Character tables of some selected groups of extension type using Fischer-Clifford matrices

Abstract

The aim of this dissertation is to calculate character tables of group extensions. There are several well developed methods for calculating the character tables of some selected group extensions. The method we study in this dissertation, is a standard application of Clifford theory, made efficient by the use of Fischer-Clifford matrices, as introduced by Fischer. We consider only extensions Ḡ of the normal subgroup N by the subgroup G with the property that every irreducible character of N can be extended to an irreducible character of its inertia group in Ḡ , if N is abelian. This is indeed the case if Ḡ is a split extension, by a well known theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of Fischer-Clifford matrices is discussed in detail. Some of the properties of these Fischer-Clifford matrices which make their calculation much easier, are also given. We restrict ourselves to split extension groups Ḡ = N:G in which N is always an elementary abelian 2-group. In this thesis we are concerned with the construction of the character tables (by means of the technique of Fischer-Clifford matrices) of certain extension groups which are associated with the orthogonal group O+10(2), the automorphism groups U₆(2):2, U₆(2):3 of the unitary group U₆(2) and the smallest Fischer sporadic simple group Fi₂₂. These groups are of the type type 2⁸:(U₄(2):2), (2⁹ : L₃(4)):2, (2⁹:L₃(4)):3 and 2⁶:(2⁵:S₆).