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The Fischer group F i24 is the largest sporadic simple Fischer group of order 1255205709190661721292800 = 2.3.5.7.11.13.17.23.29 . The group F i24 is the derived subgroup of the Fischer 3-transposition group F i24 discovered by Bernd Fischer. There are five classes of elements of order 3 in F i24 as represented in ATLAS by 3A, 3B, 3C, 3D and 3E. A subgroup of F i24 of order 3 is called of type 3X, where X ∈ {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 F i24 as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local subgroups Ḡ := Fi24 (〈N〉) ∼= 3·O7(3), where N ∼= 37 is the natural orthogonal module for Ḡ/N ∼= O7(3) with 364 subgroups of type 3B corresponding to the totally isotropic points. The group Ḡ is a nonsplit extension of N by G ∼= O7(3).

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