On 2 nonsplit extension groups associated with HS and HS:2

Authors: JAMSHID MOORI, THEKISO SERETLO

Abstract: The group HS:2 is the full automorphism group of the Higman--Sims group HS. The groups 2^{4.}S_6 and 2^{5.}S_6 are maximal subgroups of HS and HS:2, respectively. The group 2^{4.}S_6 is of order 11520 and 2^{5.}S_6 is of order 23040 and each of them is of index 3 850 in HS and HS:2, respectively. The aim of this paper is to first construct \overline{G} = 2^{5.}S_6 as a group of the form 2^{4.}S_6.2 (that is, \overline{G} = G_1.2) and then compute the character tables of these 2 nonsplit extension groups by using the method of Fischer--Clifford theory. We will show that the projective character tables of the inertia factor groups are not required. The Fischer--Clifford matrices of \overline{G}_1 and \overline{G} are computed. These matrices together with the partial character tables of the inertia factors are used to compute the full character tables of these 2 groups. The fusion of \overline{G}_1 into \overline{G} is also given.

Keywords: Group extensions, Higman--Sims group, automorphism group, character table, Clifford theory, inertia groups, Fischer--Clifford matrices

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