where A= 2((7) 4

i

1

-1 -1 -1 2 -1 -1 -1

+ 2((7) 2 + 2((7) + 1,

-2

B = 2((5) 3

+ 2((5) 2 + 1.

669

14,6 Character tables of local subgroups

14.6.5 Character table of N1 = Ne (3A)

2P 3P 5P 7P

la la la la

la 1 1 21 21 70 72 90 90 90 90 140 140 189 18 9 210 210 252 378 420 420

2a la

2a 2a

2a 1 1 5 5 6

2b I a 26 2b 2b 1 -1 -1 1

8

10 I0 -6 -6 12 10 12 -10

-3 -3

-9

9 2 10 2 -10 28

-6

4 10 4 -10 560 -16

560 -16 560 -16 560 -16 630 630 630 630 72 9 729 896 896 1260 1280 1440 14 5 8 1512 1890

22 22 -10 -10 9 9

3a

3a

1a 3a 3a 1 1

3b 3b la 3b 3b 1 1

21 70 -36 90 90

-6 -6 16 18 9 9

140 140 189 189 210 210 -126 -189 420 420 560 560 560

5 5 27 27 21 21 -18 54 -39 -39 20 20 -34

21

-45 -18 -45 -18

3c

3d 3 e 4a 4b 4c 5a

3c

3d 3c 2a la la 4a 3d 3 C 4a 3d 3 C 4a J 1 1 1 3 I 1 3 3 7 -2 6 8 : -2 -2 2 2 4 -4 -4 4

la

3c 3c

5

3

5 -2

3 -2 4

JO

0 -3

4 4

6 -3 11 2 11 2 2 2 2 2

560 -34

4b 4b 4/; 1 1 1 I -2

4c 5a 4c la 4c 5a 1 1 -1 1 3 1 -3 1

6a

6b 3a 3 b 2 a, 2a 6a 66 6a 6b

2

2 4

1 1 6 6 -3 -3 3 3

2 -4 2 2

;

'-2 1 -1 -i 1 -I I -2 -2 -2 2 4 i : -2 2 2 -2

5 5

36 630 -18 36 36

2 2

. -2 -2 6 -2 6 -2 : -3 1 -3 1

-3 -i

3 -1 1 1

4 -4

-16

: -6

: -2

-8

2

2

=

J

6g 7a 3e 7a 2 b 7a 6g 7a 6g la 1 1 1 J -1 -1 -1 . -1 1 3 -4 2 -1 J . -1 1 '-1 -1

1

-1 -1 -1 -1 3 3 -i 2 2 1 1 J 1 1 1 -3 -3 -3 -3

i

-1

Sa 8 b 4a 4a Sa Sb Sa Sb 8a 8 b 1 1 -1 1

Sc 4a 8c Sc 8C 1 -1 -3 -1 1 3 -1 -1 -2

2

-2

2

'-2 '-1 1

3 3 .5 -i -i -i i 5 -1 -1 -1 -1 1 -2 -2 4

: -4

i

: -2

2 2

2 2

-2 -2 -4 2 -4 2

E

2 2

8d 9a 4b 96 9a Sd 3c 3 C 8d 9 6 9a 8d 9a 96 1 1

-1 1

1

1

E

E

E

6

-2 -2 -2

2

;

: -i -i

-1 -1 ' -2 1 -1 J 1 -]

i

-6 -16

2

'-2 2

-9

12 15 -6

6

i

2

-2

-2

-3

' -2 3 -1 1 -3 -1

i

-i

i

1 -1

-i -i -i

- ] -1

-2 2

-3 -3

-1

_;

-2

3

Nc(3A) ( continued)

2' 1 1

; -i1

i

-i -i -i -1 -1

2

1 -1

-1

-1

-1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 1 1 1 1

B

2 -1 -1

2 1 1 -1 -1 1 1 -2 -2 1 1

4

i

1 -2

-2

-2 -1

A

-i

-i

-1 B

A

-i -1

-i

1

1

-i1

-1 1 -1 -1 1 1 -1 -1 1 1

i

i -2

-2 -2 -2 I I I J

-i1 -i

i

-1

-1

D jj

jj D

-i

-i

2 1 I

-i -1

-] -]

1

1

-2 -2

2

-i -i

-i -i

i

-1

i

-I

1

-]

-3

1

1

-i

2 -4

-4 -2 -2 -1 -1 4 4 5 5 -2 -2 -2 -5 4 4

C

-1 -1 -1 -1

i

2

0 -3 -4 0 -3

i

-1

4

C

-2

1

1 1 1

1

-1

6

2 2 4 1 2 2

4 -2 -2 -2

-4 -4

Character table of N1

6d 6e bf 3d 3d 3d 2a 2a 2a 6 C 6d 6f 6d 6 C bf I 1 1

-2 6 -3 I 1 -2 -2 -2 1 -2 -2 -2 -1 I

. -2 -2

-315

6c 3c 2a 6c 6c 1 1 2 2

-4 -4 -1 -1 -1 -4 -4 -1 -1 -1

-4 -4 12 32 18 -24 -30

2a 2a 5a

(g1 ' 92' 93' 94)

1 -1

-i

where A= 2((21)3((21)t + 2((21)3((21)~ + 2((21)3((21)7 + ((21)3 + ((21)t + ((21)~ + ((21)7 + 1, B = -A+ 1, C = 3((3) + 2, D = 2((24H((24)3 + ((24)g - 2((24)s((24)3 - ((24)s, E = -2C.

3

4 -1

Suzuki's group Suz

670

14.6.6 Character table of N2

2P 3P X.I X.2 X.3 X.4 X.5 X.6

X.7 X.8

X.9 X.10 X.11 X.12 X.13 X.14 X.15 X.16 X.17 X.18 X.19 X.20 X.21 X.22 X.23 X.24 X.25 X.26 X.27 X.28 X.29 X.30 X.31 X .32 X.33 X.34 X.35 X.36 X.37 X.38 X.39 X.40 X.41 X.42 X.43 X.44 X.45 X.46 X.47 X.48 X.49

4 4 2b 2c la la 2b 2c I I -1 I -1 I I I -2 -2 2 2 2 . -2 . -2 3 -1 -3 -1 -3 -1 3 -1

la 2a 3a 3b 3c la la 3a 3b 3c la 2a la la la I I I I I I I I I I I I I I I I I I I I 2 2 2 2 2 2 2 2 -1 2 2 2 2 -1 2 2 2 -1 2 2 2 2 -1 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 -4 4 4 4 -2 4 -4 4 4 -2 6 6 6 6 8 -8 8 8 2 32 32 8 32 8 -8 32 32 32 8 -8 32 32 32 -4 32 8 32 32 -4 -8 32 32 32 -4 32 32 32 -4 8 9 -18 -12 36 12 36 -6 : -18 9 12 . -18 36 6 9 9 -18 -12 36 . -18 36 6 9 -6 -6 . -18 36 9 -6 . -18 36 6 9 -6 -6 36 . -18 9 -6 48 48 48 8 48 8 48 48 48 -8 48 48 48 -8 48 48 18 -36 72 12 8 72 -8 . -36 18 6 72 -8 . -36 18 6 72 -8 . -36 18 -12 . -36 18 -12 72 -8 6 72 -8 . -36 18 6 72 -8 . -36 18 18 -36 72 -8 12 18 -36 72 -8 12 27 -54 108 12 108 12 18 : -54 27 12 -18 108 . -54 27 27 -54 108 12 144 -16 36 -72 -12

3d 3e la I I I I 2 -1 -1 -1 -1

3e 3d la I I I I 2 -1

-1

-1 -1

-2 -2

-2 -2 2 8 8 -4 -4 -4 -4 6 3 3 6 E E E E

8 -4 -4 -4 -4 6 3 3 6 E E E E

-6 -6

-E_ -E -E -3 -3 -E -E -6 -6

= Nc(Z) = (t1, t2, t3, t4)

I 5 3/ 3g 3h 4a 4b 4c 4d 3f 3g 3h 2a 2c 2a 2a la la la 4a 4b 4c 4d I I I I I I I -1 I -1 I I I -1 I I I I I I -1 I I I . -2 2 2 2 -2 -1 2 -1 2 2 2 -1 2 . -2 -1 -2 2 2 3 -i 3 -1 -i 3 -1 -1 I 3 I I 3 -1 3 3 -1 3 -1 I -2 -2 . 4 -2 2 4 6 -2 . -6 2 8 8 5 -4 8 5 -4 8 -4 5 -4 -4 5 -4 -4 5 -4 5 -4 -4 -3 4 -6 4 2 -2 -6 4 . -3 4 -2 4 3 4 2 3 4 . -2 3 3 4 2 . -6 8 . -6 : -8 . -6 . -6 3 -3 -3

-E -3 -3 6 -E -3 -E -3 -6 3 -6 3

6 -:i

4e 4/ 6a 6b 6c 6d 2a 2c 3b 3b 3a 3b 4e 4/ 2b 2b 2a 2a I I I I I I 1 -1 -1 -1 I I -1 I -1 -1 I I -1 -1 I I I I 2 2 -2 -2 2 2 2 2 2 2 2 2 2 2 i -i :i :i 3 3 -1 -1 -3 -3 3 3 1 -3 -3 I 3 3 -1 I 3 3 3 3 . -4 -4 2 . -4 -4 : -2 6 6 : -8 -8 8 8 : -8 -8 . -8 -8 8 8 . -8 -8 8 8 i -2 :i :i -2 I : -3 -3 -2 I -2 I -2 . -3 -3 -2 I I 3 3 -2 I . -3 -3 -2 I 3 -2 3

H

H H H H

H

: -4

. -4 . -4 . -4

H H

H H H

. .

: -9 -9

2

: -2

ii

-2

9

9 :

2 4 4 4 4 4 4 -2 -2 3 -6 -6 3 -4

2 3 6e 6f 6g 6h 6i 3c 3d 3e 3c 3c 2a 2b 2b 2b 2b I I I I I I -1 -1 -1 -1 I -1 -1 -1 -1 I I I I I 2 -1 i i i i -1 -1 -1 -1 -1 -1 A A A A -1 A A A A

2 3 6k 3e 2a I I I I 2 -! -! -! -1

2

-2 -2 2 2 2 2 -2 -1 -2 -2 B B B B . -B -B -B -B B B B B . -B -B -B -B 4 -2 -2 4 :i :i I I 4 -3 -3 I I 4 -2 -2 -2 F ff' . -CJ. -c -2 -F' -F . -CJ. -c -2 ff' F . -c -c -2 -F -ff' . -G -G -2

-4 -4 -2 -2 -2 -2 -2 -2 4 4 -6 3 3 -6 8

6j 3d 2a I I I I 2 -1 -1 -1 -1

i

j J ./.C -CJ. -c2 -2 I -2 - [ -f -J -! -G -G 4 A A K K I I 4 A A K K I I -2 -f - [ -J -J -c -c -2 t I J J -c -c 4 . -2 -2 4 -2 -2

-4

14.6 Character tables of local subgroups Character table of N 2 6! 2P 3d 3P 2b X.l 1 X.2 -1 X.3 -1 X.4 1 X.5 X.6 i X.7 -1 X.S A X.9 A X .IO X.11 X.12 X.13 X.14 X.15

2 2 6p 3h 2c 1 1 1 1 -2 2 2 -2 -2 3 -1 -3 -1 : -3 -1 3 -1

ii

X.16

X.17 X.lS X.19

X.20 X.21 X.22 X.23 X.24 X .25 X.26 X.27 X.2S X.29 X .30 X.31 X .32 X.33 X.34 X .35 X.36 X.37 X.3S X .39 X.40 X.41

i

-2

i

-~B

B -B -ti ti B -ti -B

-3 -3 3

= Na(Z)

2 1 1 3 3 3 6m 6n 60 3e 3f 2b 2a 1 1 1 -1 1 -1 -1 1 -1 1 1 1 2 i -1 -2 -1 -1 2 A -1 A -1

3

-F -ff' ff' F -ff' -F F ff'

3 1 Sa 4a Sa 1 1 -1 -1

-i

I I 1 - I -I 1 A A -2 A A -2 -I - I 1 X .42 1 I I 1 X.43 1 X.44 X.45 X.46 X.47 X.4S X.49 -i

Sb 4c Sb 1 -1 1 -1

.

-i

i

1 1 -1 -1 1 -1

-i

-1 1 1 -1 1 . -1 1 -2 -2 1 1 1 1 1

3

1 3 9a 9b 3b 1 1 1 1 2 -1 -1 -1 -1

-2

-i

-1 1 1

{continued) 9b 12a 9a 6d 3b 4a 1 1 1 1 1 2 2 -1 2 -1 2 -1 2 -1 2 -1 -1 -1 -1

-2 -2 -2 -2 i

2

-1 -1 -1 -1 C Q C C CJ C CJ C

12b 6c 4a 1 1 1 1 2 2 2 2 2 -1 -1 -1 -1

-i -2

12c 12d 12e 12! 12g 6d 6c 6e 4d 4e 4c 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -2 -1 -22 -1 1 1

1~ 1~

-i -i -i -1 -1

i

lSa I8b 24a 24b 9a 9b 12b 12b 6a 6b Sa Sa 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1

i

-1

A

A

i

-1 A

A

1 -1 1 1 1 1 1 1 -1 -1 -2 -2 2 2

-i1 -i

1 -1 -1 1 1

-i -i

-2

1 1 -2 1 1 1 1

i

-2 -2 1 -2 -2 -2 -2

-i1

-i -i 1 G

-4

2 -i -1 2 -2

671

-1 2 2 -1

a

-i

1 1 jj D . -D -fl jj D -fl -D

-i -i

1 -1 1 -1

1

a

G

-i

-i

i, L [, L -1 -1

1

where A = 2((3) + 1, B 2((3) + 2, C = -3((3) 1, D = ((3), E = -9((3) - 6, F = 3((3) + 3, G = 3((4), H = -12((3) - 6, I = ((3) + 2, J = 2((3) - 2, K = 4((3) + 2, L = -2((24)~((24)3 - ((24)~ + 2((24) 8 ((24)3 + ((24)s

15 O'Nan's group ON

The simple group ON was discovered by M. O'Nan in [104]. He determined its order IONI = 29 · 34 · 5 · 73 · 11 · 19 · 31, character table and the group structure of the normalizers Ne ( (p)) of elements p of prime order. O'Nan also described a stabilizer of a faithful permutation representation of degree 122760 of ON. He derived all his results from a presentation of a given Sylow 2-subgroup of ON due to Alperin [l]. However, he neither proved the existence nor the uniqueness of ON. The first published existence proof is due to L. H. Soicher [121]. He also showed that a simple group G is isomorphic to ON if and only if a Sylow 2-subgroup of G is isomorphic to the one of ON. Soicher's proofs are not self-contained. They depend on O'Nan's work. It is the purpose of this chapter to present Previtali's and the author's existence and uniqueness proof for the sporadic simple group ON; see [94]. A finite simple group G is called a simple group of ON-type if it possesses a 2-central involution z such that Cc (z) ~ H, where H is the finitely presented group given in Lemma 15.1.1 of Section 15.2. This lemma also describes the group structure of H. In particular, a Sylow 2subgroup S of H contains a unique maximal elementary abelian normal subgroup A= (z, t, u) of order 8 such that C = CH (A) is homocyclic of order 64. In Section 15.2 it is shown that in any simple group G of ON-type there exists an element g of order 3 which is not contained in H such that zY = t, tY = zt and uY = u and E = Ne (A) = (r, s, g) is uniquely determined by H up to isomorphism. A set R(E) of defining relations for E with respect to the generating set {r, s, g} is also given in Proposition 15.2.2. In particular, E is an iterated extension E of T = GL3(2) by the natural F-vector space V of dimension 3 over F = GF(2) such that C = CE(V) is a homocyclic group of order 64 with invariants (4,4,4). 672

O'Nan's group ON

673

Furthermore, all conditions of the first three steps Algorithm 1.3.15 are satisfied; see Proposition 15.2.l. Therefore the centralizer H can also be constructed from the non-split extension E of T = PSL 3 (2) by C by application of Algorithm 1.3.15. This construction is not presented here. The subgroup E 1 = (z, t, u, r, g) of E is a split extension of A by a Frobenius group F21 of order 21; see Lemma 15.4.l. Applying Algorithm 1.3.8 to E 1 it follows that there is an element f of order 5 in H such that H1 = (z) X (r, uz, tz, j) is the unique subgroup of H which is isomorphic to (z) x A 5 in H and contains the subgroup D 1 = (A, r) ~ S4 such that the amalgam H 1 +- D 1 ---+ E 1 satisfies all conditions of Step 5 of Algorithm 1.3.8. Proposition 15.4.2 due to Previtali and the author [94] provides a set R(J) of defining relations for the subgroup J = (H1 , E 1 ) = (r, f, g) of any finite simple group G of ON-type. By Janko's uniqueness Theorem 9.4.2 of [92] it follows that J is isomorphic to the smallest Janko group J1; see also [66]. In Section 15.5 it is shown that H = (r, s, c) = (r, s, !). This fact enables us to derive a set R 1 (H) of defining relations of H = (r, s, !). Theorem 15.5.2 due to Previtali and the author [94] states that any finite simple group G of ON-type is generated by r, s, f and g and has the set of defining relations

R(G)

= R1 (H)

U R(E) U R(J).

Furthermore, the finitely presented group G has a faithful permutation representation of degree 2624832 with stabilizer J ~ J 1 . It is simple and Cc (z) = {r, s, f} ~ H. Hence, this theorem provides an existence and uniqueness proof for the O'Nan group ON. In Corollary 15.6.1 a permutation representation of G of degree 122760 is derived from this theorem. It is used to calculate the character table of G, determine the conjugacy classes and local subgroups of G. Soicher's uniqueness theorem is stronger than Previtali's and the author's uniqueness theorem. But it is an immediate consequence of Theorem 15.5.2 and O'Nan's Lemma 4.8 of [104], because Previtali has shown that there are exactly 36 non-isomorphic groups Hi of shape 4L 3 ( 4) : 2, and that the isomorphism type of Hi is uniquely determined by the isomorphism type of a Sylow 2-subgroup Si of Hi. Furthermore, these examples show that the given centralizer H = Cc (z) of a 2-central involution of a simple group G of ON-type has to be defined explicitly by generators and relations.

O'Nan's group ON

674

15.1 The centralizer of a 2-central involution

In this section we give a presentation of the finite group H which is assumed to be the centralizer of a 2-central involution in a group of ON-type. We determine a Sylow 2-subgroup S of Hand an elementary abelian normal subgroup A of S such that D = NH(A) is of maximal order among all normalizers NH (B) of the elementary abelian maximal subgroups B of S.

Lemma 15.1.1 Let H = (r, s, c) be the finitely presented group with the following set R(H) of defining relations:

r 3 = s 16 = c5 = (sc- 1) 4 = (rs 2 r- 1 s- 1)2 = (r- 1s2rs- 1 )2 = 1, s- 1r- 1c- 2s- 1r- 1crc2 = (r- 1c- 1s- 1cs) 2 = rc2 rs- 1 r- 1 s- 1c- 2rc- 1 = 1, cr- 1s- 3r- 1c- 1r- 1scr- 1 = sc- 1sr- 1 sr- 1c- 1sc- 1r- 1c = 1, s 5 r- 1cr- 1c- 1sc- 1 = sr- 1 s- 1c- 1sr- 1 s- 1 rcr- 1 c = 1, cs 2c- 1sr- 1 cr- 1 scr = (rsr- 1 s- 1r- 1s- 1)2 = (r- 1s- 3 r- 1 s- 1)2 s 2r- 1s- 1r- 1s- 1r- 1s 3 r- 1 s

= 1, = r- 1c- 2sr- 1 s- 1r- 1sc- 1r- 1s- 1c = 1.

Then z = (rsrs- 1 ) 2 , t = (rs 2 )3 s- 4 and u = (sr) 4 are involutions and V1 = (cr)- 3 , v2 = r- 1 s- 1 rs- 1 r- 1 s- 2 and V3 = (sr)- 2 have order 4. Furthermore, the following assertions hold.

(a) Aut(H) = Inn(H) : K, where K is a Klein 4-group and there is "( E Aut(H) - Inn(H) fixing r.

= (s,s- 1 rsr,s 2 r- 1 s- 2 rs- 1 ) is a Sylow 2-subgroup of H with = z, center Z(S) = (z) such that "f(S) = s. Moreover, s8 = v~ = t and v~ = u, z, t, u are involutions. P = 02(H) = (v1) is cyclic of order 4. A= (z, t, u) is the unique elementary abelian normal subgroup of rank 3 of S. V = (z, t) is the unique elementary abelian normal 4-group of S. CH(V) = Cs(V) is a maximal subgroup of S, andV = Z(Cs(V)). S = NH(V). C = CH(A) = Cs(A) = (v1,v2,v3) is a homocyclic subgroup of rank 3 and order 64 of S. C is the unique abelian subgroup of its order in S. D = NH(A) = (r,s) is a non-split extension ofC by 84. H has a faithful permutation representation of degree 448 with stabilizer (sr- 1c2sr- 1c- 1rc, cs- 1r- 1cs- 3c) ~ A 6.

(b) S

(c) (d)

(e) (f) (g) (h)

(i) (j) (k)

vr

15.2 Fusion

675

(1) A system of representatives hi and the corresponding centralizer orders ICH(hi)I of the 31 conjugacy classes hf of H is given in Table 15. 7.1.

(m) The character table of H is given in Table 15.8.1. (n) Inn(D) has an elementary abelian complement of order 8

in

Aut(D).

(o) A system of representatives di and the corresponding centralizer orders ICn (di)I of the classes of D 15.7.2.

= (r, s) is given in Table

(p) The character table of D is given in Table 15. 8. 2. Proof All these statements have been checked by means of MAGMA and the given presentation of H. The system of representatives of the conjugacy classes of H and D have been calculated by means of Kratzer's Algorithm 5.3.18 of [92]. D Definition 15.1.2 A finite simple group G is said to be of ON-type, if it has a 2-central involution z such that Cc ( z) is isomorphic to the finitely presented group H stated in Lemma 15.1.1.

15.2 Fusion

In this section it is shown that any finite simple group G of ON-type has one conjugacy class of involutions and the conjugacy classes of elements of even order are classified. Most of the results are originally due to O'Nan; see [104]. Proposition 15.2.1 Let G be a finite simple group having a 2-central involution z with centralizer Cc (z) = H given in Lemma 15.1.1. The fallowing statements hold.

(a) There exists an element g of order 3 in N c (A), g t}. H, such that z 9 = t, t 9 = zt and u 9 = u. Such an element is unique up to conjugation in C = CH(A). (b) E = N c (A) is the unique non-split extension of C = Cc (A) = CH (A) by GL 3 (2) which has a Sylow 2-subgroup isomorphic to the ones of H. (c) The Goldschmidt index of the amalgam H ,-- D ---+ E is l.

O'Nan's group ON

676

(d) E = ('r, s, g) has the following set R(E) of defining relations:

r3 = g3 =

S16

= 1,

sr-1sr-1g-1s-1g-1 = (r-1g-1r-1s-1)2 = (sgs2)2 = 1, (sg-1r-1 g-1 )2 = (gr-1 s-1 )3 = (rs2r-1 s-1 )2 = 1, s-2g-1r-1 s-1rs-2rg = s-1r-1gr-1 s-1rsgr-1g-1 = l.

(e) z = s 8 is a 2-central involution of E such that CE(z)

c:,, D NH (A). In particular, all conditions of Step 5 of Algorithm 1. 3. 8 are satisfied by the amalgam H .- D ---+ E.

(f) E has a faithful permutation representation PE of degree 112 with stabilizer (r- 1sgs 2 r,s 3 g- 1s- 1rg- 1) whose generators have order 8.

(g) A system of representatives ek and the corresponding centralizer orders ICE(ek)I of the 18 conjugacy classes of ef in E are given in Table 15. 7. 3.

(h) The character table of E is stated in Table 15.8.3. Proof (a) Let S be the Sylow 2-subgroup of H given in Lemma 15.1.l(b). Using the faithful permutation representation of H described in Lemma 15.1.l(k) and MAGMA it has been checked that w = rs- 2 r 2 s is an involution of S that does not belong to H'. Therefore z E H', t E H' and w E H - H' are representatives of all conjugacy classes of involutions of Hand ICH(w)I = 144 by Lemma 15.1.l(k). Suppose that z and tare not conjugate in G. Then Glauberman's Z* Theorem implies that zq = w for some q E G. Hence CH (w) = H n Hq. Now divides 4 because IH: H'I = 2. As 16 divides ICH(w)I it follows that IH' n H'q I is even. Hence there is an involution b1 E H' n H'q and v = wb 1 is an involution in H - H'. Thus v and ware H-conjugate. Furthermore, v E H'q, because w = zq and b1 belong to H'q. Therefore vh = tq for some h E Hq. In particular, z and t are G-conjugate. This contradiction to our assumption proves that there is an x in G such that zx = t and x (/. H.

iff~!'.)

1

Let K = Ca (t), then sx is a Sylow 2-subgroup of K = Hx. By Lemma 15.1.1 X = CH(V) = CH(t) has index 2 in S. Let Y be a Sylow 2subgroup of K containing X. Then Xis normal in Y. By Sylow's Theorem there is a y E K such that Y = sxy. Now V = Z(X) char X