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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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The Fischer-Clifford matrices of a maximal subgroup of $Fi^{\prime }_{24}$
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by Faryad Ali and Jamshid Moori
Represent. Theory 7 (2003), 300-321
DOI: https://doi.org/10.1090/S1088-4165-03-00175-4
Published electronically: July 29, 2003
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Abstract:

The Fischer group $Fi_{24}^{\prime }$ is the largest sporadic simple Fischer group of order \[ 1255205709190661721292800 = 2^{21}.3^{16}.5^2.7^3.11.13.17.23.29 \;\;.\] The group $Fi_{24}^{\prime }$ is the derived subgroup of the Fischer $3$-transposition group $Fi_{24}$ discovered by Bernd Fischer. There are five classes of elements of order 3 in $Fi_{24}^{\prime }$ as represented in ATLAS by $3A$, $3B$, $3C$, $3D$ and $3E$. A subgroup of $Fi_{24}^{\prime }$ of order $3$ is called of type $3X$, where $X \in \{A,B,C,D,E \}$, if it is generated by an element in the class $3X$. There are six classes of maximal 3-local subgroups of $Fi_{24}^{\prime }$ as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local subgroups $\bar {G} := N_{Fi_{24}^{\prime }}(\langle N\rangle ) \cong 3^7{\cdot }O_7(3)$, where $N \cong 3^7$ is the natural orthogonal module for $\bar {G}/N \cong O_7(3)$ with $364$ subgroups of type $3B$ corresponding to the totally isotropic points. The group $\bar {G}$ is a nonsplit extension of $N$ by $G \cong O_7(3)$.
References
    Ali F. Ali, Fischer-Clifford Matrices for Split and Non-Split Group Extensions, PhD Thesis, University of Natal, Pietermaritzburg, 2001. AliMoo1 F. Ali and J. Moori, Fischer-Clifford Matrices of the Group $2^7{:}Sp_6(2)$, In preparation. AliMoo2 F. Ali and J. Moori, Fischer-Clifford Matricesand Character Table of the Group $2^8{:}Sp_6(2)$, In preparation. AliMoo3 F. Ali and J. Moori, The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of $Fi_{24}$ , In preparation. magma Wieb Bosma and John Cannon. Handbook of Magma Functions, Department of Mathematics, University of Sydney, November 1994.
  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
  • M. R. Darafsheh and A. Iranmanesh, Computation of the character table of affine groups using Fischer matrices, Groups ’93 Galway/St. Andrews, Vol. 1 (Galway, 1993) London Math. Soc. Lecture Note Ser., vol. 211, Cambridge Univ. Press, Cambridge, 1995, pp.Β 131–137. MR 1342786, DOI 10.1017/CBO9780511629280.013
    • Fis1 B. Fischer, Finite Groups Generated by 3-Transpositions, Notes, Mathematics Institute, University of Warwick, 1970.
  • Bernd Fischer, Clifford-matrices, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, BirkhΓ€user, Basel, 1991, pp.Β 1–16. MR 1112155, DOI 10.1007/s10107-002-0338-6
    • Fis3 B. Fischer, Character tables of maximal subgroups of sporadic simple groups -III, Preprint. Fis4 B. Fischer, unpublished manuscript (1985).
  • P. X. Gallagher, Group characters and normal Hall subgroups, Nagoya Math. J. 21 (1962), 223–230. MR 142671, DOI 10.1017/S0027763000023849
  • Patrick X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970), 175–179. MR 276318, DOI 10.1007/BF01113339
  • Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
  • D. F. Holt, A computer program for the calculation of a covering group of a finite group, J. Pure Appl. Algebra 35 (1985), no.Β 3, 287–295. MR 777260, DOI 10.1016/0022-4049(85)90046-5
  • I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
  • Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson, An atlas of Brauer characters, London Mathematical Society Monographs. New Series, vol. 11, The Clarendon Press, Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton; Oxford Science Publications. MR 1367961
  • Gregory Karpilovsky, Group representations. Vol. 1. Part A, North-Holland Mathematics Studies, vol. 175, North-Holland Publishing Co., Amsterdam, 1992. Background material. MR 1183469
  • R. J. List, On the characters of $2^{n-\epsilon }\cdot S_n$, Arch. Math. (Basel) 51 (1988), no.Β 2, 118–124. MR 959386, DOI 10.1007/BF01206468
  • R. J. List and I. M. I. Mahmoud, Fischer matrices for wreath products $G\,w\,S_n$, Arch. Math. (Basel) 50 (1988), no.Β 5, 394–401. MR 942535, DOI 10.1007/BF01196499
  • Jamshid Moori and Zwelethemba Mpono, The Fischer-Clifford matrices of the group $2^6\colon \!\textrm {SP}_6(2)$, Quaest. Math. 22 (1999), no.Β 2, 257–298. MR 1728499, DOI 10.1080/16073606.1999.9632080
  • Jamshid Moori and Zwelethemba Mpono, The centralizer of an involutory outer automorphism of $F_{22}$, Math. Japon. 49 (1999), no.Β 1, 93–113. MR 1671961, DOI 10.1080/16073606.1999.9632080
  • Jamshid Moori and Zwelethemba Mpono, The centralizer of an involutory outer automorphism of $F_{22}$, Math. Japon. 49 (1999), no.Β 1, 93–113. MR 1671961
  • Jamshid Moori and Zwelethemba Mpono, Fischer-Clifford matrices and the character table of a maximal subgroup of $\overline F_{22}$, Int. J. Math. Game Theory Algebra 10 (2000), no.Β 1, 1–12. MR 1756359
    • Mpo Z. E. Mpono, Fischer-Clifford Theory and Character Tables of Group Extensions, PhD thesis, University of Natal, Pietermaritzburg, 1998. NagTsu H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, San Diego, 1987. Sal R. B. Salleh, On the Construction of the Character Tables of Extension Groups, PhD thesis, University of Birmingham, 1982. Sch U. Schiffer, Cliffordmatrizen, Diplomarbeit, Lehrstul D Fur Matematik, RWTH, Aachen, 1995. gap The GAP Group, GAP - Groups, Algorithms and Programming, Version 4.2 , Aachen, St Andrews, 2000, (http://www-gap.dcs.st-and.ac.uk/~gap). Whi N. S. Whitley, Fischer Matrices and Character Tables of Group Extensions, MSc thesis, University of Natal, Pietermaritzburg, 1994.
  • Robert A. Wilson, The local subgroups of the Fischer groups, J. London Math. Soc. (2) 36 (1987), no.Β 1, 77–94. MR 897676, DOI 10.1112/jlms/s2-36.1.77
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Bibliographic Information
  • Faryad Ali
  • Affiliation: School of Mathematics, Statistics and I.T., University of Natal, Private Bag X 01, Scottsville, Pietermaritzburg 3209, South Africa
  • ORCID: setImmediate$0.9265022794625328$2
  • Jamshid Moori
  • Affiliation: School of Mathematics, Statistics and I.T., University of Natal, Private Bag X 01, Scottsville, Pietermaritzburg 3209, South Africa
  • 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)
  • © Copyright 2003 American Mathematical Society
  • Journal: Represent. Theory 7 (2003), 300-321
  • MSC (2000): Primary 20C15, 20D08, 20E22
  • DOI: https://doi.org/10.1090/S1088-4165-03-00175-4
  • MathSciNet review: 1993362