The Character Table of a Split Extension of Shape 28:U4(2) Using Fischer-Clifford Matrices

This study aimed to construct the character table of a split extension with shape 2⁸: U₄(2) using Fischer-Clifford matrices. A significant research gap existed in applying Fischer-Clifford matrices to characterize this specific split extension. While previous research had demonstrated the method’s effectiveness in deciphering character tables, its application to the 2⁸: U₄ (2) split extension remained unexplored. This study addressed the knowledge gap by investigating the construction of the character table for this particular split extension. The method employed the standard application of Clifford theory, enhanced by Fischer-Clifford matrices as developed by Bernd Fischer. The study focused on split extensions of groups N:G, where N is an elementary abelian 2-group where every irreducible character of N was extended to an irreducible character of its inertia group in 2⁸:U₄ (2). This property holds for split extensions, as per Mackey’s theorem. Coset analysis method was used to calculate conjugacy classes while Fischer Clifford Matrices technique together with the character table of inertia factor groups was used to construct the character table. Computations were primarily performed using the computer algebra systems GAP and MAGMA. The subgroup 2⁸:U₄ (2) whose order is 6,635,520 was found to have 49 conjugacy classes and 49 irreducible representations which are structured into 3 blocks; H₁ with 20 conjugacy classes, H₂ with 16 conjugacy classes and H₃ with 13 conjugacy classes whose structure descriptions were found out to be U₄(2), 3₊²⁺¹ :2(D₈) and 2⁴:S₄ respectively. The findings have potential applications in various scientific and engineering fields. Based on the study, future research emerges which include: extending the Fischer-Clifford matrices technique to group extensions with non-abelian kernels, writing GAP or MAGMA routines to assist in construction of character tables of challenging group extensions and investigation the applications of character tables of alike extensions in coding theory, cryptography and symmetry studies.