Fischer Clifford Matrices and Character Table of the Split Extension Group 28: A10

Character tables form a central tool in the representation theory of finite groups, yet their construction especially for large and complex groups remains a significant challenge. The primary goal of the research was to compute the Fischer matrices and ordinary character table for a specific group extension, namely 2⁸:A₁₀, which was a subgroup of the affine group with structure Sp(8, 2). The computation of character tables for group extensions had been a significant area of focus in group theory, with various methods developed to tackle this complex task. In this study, the Fischer-Clifford matrices technique, an effective and robust method introduced by Fischer, was used. The technique operated on the principle that, for an extension G̅ = K:Q where K ⊴ G̅, every ordinary irreducible character of K could either extend to an ordinary irreducible character or an irreducible projective character of the corresponding inertia subgroup in G̅, provided that K was abelian. The method involved calculating an invertible matrix, known as a Fischer matrix, for each conjugacy class of Q. These matrices, in conjunction with the ordinary character tables or projective characters of subgroups of Q, known as inertia factor groups, were then used to construct the complete ordinary character table of G̅. To determine the conjugacy classes of the study group, coset analysis technique was used, which had been pioneered by Moori. The purpose of this research was to calculate the Fischer matrices and character table for the split extension 2₈:A₁₀. This group was particularly interesting because it acted as a maximal subgroup within the affine group Sp(8, 2), which itself was a subgroup of the symplectic group Sp(8, 2). Given the complexity of such large groups, the Fischer-Clifford technique offered a structured approach to systematically derive their character tables. The computations heavily relied on the computer algebra systems MAGMA and GAP, which were invaluable tools for such algebraic and combinatorial calculations. By focusing on the group 2⁸:A₁₀, this study aimed to provide a deeper understanding of the Fischer matrices and character tables of specific extensions, thereby contributing to the broader efforts in classifying finite groups and understanding their internal structures.