The Fischer-Clifford matrices of a maximal subgroup of

Authors:
Faryad Ali and Jamshid Moori

Journal:
Represent. Theory **7** (2003), 300-321

MSC (2000):
Primary 20C15, 20D08, 20E22

DOI:
https://doi.org/10.1090/S1088-4165-03-00175-4

Published electronically:
July 29, 2003

MathSciNet review:
1993362

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Abstract | References | Similar Articles | Additional Information

Abstract: The Fischer group is the largest sporadic simple Fischer group of order

The group is the derived subgroup of the Fischer -transposition group discovered by Bernd Fischer. There are five classes of elements of order 3 in as represented in ATLAS by , , , and . A subgroup of of order is called of type , where , if it is generated by an element in the class . There are six classes of maximal 3-local subgroups of as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local subgroups , where is the natural orthogonal module for with subgroups of type corresponding to the totally isotropic points. The group is a nonsplit extension of by .

**1.**F. Ali,*Fischer-Clifford Matrices for Split and Non-Split Group Extensions*, PhD Thesis, University of Natal, Pietermaritzburg, 2001.**2.**F. Ali and J. Moori,*Fischer-Clifford Matrices of the Group*, In preparation.**3.**F. Ali and J. Moori,*Fischer-Clifford Matricesand Character Table of the Group*, In preparation.**4.**F. Ali and J. Moori,*The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of*, In preparation.**5.**Wieb Bosma and John Cannon.*Handbook of Magma Functions*, Department of Mathematics, University of Sydney, November 1994.**6.**J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson.*An Atlas of Finite Groups*, Oxford University Press, 1985. MR**88g:20025****7.**M. R. Darafsheh and A. Iranmanesh,*Computation of the character table of affine groups using Fischer matrices*, London Mathematical Society Lecture Note Series**211**, Vol. 1, C. M. Campbell et al., Cambridge University Press (1995), 131 - 137. MR**96j:20011****8.**B. Fischer,*Finite Groups Generated by 3-Transpositions*, Notes, Mathematics Institute, University of Warwick, 1970.**9.**B. Fischer,*Clifford-matrices*, Progr. Math.**95**, Michler G. O. and Ringel C. M. (eds), Birkhauser, Basel (1991), 1 - 16. MR**92i:20012****10.**B. Fischer,*Character tables of maximal subgroups of sporadic simple groups -III*, Preprint.**11.**B. Fischer, unpublished manuscript (1985).**12.**P. X. Gallagher,*Group characters and normal Hall subgroups*, Nagoya Math. J.**21**(1962), 223 - 230. MR**26:240****13.**P. X. Gallagher,*The number of conjugacy classes in a finite group*, Math. Z.**118**(1970), 175 - 179. MR**43:2065****14.**D. Gorenstein,*Finite Groups*, Harper and Row Publishers, New York, 1968. MR**38:229****15.**D. F. Holt,*A computer program for the calculation of a covering group of a finite group*, J. Pure Applied Alg.**35**(1985), 287 - 295. MR**87i:20004****16.**I. M. Isaacs,*Character Theory of Finite Groups*, Academic Press, San Diego, 1976. MR**57:417****17.**C. Jansen, K. Lux, R. Parker and R. Wilson,*An Atlas of Brauer Characters*, London Mathematical Society Monographs New Series 11, Oxford University Press, Oxford, 1995. MR**96k:20016****18.**G. Karpilovsky,*Group Representations: Introduction to Group Representations and Characters*, Vol 1 Part B, North-Holland Mathematics Studies 175, Amsterdam, 1992. MR**93j:20001b****19.**R. J. List,*On the characters of*, Arch. Math.**51**(1988), 118 - 124. MR**89i:20022****20.**R. J. List and I. M. I. Mahmoud,*Fischer matrices for wreath products*, Arch. Math.**50**(1988), 394-401. MR**89e:20031****21.**J. Moori,*On the Groups**and**of the forms**and*, PhD thesis, University of Birmingham, 1975. MR**2001a:20027****22.**J. Moori and Z.E. Mpono,*The Fischer-Clifford matrices of the group*, Quaestiones Math.**22**(1999), 257-298. MR**2000a:20013****23.**J. Moori and Z.E. Mpono,*The centralizer of an involutory outer automorphism of*, Math. Japonica**49**(1999), 93 - 113. MR**2000a:20013****24.**J. Moori and Z.E. Mpono,*Fischer-Clifford matrices and the character table of a maximal subgroup of*, Intl. J. Maths. Game Theory, and Algebra**10**(2000), 1 - 12. MR**2001b:20023****25.**Z. E. Mpono,*Fischer-Clifford Theory and Character Tables of Group Extensions*, PhD thesis, University of Natal, Pietermaritzburg, 1998.**26.**H. Nagao and Y. Tsushima,*Representations of Finite Groups*, Academic Press, San Diego, 1987.**27.**R. B. Salleh,*On the Construction of the Character Tables of Extension Groups*, PhD thesis, University of Birmingham, 1982.**28.**U. Schiffer,*Cliffordmatrizen*, Diplomarbeit, Lehrstul D Fur Matematik, RWTH, Aachen, 1995.**29.**The GAP Group,*GAP - Groups, Algorithms and Programming, Version 4.2*, Aachen, St Andrews, 2000,`(http://www-gap.dcs.st-and.ac.uk/~gap)`

.**30.**N. S. Whitley,*Fischer Matrices and Character Tables of Group Extensions*, MSc thesis, University of Natal, Pietermaritzburg, 1994.**31.**R. A. Wilson,*The local subgroups of the Fischer groups*, J. London. Math. Soc. (2)**36**(1987), 77 - 94. MR**88k:20037**

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Additional Information

**Faryad Ali**

Affiliation:
School of Mathematics, Statistics and I.T., University of Natal, Private Bag X 01, Scottsville, Pietermaritzburg 3209, South Africa

**Jamshid Moori**

Affiliation:
School of Mathematics, Statistics and I.T., University of Natal, Private Bag X 01, Scottsville, Pietermaritzburg 3209, South Africa

DOI:
https://doi.org/10.1090/S1088-4165-03-00175-4

Received by editor(s):
August 29, 2002

Received by editor(s) in revised form:
April 7, 2003

Published electronically:
July 29, 2003

Additional Notes:
The first author was supported by a postgraduate bursary from the NRF(SA)

The second author was supported by a research grant from University of Natal and NRF(SA)

Article copyright:
© Copyright 2003
American Mathematical Society