Clifford-Fischer Matrices and character tables of certain group extensions associated with M₂₂:2, M₂₄ and HS:2

Abstract

Character tables of finite groups provide substantial information about a group and their use is of great importance in physical sciences and pure mathematics. Any finite group is either simple or has a proper normal subgroup and hence may be of extension type Ḡ = N.G. There are several methods for constructing character tables of group extensions especially when the kernel of the extension, N is an elementary abelian p-group. In this dissertation, we use a more natural approach to the study of the character table of Ḡ called the Clifford-Fischer matrices due to Bernd Fischer. This method is based on Clifford's Theory. For each conjugacy class [g]G, we construct an invertible matrix M(g), called a Fischer matrix. We have employed in this dissertation a new approach of calculating these Fischer matrices. For the determination of the conjugacy classes, we use coset analysis developed by Moori. Having all the conjugacy classes, Fischer matrices, character tables and fusions of the inertia factor groups into G, we can easily construct the character table of Ḡ. We will apply the method of Fischer matrices to construct character tables of group extensions associated with the full automorphism group of the Mathieu group, M₂₂ denoted by M₂₂:2, the full automorphism group of the Higman-Sims group, HS denoted by HS:2 and the largest of the five Mathieu groups, M₂₄: These groups are 2⁴:S₆ a maximal subgroup of M₂₂:2, a subgroup 2⁵:A₆ of HS:2 and a maximal subgroup 2⁶:(3'S₆) of M₂₄.