The Fischer-Clifford matrices of a maximal subgroup of 𝐹𝑖′₂₄

Abstract

The Fischer group F i 24 ′ Fi_{24}^{\prime } is the largest sporadic simple Fischer group of order \[ 1255205709190661721292800 = 2 21 .3 16 .5 2 .7 3 .11 .13 .17 .23 .29 . 1255205709190661721292800 = 2^{21}.3^{16}.5^2.7^3.11.13.17.23.29 \;\;. \] The group F i 24 ′ Fi_{24}^{\prime } is the derived subgroup of the Fischer 3 3 -transposition group F i 24 Fi_{24} discovered by Bernd Fischer. There are five classes of elements of order 3 in F i 24 ′ Fi_{24}^{\prime } as represented in ATLAS by 3 A 3A , 3 B 3B , 3 C 3C , 3 D 3D and 3 E 3E . A subgroup of F i 24 ′ Fi_{24}^{\prime } of order 3 3 is called of type 3 X 3X , where X ∈ { A , B , C , D , E } X \in \{A,B,C,D,E \} , if it is generated by an element in the class 3 X 3X . There are six classes of maximal 3-local subgroups of F i 24 ′ 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 G ¯ := N F i 24 ′ ( ⟨ N ⟩ ) ≅ 3 7 ⋅ O 7 ( 3 ) \bar {G} := N_{Fi_{24}^{\prime }}(\langle N\rangle ) \cong 3^7{\cdot }O_7(3) , where N ≅ 3 7 N \cong 3^7 is the natural orthogonal module for G ¯ / N ≅ O 7 ( 3 ) \bar {G}/N \cong O_7(3) with 364 364 subgroups of type 3 B 3B corresponding to the totally isotropic points. The group G ¯ \bar {G} is a nonsplit extension of N N by G ≅ O 7 ( 3 ) G \cong O_7(3) .