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Ali, Faryad; Moori, Jamshid

The Fischer-Clifford matrices of a maximal subgroup of \(Fi_{24}'\). (English) Zbl 1065.20020

Represent. Theory 7, 300-321 (2003).
Summary: The Fischer group \(Fi_{24}'\) is the largest sporadic simple Fischer group of order \[ 1.255.205.709.190.661.721.292.800=2^{21}\cdot 3^{16}\cdot 5^2\cdot 7^3\cdot 11\cdot 13\cdot 17\cdot 23\cdot 29. \] The group \(Fi_{24}'\) 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 \(Fi_{24}'\) as represented in ATLAS by \(3A\), \(3B\), \(3C\), \(3D\) and \(3E\). A subgroup of \(Fi_{24}'\) 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 \(Fi_{24}'\) as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local subgroups \(\overline G:=N_{Fi_{24}'}(\langle N\rangle)\cong 3^7\cdot O_7(3)\), where \(N\cong 3^7\) is the natural orthogonal module for \(\overline G/N\cong O_7(3)\) with \(364\) subgroups of type \(3B\) corresponding to the totally isotropic points. The group \(\overline G\) is a nonsplit extension of \(N\) by \(G\cong O_7(3)\).

Cited in 1 Review
Cited in 9 Documents

MSC:

20C34 Representations of sporadic groups
20D08 Simple groups: sporadic groups
20E45 Conjugacy classes for groups

Keywords:

Fischer group \(Fi_{24}'\); 3-transposition groups; maximal 3-local subgroups; Fischer-Clifford matrices; conjugacy classes

Software:

GAP; Magma
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Full Text: DOI

References:

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