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The Fischer-Clifford matrices of a maximal subgroup of $Fi^{\prime }_{24}$


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: 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 $G \cong O_7(3)$.


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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
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)
Article copyright: © Copyright 2003 American Mathematical Society