Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups
Clifford-Fischer Theory Applied to Certain Groups Associated with Symplectic, Unitary and Thompson Groups By Ayoub Basheer Mohammed Basheer ayoubac@aims.ac.za ayoubbasheer@gmail.com Supervisor : Professor Jamshid Moori jamshid.moori@nwu.ac.za School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg, South Africa A thesis submitted in the fulfillment of the requirements for Philosophiæ Doctor (PhD) in Science at the School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg April 2012 i Abstract The character table of a finite group is a very powerful tool to study the groups and to prove many results Any finite group is either simple or has a normal subgroup and hence will be of extension type The classification of finite simple groups, more recent work in group theory, has been completed in 1985 Researchers turned to look at the maximal subgroups and automorphism groups of simple groups The character tables of all the maximal subgroups of the sporadic simple groups are known, except for some maximal subgroups of the Monster M and the Baby Monster B There are several well-developed methods for calculating the character tables of group extensions and in particular when the kernel of the extension is an elementary abelian group Character tables of finite groups can be constructed using various theoretical and computational techniques In this thesis we study the method developed by Bernd Fischer and known nowadays as the theory of Clifford-Fischer matrices This method derives its fundamentals from the Clifford theory Let G = N ·G, where N ⊳ G and G/N ∼ = G, be a group extension For each conjugacy class [gi ]G , we construct a non-singular square matrix Fi , called a Fischer matrix Once we have all the Fischer matrices together with the character tables (ordinary or projective) and fusions of the inertia factor groups into G, the character table of G is then can be constructed easily In this thesis we apply the coset analysis technique (this is a method to find the conjugacy classes of group extensions) together with theory of Clifford-Fischer matrices to calculate the ordinary character tables of seven groups of extensions type, in which four are non-split and three are split extensions These groups 1+6 · 6· 5· 10 1+2 :8):2) are of the forms: 21+8 + A9 , :Sp(6, 2), Sp(6, 2), GL(5, 2), :(U5 (2):2), 2− :((3 and 22n· Sp(2n, 2) and 28· Sp(8, 2) In addition we give some general results on the non-split group 22n· Sp(2n, 2) ii Preface The work described in this thesis was carried out under the supervision and direction of Professor Jamshid Moori, School of Mathematics, Statistics and Computer Sciences, University of KwaZuluNatal, Pietermaritzburg, from January 2009 to April 2012 The Thesis represent original work of the author and has not been otherwise been submitted in any form for any degree or diploma to any University Where use has been made of the work of others it is duly acknowledged in the text Signature (Student) Date: 5th of April 2012 Signature (Supervisor) Date: 5th of April 2012 iii Dedication TO MY PARENTS, MY LOVELY WIFE MUNA, MY LOVELY DAUGHTER FATIMA, MY FAMILY AND TO THE BEST FRIEND I HAVE EVER GOT MUSA COMTOUR, I DEDICATE THIS WORK iv Acknowledgements First of all, I thank ALLAH for his Grace and Mercy showered upon me I heartily express my profound gratitude to my supervisor, Professor Jamshid Moori, for his invaluable learned guidance, advises, encouragement, understanding and continued support he has provided me throughout the duration of my studies which led to the compilation of • the Postgraduate Diploma project at AIMS - Cape Town 2006, • the MSc in Mathematics at the University of KwaZulu-Natal 2009, • this PhD thesis and hopefully I aim to continue with Prof J Moori for a Postdoctoral research I will be always indebted to him for introducing me to this fascinating area of Mathematics and creating my interest in Group Theory Professor Moori is a unique encyclopaedia and I have learnt so much from him, not only in the academic orientation, but in various walks of life May ALLAH gives him the power to enrich furthermore this interesting domain of Mathematics, and in general to advance the wheel of life forward I lovingly thank my precious wife Muna, who supported me each step of the way and without her help and encouragement it simply never would have been possible to finish this work I am grateful for the facilities made available to me by the School of Mathematics, Statistics and Computer Sciences of the University of KwaZulu-Natal, Pietermaritzburg I am also grateful for the financial support that I have received from the University of Khartoum through the Ministry of Higher Education of Sudan, the National Research Foundation of South Africa (NRF) for a v grant holder bursary through Professor Moori and to the University of KwaZulu-Natal for the graduate assistantship and the doctoral research scholarship for the year 2009 My thanks extend to the administration of University of Khartoum (UofK), in particular to Mrs Islah Shaaban the deputy head of teaching assistants and training department, Dr Mohsin the principal of UofK, Dr Manar the dean to the Faculty of Mathematical Sciences and Dr Eltayib Yousif head of Applied Mathematics Department at UofK I would like to thank my officemates Muna Elshareef, T T Seretlo and Kassahun M Tessema for creating a pleasant working environment Finally I sincerely thank my entire extended family represented by Basheer, Suaad, Muna, Fatima, Eihab, Adeeb, Nada, Balla, Hanan, Mujtaba, Ahmed, Khalid, Tayseer, Samah, Rana, Amro, Ahmed, Mustafa, Iyad, Mohsin and Mohammed vi Table of Contents Abstract ii Preface iii Dedication iv Acknowledgements v Table of Contents vii List of Notations xii Introduction Group Extensions 2.1 Introduction 2.2 Semidirect Products and Split Extensions 10 2.3 The Conjugacy Classes of Group Extensions 13 vii TABLE OF CONTENTS TABLE OF CONTENTS Elementary Theories of Representations and Characters 16 3.1 Preliminaries 17 3.2 Character Tables and Orthogonality Relations 19 3.3 Tensor Product of Characters 22 3.4 Lifting of Characters 24 3.5 Restriction and Induction of Characters 26 3.6 3.5.1 Restriction of Characters 26 3.5.2 Induction of Characters 27 Permutation Character 31 Schur Multiplier, Projective Representations and Characters 35 4.1 Schur Multiplier 36 4.2 Projective Representations 37 4.3 Projective Characters 43 The Theory of Clifford-Fischer Matrices 46 5.1 The Clifford Theory 46 5.2 The Fischer Matrices 51 5.3 The Character Tables of Group Extensions 54 A Group of the Form 37 :Sp(6, 2) 57 6.1 Introduction 57 6.2 The Conjugacy Classes of G = 37 :Sp(6, 2) 60 viii TABLE OF CONTENTS 6.3 TABLE OF CONTENTS Inertia Factor Groups of G = 37 :Sp(6, 2) 65 6.3.1 First, Second, Third and Fourth Inertia Factor Groups 67 6.3.2 Fifth and Sixth Inertia Factor Groups 71 6.3.3 Fusions of Inertia Factor Groups into Sp(6, 2) 75 6.4 Character Tables of the Inertia Factor Groups 82 6.5 Fischer Matrices of G = 37 :Sp(6, 2) 83 6.6 The Character Table of G = 37 :Sp(6, 2) 89 Two Maximal Subgroups of Thompson Group Th 7.1 7.2 95 Dempwolff Group 25· GL(5, 2) 95 7.1.1 Introduction 96 7.1.2 The Conjugacy Classes of G = 25· GL(5, 2) 97 7.1.3 The Inertia Groups of G = 25· GL(5, 2) 99 7.1.4 Fusion of Classes of H2 into Classes of GL(5, 2) 103 7.1.5 Fischer Matrices of 25· GL(5, 2) 105 7.1.6 The Character Table of Dempwolff Group G = 25· GL(5, 2) 108 · A Group of the Form 21+8 + A9 111 7.2.1 Introduction 111 7.2.2 · Conjugacy Classes of Group Extensions and of G = 21+8 + A9 112 7.2.3 · Inertia Factor Groups of G = 21+8 + A9 115 7.2.4 The Character Table of the Inertia Factor Group H2 115 ix Table 11.17 (continued) [x]G 6E 6F 6h 6C 6i 12b 6j 6k 6l 6m 6n 12c 6o 6p 6q 6r 6s 12d 6t 12e 12f 12g 8d 8e 8f 8g 8h 8i 8 8 6 4 5 3 5 5 9 9 3 2 2 2 2 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 χ1 1 1 1 1 1 1 1 1 1 1 1 1 χ2 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 χ3 0 −3 −3 −3 −1 −1 −1 0 2 0 0 0 −A −A −A A A A χ4 0 −3 −3 −3 −1 −1 −1 0 2 0 0 0 A A A −A −A −A χ5 −4 −4 −4 −1 −1 −1 3 2 0 0 0 0 0 0 0 χ6 −3 −3 −3 3 −1 −1 0 2 2 −1 −1 −1 −1 −4 −4 −4 −4 −4 −4 χ7 −3 −3 −3 3 −1 −1 0 2 −2 −2 1 1 4 4 4 χ8 2 5 −2 1 −1 −1 1 −1 −1 2 2 −3 −3 −3 −3 −3 −3 [g]G |CG (g)| 6D 6G 6H 6I 6J 8A 8B 340 χ9 2 5 −2 1 −1 −1 1 1 −2 −2 −2 −2 3 3 χ10 2 2 2 −5 4 −4 −4 −2 −2 0 0 0 0 0 0 χ11 −2 −2 −2 −2 −2 −2 −1 2 −2 −2 2 0 0 0 −A −A −A A A A χ12 −2 −2 −2 −2 −2 −2 −1 2 −2 −2 2 0 0 0 A A A −A −A −A χ13 3 0 2 3 −1 −1 1 1 1 2 2 2 χ14 3 0 2 3 −1 −1 −1 −1 −1 −1 −1 −1 −2 −2 −2 −2 −2 −2 χ15 6 −3 −3 −3 −1 −1 −1 0 2 0 0 0 0 0 0 χ16 3 3 3 −1 −1 0 2 −2 −2 1 1 −1 −1 −1 −1 −1 −1 3 3 3 −1 −1 0 2 2 −1 −1 −1 −1 1 1 1 χ18 −4 −4 −4 2 −2 −2 −1 −1 1 1 −2 −2 −2 −2 0 0 0 χ19 −4 −4 −4 2 −2 −2 −1 −1 1 −1 −1 2 2 0 0 0 χ20 −2 −2 −2 −8 −8 −8 0 −2 −2 0 0 0 0 0 0 0 χ21 0 3 −3 −3 0 0 0 0 0 0 0 0 χ22 2 −4 −4 −4 0 2 0 0 0 0 −B −B −B B B B χ23 2 −4 −4 −4 0 2 0 0 0 0 B B B −B −B −B χ24 6 −3 −3 −3 −2 −5 −5 0 −2 −2 0 0 0 0 0 0 χ25 −2 −2 −2 1 −6 −3 −3 4 0 0 0 0 0 0 0 χ26 1 4 2 1 −1 −1 −1 −1 −1 −1 −1 −1 2 2 2 χ27 1 4 2 1 −1 −1 1 1 1 −2 −2 −2 −2 −2 −2 χ28 6 0 2 0 2 0 0 0 0 0 0 χ29 3 −3 −3 −3 −1 −1 −3 −3 −1 −1 1 1 1 2 2 2 χ30 −6 −6 −6 −3 −3 −3 1 0 −2 −2 0 0 0 0 0 0 χ31 3 −3 −3 −3 −1 −1 −3 −3 −1 −1 −1 −1 −1 −1 −1 −1 −2 −2 −2 −2 −2 −2 χ32 −2 −2 −2 4 −2 −2 −2 −2 −2 −2 −2 0 0 0 0 0 0 χ33 0 0 0 −3 0 0 0 0 0 0 −3 −3 −3 −3 −3 −3 Continued on next page Appendix χ17 Table 11.17 (continued) [x]G 341 6F 6h 6C 6i 12b 6j 6D 6k 6l 6m 6n 12c 6G 6o 6p 6H 6q 6r 6I 6s 12d 6t 12e 6J 12f 12g 8d 8A 8e 8f 8g 8B 8h 8i χ34 0 0 0 −3 0 0 0 0 0 0 3 3 3 χ35 −6 −6 −6 3 −3 −3 −3 0 0 0 0 0 0 0 0 χ36 −6 −6 −6 3 −3 3 0 0 0 0 0 0 0 0 χ37 0 0 0 0 0 0 2 2 2 0 0 0 χ38 0 0 0 0 0 0 −2 −2 −2 −2 −2 −2 0 0 0 χ39 −4 −4 −4 −4 −4 −4 0 2 0 0 0 0 0 0 0 χ40 0 −6 −6 −6 −4 2 0 2 0 0 0 0 0 0 χ41 0 0 0 0 0 0 0 0 0 0 0 0 χ42 6 6 6 0 0 0 0 0 0 0 0 0 χ43 0 0 0 0 0 0 0 0 0 0 0 0 χ44 3 −1 −3 −1 0 0 0 −3 −3 1 −7 1 −7 1 χ45 3 −1 −3 −1 0 0 0 3 −1 −1 −1 −1 −1 −1 χ46 3 −1 −3 −1 0 0 0 −3 −3 1 −3 −3 χ47 3 −1 −3 −1 0 0 0 3 −1 −1 −5 −1 −5 −1 χ48 6 −2 18 −6 −2 0 0 0 0 0 0 0 0 χ49 6 −2 −18 −2 −6 0 0 0 0 0 −A A C −C χ50 6 −2 −18 −2 −6 0 0 0 0 0 C −C −A A χ51 0 −3 −1 0 0 0 0 0 0 0 0 χ52 0 −3 −1 0 0 0 0 0 0 0 0 χ53 0 −3 −1 0 0 0 0 0 0 0 0 χ54 0 −3 −1 0 0 0 0 0 0 0 0 χ55 −3 −3 0 0 0 0 0 0 −3 −3 1 2 −2 2 −2 χ56 −3 −3 0 0 0 0 0 3 −1 −1 −2 −2 −2 −2 χ57 −6 −6 −9 −1 −1 0 0 0 0 0 −D A D −A χ58 −6 −6 −9 −6 −1 −1 0 0 0 0 0 D −A −D A χ59 −6 −6 −9 −6 −1 −1 0 0 0 0 0 −B B A B −B −A χ60 −6 −6 −9 −1 −1 0 0 0 0 0 B −B −A −B B A χ61 0 −18 −2 −6 0 0 0 0 0 0 0 0 χ62 −12 −12 18 −6 0 0 0 0 0 12 −4 12 −4 χ63 −12 −12 18 −6 0 0 0 0 0 −12 −12 χ64 −12 −12 −18 −2 −2 0 0 0 0 0 0 0 0 χ65 0 0 0 0 0 0 0 0 0 −9 −1 −9 −1 χ66 0 0 0 0 0 0 0 0 0 −3 −3 χ67 0 0 0 0 0 0 0 0 0 −5 3 −5 χ68 0 0 0 0 0 0 0 0 0 −3 −3 −3 −3 χ69 9 −3 0 0 0 0 0 3 −1 −1 2 −2 2 −2 χ70 9 −3 0 0 0 0 0 3 −1 −1 −4 −4 −4 −4 χ71 9 −3 0 0 0 0 0 −3 −3 1 −2 −2 −2 −2 Continued on next page Appendix 6E [g]G Table 11.17 (continued) [x]G 6E 6F [g]G 6h 6C 6i 12b 6j 6D 6k 6l 6m 6n 12c 6G 6o 6p 6H 6q 6r 6I 6s 12d 6t 12e 6J 12f 12g 8d 8A 8e 8f 8g 8B 8h 8i χ72 9 −3 0 0 0 0 0 −3 −3 1 4 −4 4 −4 χ73 0 −9 −1 −1 0 0 0 0 0 0 0 0 χ74 0 −9 −1 −1 0 0 0 0 0 0 0 0 χ75 0 −9 −1 −1 0 0 0 0 0 0 0 0 χ76 0 −9 −1 −1 0 0 0 0 0 0 0 0 χ77 0 0 0 0 0 0 0 0 0 0 0 0 χ78 0 0 0 0 0 0 0 0 0 −A A C −C χ79 0 0 0 0 0 0 0 0 0 C −C −A A χ80 6 −2 −6 0 0 0 0 0 0 0 −B −B B B B −B 342 6 −2 0 0 0 −1 0 0 0 B B −B −B −B B χ82 −9 −9 0 0 0 −1 0 3 −1 −1 2 −2 2 −2 χ83 −9 −9 0 0 0 −3 0 −3 −3 1 −2 −2 −2 −2 χ84 0 18 2 −6 0 −3 0 0 0 0 0 0 χ85 12 −4 −2 0 −1 0 −1 −4 0 0 0 0 χ86 12 −4 −2 0 −1 0 −1 −4 0 0 0 0 χ87 −24 −2 0 −1 0 −1 −2 −2 0 0 0 χ88 −24 −2 0 −1 0 −1 −2 −2 0 0 0 χ89 0 −18 −6 0 0 −2 0 0 0 0 0 0 χ90 12 −4 −2 0 −6 0 −2 −2 −2 0 0 0 χ91 12 −4 −2 0 −6 0 −2 −2 2 −2 0 0 0 χ92 24 −8 12 −4 0 −3 −3 −1 −2 −2 0 0 0 χ93 24 −8 12 −4 0 −3 −3 1 −1 −2 −2 0 0 0 χ94 −24 −2 0 −6 −2 0 0 0 0 0 0 χ95 12 −4 −12 −4 0 −1 −1 −1 −2 2 −2 0 0 0 χ96 12 −4 −12 −4 0 −1 −1 −1 −2 −2 0 0 0 χ97 −12 12 −4 0 −2 0 −2 −4 0 0 0 0 χ98 −12 12 −4 0 −2 0 −2 −4 0 0 0 0 χ99 −12 −6 −2 0 −3 −3 1 −1 −4 0 0 0 0 χ100 −12 −6 −2 0 −3 −3 −1 −4 0 0 0 0 χ101 0 −18 −6 0 0 −6 0 0 0 0 0 0 χ102 −12 12 −4 0 −3 −1 −1 −2 2 −2 0 0 0 χ103 −12 12 −4 0 −3 −1 −1 −2 −2 0 0 0 χ104 0 0 0 0 0 0 −2 −2 −2 0 0 0 χ105 0 0 0 0 0 0 −2 −2 −2 0 0 0 χ106 24 −8 12 −4 0 −2 0 0 0 0 0 0 0 χ107 0 0 0 0 0 0 0 0 0 0 0 0 χ108 0 0 0 0 0 0 0 0 0 0 0 0 χ109 0 −18 −6 0 0 0 0 0 0 0 0 0 Continued on next page Appendix χ81 Table 11.17 (continued) [x]G 9A 9B 11A 12A 12B 8j 8k 8C 8l 8m 8n 8o 8p 8q 9a 9b 18a 10b 20a 11a 12h 12i 12j 12k 12l 12m 12n 12o 8 7 7 2 2 5 7 7 0 0 0 0 3 0 2 1 1 0 0 0 0 0 1 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 χ1 1 1 1 1 1 1 1 1 1 1 1 χ2 −1 −1 −1 −1 −1 1 1 1 −1 −1 1 1 1 1 χ3 0 0 0 0 −2 1 0 −1 −1 2 −1 −1 −1 −1 −1 χ4 0 0 0 0 −2 1 0 −1 −1 2 −1 −1 −1 −1 −1 χ5 0 0 2 1 0 0 0 −3 −3 −3 −3 −3 χ6 0 0 0 0 −1 −1 1 −2 1 1 1 χ7 0 0 0 0 −1 −1 −1 −1 −2 1 1 1 χ8 1 1 −1 −1 −1 1 0 2 −1 −1 −1 −1 −1 [g]G |CG (g)| 8D 10A 12C 343 −1 −1 −1 −1 −1 −1 −1 −1 1 0 2 −1 −1 −1 −1 −1 0 0 2 −1 −1 −1 0 −1 2 2 2 χ11 0 0 0 0 −1 −1 −1 0 0 0 0 χ12 0 0 0 0 −1 −1 −1 0 0 0 0 χ13 2 2 0 0 0 0 −1 −1 −1 2 2 χ14 −2 −2 −2 −2 −2 0 0 0 0 −1 −1 −1 2 2 χ15 0 0 0 0 0 0 0 −5 −2 −2 1 1 χ16 −1 −1 −1 −1 −1 1 0 0 0 −1 −1 −1 −1 −1 −1 −1 χ17 1 1 1 1 0 0 0 −1 −1 −1 −1 −1 −1 −1 χ18 0 0 0 0 −1 −1 −1 −1 −1 0 0 0 0 χ19 0 0 0 0 −1 −1 −1 1 0 0 0 0 χ20 0 0 0 0 1 0 −1 2 2 2 χ21 0 0 0 0 −2 1 0 0 0 3 3 χ22 0 0 0 0 −1 −1 −1 0 0 0 0 0 χ23 0 0 0 0 −1 −1 −1 0 0 0 0 0 χ24 0 0 0 0 −1 −1 0 2 −1 −1 −1 −1 −1 χ25 0 0 0 0 −1 −1 −1 0 2 −1 −1 −1 −1 −1 χ26 2 2 0 −1 −1 −1 0 −2 1 −2 −2 −2 −2 −2 χ27 −2 −2 −2 −2 −2 0 −1 −1 −1 0 −2 1 −2 −2 −2 −2 −2 χ28 0 0 0 0 0 0 0 −2 −2 −2 −2 −2 −2 −2 −2 χ29 −2 −2 −2 −2 −2 0 0 0 0 −1 −1 −1 −1 −1 −1 −1 χ30 0 0 0 0 0 0 0 −1 2 −1 −1 −1 −1 −1 χ31 2 2 0 0 0 0 −1 −1 −1 −1 −1 −1 −1 χ32 0 0 0 0 1 0 −2 −2 −2 −2 −2 −2 −2 −2 χ33 1 1 −1 −1 −1 0 −1 −1 −3 0 0 0 Continued on next page Appendix χ9 χ10 Table 11.17 (continued) [x]G 344 9A 9B 11A 12A 12B [g]G 8j 8k 8C 8l 8m 8n 8o 8D 8p 8q 9a 9b 18a 10A 10b 20a 11a 12h 12i 12j 12k 12l 12m 12C 12n 12o χ34 −1 −1 −1 −1 −1 −1 −1 −1 0 1 −3 0 0 0 χ35 0 0 2 0 0 0 −2 −2 1 1 χ36 0 0 −2 −2 −2 0 0 0 −2 −2 1 1 χ37 0 0 0 0 −2 1 −1 −1 0 0 0 0 χ38 0 0 0 0 −2 1 1 0 0 0 0 χ39 0 0 0 0 1 0 0 0 0 0 χ40 0 0 0 0 −1 −1 0 0 0 0 0 χ41 0 0 −2 −2 −2 0 0 0 0 0 0 χ42 0 0 0 0 0 0 0 −1 2 2 2 χ43 0 0 2 0 0 −1 0 0 0 0 χ44 −3 −3 1 −1 −1 0 0 0 −1 1 −3 χ45 3 −1 −1 −1 −1 −1 0 0 0 −1 1 −3 χ46 1 −3 −1 −1 0 0 0 −1 −3 −3 −3 χ47 −1 −1 −1 −1 −1 −1 0 0 0 −1 −3 −3 −3 χ48 0 0 −2 −2 0 0 0 −2 −6 2 −6 χ49 0 A −A 0 0 0 0 0 −2 0 0 χ50 0 −A A 0 0 0 0 0 −2 0 0 χ51 0 0 −2 −2 0 0 0 0 −9 −1 −1 −1 χ52 0 0 −2 −2 0 0 0 0 3 −5 −1 χ53 0 0 −2 −2 0 0 0 0 E E −1 χ54 0 0 −2 −2 0 0 0 0 E E −1 χ55 2 −2 −2 0 0 0 0 0 −3 0 0 χ56 0 −A A 0 0 0 0 0 −3 0 0 χ57 0 0 0 0 0 0 0 0 −9 −1 −1 −1 χ58 −4 −4 0 0 0 0 0 0 −9 −1 −1 −1 χ59 4 −4 0 0 0 0 0 0 3 −5 −1 0 0 0 0 0 0 0 0 3 −5 −1 χ61 −1 −1 −1 −1 −3 1 0 0 0 0 0 0 0 χ62 1 −3 −3 1 0 0 0 0 0 0 0 χ63 3 −1 −1 −1 −3 0 0 0 0 0 0 0 χ64 −3 −3 1 1 −3 0 0 0 0 −2 −2 −2 χ65 −2 −2 −2 0 0 0 0 0 0 0 0 χ66 0 0 0 0 0 0 0 0 0 0 0 χ67 2 −2 −2 0 0 0 0 0 0 0 0 χ68 0 0 0 0 0 0 0 0 0 0 0 χ69 0 0 0 0 0 0 0 −3 0 0 χ70 0 0 0 0 0 0 0 −3 0 0 χ71 0 0 0 0 0 0 0 −3 0 0 Continued on next page Appendix χ60 Table 11.17 (continued) [x]G 345 9A 9B 11A 12A 12B [g]G 8j 8k 8C 8l 8m 8n 8o 8D 8p 8q 9a 9b 18a 10A 10b 20a 11a 12h 12i 12j 12k 12l 12m 12C 12n χ72 0 0 0 0 0 0 0 −3 0 0 χ73 0 0 2 −2 0 0 0 0 1 −3 χ74 0 A −A 0 0 0 0 0 0 −3 −3 −3 χ75 0 −A A 0 0 0 0 0 0 −3 −E −E −3 χ76 0 0 0 0 0 0 0 0 −3 −E −E −3 χ77 0 0 0 0 0 0 0 0 0 0 0 χ78 2 −2 −2 0 0 0 0 0 0 0 0 χ79 −2 −2 −2 0 0 0 0 0 0 0 0 χ80 0 0 0 0 0 0 0 0 0 0 0 χ81 −4 0 0 0 0 0 0 0 0 0 0 χ82 −4 0 0 0 0 0 0 −1 0 0 χ83 0 0 0 0 0 0 0 −1 0 0 χ84 0 0 0 0 0 0 0 0 0 0 0 χ85 −2 −2 −2 0 0 −1 −1 0 0 −2 −2 −2 χ86 0 −A A 0 0 −1 −1 0 0 −2 −2 −2 χ87 0 A −A 0 0 0 −1 0 0 −6 2 −2 χ88 0 A −A 0 0 0 −1 0 0 −6 2 −2 χ89 0 0 0 0 −1 0 0 0 −6 2 −2 χ90 −4 0 0 0 0 0 0 0 −6 2 −2 χ91 −4 0 0 0 0 0 0 0 −6 2 −2 χ92 −4 0 0 0 0 0 0 0 0 0 0 χ93 −4 0 0 0 0 0 0 0 0 0 0 χ94 0 0 0 0 −3 0 0 0 −2 −2 −2 χ95 0 0 0 0 −3 0 0 0 0 0 χ96 0 0 0 0 −3 0 0 0 0 0 χ97 0 0 0 0 0 −1 0 0 0 0 χ98 0 0 0 0 0 −1 0 0 0 0 χ99 0 0 0 0 0 0 0 0 −2 −2 −2 χ100 0 0 0 0 0 0 0 0 −2 −2 −2 χ101 0 0 0 0 0 0 0 0 −2 −2 −2 χ102 0 0 0 0 0 0 0 0 0 0 0 χ103 0 0 0 0 0 0 0 0 0 0 0 χ104 0 0 0 0 −1 −1 0 0 0 0 χ105 0 0 0 0 −1 −1 0 0 0 0 χ106 0 0 0 0 −3 0 0 0 0 0 χ107 −4 0 0 0 0 −1 0 0 0 0 χ108 −4 0 0 0 0 −1 0 0 0 0 χ109 0 0 0 0 0 0 0 0 −6 2 −2 12o Appendix Continued on next page Appendix Table 11.17 (continued) [x]G 12D [g]G 12E 12F 15A 16A 16B 18A 24a 24B 12p 12q 12r 12s 12t 12u 12v 15a 16a 16b 16c 16d 18b 24a 24b 24c 24d 5 5 3 5 5 4 4 1 1 1 1 0 0 1 1 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 χ1 1 1 1 1 1 1 1 1 χ2 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 χ3 1 1 0 F F −F −F F F −F −F χ4 1 1 0 −F −F F F −F −F F F χ5 −2 1 1 0 −1 0 0 −1 0 0 χ6 1 1 0 −1 0 0 −1 −1 −1 −1 |CG (g)| χ7 1 1 0 −1 0 0 1 χ8 −1 −1 −1 −1 1 −1 −1 −1 −1 −1 0 0 χ9 −1 −1 −1 −1 −1 −1 1 1 −1 0 0 χ10 0 0 0 0 0 −1 0 0 χ11 −1 0 0 0 −F −F F F F F −F −F χ12 −1 0 0 0 F F −F −F −F −F F F χ13 0 0 −1 −1 0 0 0 −1 −1 −1 −1 χ14 0 0 1 0 0 0 1 1 χ15 −1 −1 −1 −1 0 −1 0 0 0 0 χ16 −1 −1 −1 −1 0 1 1 −1 −1 −1 −1 χ17 −1 −1 −1 −1 0 −1 −1 −1 −1 1 1 χ18 0 0 1 0 0 −1 0 0 χ19 0 0 −1 −1 0 0 −1 0 0 χ20 −1 −2 −2 −2 −2 0 0 0 0 0 χ21 −2 1 1 0 0 0 0 0 0 χ22 0 0 0 0 0 0 −1 F F −F −F χ23 0 0 0 0 0 0 −1 −F −F F F χ24 1 1 0 0 0 0 0 0 χ25 −1 −1 −1 −1 0 0 0 0 0 χ26 0 0 1 0 0 −1 −1 −1 −1 χ27 0 0 −1 −1 0 0 1 1 χ28 0 0 0 0 0 0 0 χ29 1 1 1 0 0 0 −1 −1 −1 −1 χ30 −1 −1 −1 −1 0 0 0 0 0 0 χ31 1 1 −1 −1 0 0 0 1 1 χ32 0 0 0 0 0 0 0 0 χ33 −1 0 0 0 1 1 0 0 χ34 −1 0 0 0 −1 −1 −1 −1 0 0 χ35 −1 −1 −1 −1 0 0 0 0 0 0 χ36 1 1 0 0 0 0 0 0 χ37 0 0 0 −1 0 0 0 0 χ38 0 0 0 −1 0 0 0 0 χ39 0 0 0 0 0 −1 0 0 χ40 0 0 0 0 0 0 0 0 χ41 0 0 0 −1 0 0 0 0 χ42 −1 0 0 0 0 0 0 0 0 χ43 0 0 0 0 0 0 0 0 χ44 −1 −1 −1 0 −1 −1 −1 −1 χ45 −1 −1 −1 0 −1 −1 1 −1 −1 χ46 −1 −1 −1 0 −1 −1 −1 −1 χ47 −1 −1 −1 0 −1 −1 −1 −1 χ48 −2 −2 2 0 0 0 0 0 0 χ49 0 0 0 0 −F F F −F 0 0 χ50 0 0 0 0 F −F −F F 0 0 χ51 −3 1 0 0 0 0 0 0 χ52 −3 1 0 0 0 0 0 0 Continued on next page 346 Appendix Table 11.17 (continued) [x]G 12D 15A 16A 18A 24a [g]G 12p 12q 12E 12r 12s 12t 12F 12u 12v 15a 16a 16b 16c 16d 18b 24a 24b 24c 24d χ53 1 G G 0 0 0 0 0 0 χ54 1 G G 0 0 0 0 0 0 χ55 0 0 0 0 0 0 −1 −1 χ56 −1 −1 −1 0 0 0 0 −1 −1 χ57 0 0 0 0 0 0 F −F −F F χ58 0 0 0 0 0 0 −F F F −F χ59 0 0 0 0 0 0 F −F −F F χ60 −2 −2 2 0 0 0 0 −F F F −F χ61 0 0 0 0 −1 −1 0 0 χ62 0 0 0 0 −1 −1 0 0 χ63 0 0 0 0 −1 −1 0 0 χ64 0 0 0 0 −1 −1 0 0 χ65 0 0 0 0 0 0 0 0 χ66 0 0 0 0 0 0 0 0 χ67 0 0 0 0 0 0 0 0 χ68 0 0 0 0 0 0 0 0 χ69 −3 1 0 0 0 0 −1 −1 χ70 −3 1 0 0 0 0 −1 −1 χ71 1 G G 0 0 0 0 −1 −1 χ72 1 G G 0 0 0 0 −1 −1 χ73 0 0 0 0 0 0 0 0 χ74 0 0 0 0 F −F −F F 0 0 χ75 0 0 0 0 −F F F −F 0 0 χ76 0 0 0 0 0 0 0 0 χ77 0 0 0 0 0 0 0 0 χ78 0 0 0 0 0 0 0 0 χ79 0 0 0 0 0 0 0 0 χ80 0 0 0 0 0 0 F −F −F F χ81 0 0 −1 0 0 0 −F F F −F χ82 0 0 −1 0 0 0 −1 −1 χ83 0 0 −1 0 0 0 −1 −1 χ84 0 0 −1 0 0 0 0 0 χ85 0 0 0 0 0 0 0 0 χ86 −1 −1 −1 0 0 0 0 0 0 χ87 −1 −1 −1 0 0 0 0 0 0 χ88 −1 −1 −1 0 0 0 0 0 0 χ89 0 0 0 0 0 0 0 0 χ90 0 0 0 0 0 0 0 0 χ91 0 0 0 0 0 0 0 0 χ92 0 0 −1 0 0 0 0 0 χ93 0 0 −1 0 0 0 0 0 χ94 0 0 0 0 0 0 0 0 χ95 0 0 −1 0 0 0 0 0 χ96 0 0 −1 0 0 0 0 0 χ97 0 0 0 0 0 0 0 0 χ98 0 0 0 0 0 0 0 0 χ99 0 0 −1 0 0 0 0 0 χ100 0 0 −1 0 0 0 0 0 χ101 0 0 0 0 0 0 0 0 χ102 0 0 −1 0 0 0 0 0 χ103 0 0 −1 0 0 0 0 0 χ104 0 0 0 0 0 0 0 0 χ105 0 0 0 0 0 0 0 0 χ106 0 0 0 0 0 0 0 0 χ107 0 0 0 0 0 0 0 0 χ108 0 0 0 0 0 0 0 0 χ109 0 0 0 0 0 0 0 0 347 16B 24B Appendix where in Table 11.17, √ • A = ∗ E(8) + ∗ E(8)3 = ∗ ER(−2) = 2i, √ • B = −4 ∗ E(8) − ∗ E(8)3 = −4 ∗ ER(−2) = −4 2i = −2A, √ • C = ∗ E(8) + ∗ E(8)3 = ∗ ER(−2) = 2i, √ • D = ∗ E(8) + ∗ E(8)3 = ∗ ER(−2) = 2i, • E = ∗ E(3) − ∗ E(3)2 = −1 + ∗ ER(−3) = + 8b3, √ • F = −E(8) − E(8)3 = −ER(−2) = − 2i, √ • G = −E(3) + ∗ E(3)2 = −1 − ∗ ER(−3) = −1 − 3i 348 Bibliography [1] F Ali, Fischer-Clifford Theory for Split and 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