On two groups of the form 28:A9

Abstract

This paper is dealing with two split extensions of the form 28:A9. We refer to these two groups by G¯¯¯¯1 and G¯¯¯¯2. For G¯¯¯¯1, the 8-dimensional GF(2)-module is in fact the deleted permutation module for A9. We firstly determine the conjugacy classes of G¯¯¯¯1 and G¯¯¯¯2 using the coset analysis technique. The structures of inertia factor groups were determined for the two extensions. The inertia factor groups of G¯¯¯¯1 are A9,A8,S7,(A6×3):2 and (A5×A4):2, while the inertia factor groups of G¯¯¯¯2 are A9,PSL(2,8):3 and 23:GL(3,2). We then determine the Fischer matrices for these two groups and apply the Clifford–Fischer theory to compute the ordinary character tables of G¯¯¯¯1 and G¯¯¯¯2. The Fischer matrices of G¯¯¯¯1 and G¯¯¯¯2 are all integer valued, with sizes ranging from 1 to 9 and from 1 to 4 respectively. The full character tables of G¯¯¯¯1 and G¯¯¯¯2 are 84×84 and 40×40 complex valued matrices respectively.