On two groups of the form $2^{8}{:}A_{9}$

Abstract

This paper is dealing with two split extensions of the form $2^{8}{:}A_{9}.$ We refer to these two groups by $\overline{G}_{1}$ and $\overline{G}_{2}.$ For $\overline{G}_{1},$ the 8-dimensional GF(2)-module is in fact the deleted permutation module for $A_{9}.$ We firstly determine the conjugacy classes of $\overline{G}_{1}$ and $\overline{G}_{2}$ using the coset analysis technique. The structures of inertia factor groups were determined for the two extensions. The inertia factor groups of $\overline{G}_{1}$ are $A_{9},\,A_{8},\, S_{7},\,(A_{6} \times 3){:}2$ and $(A_{5} \times A_{4}){:}2,$ while the inertia factor groups of $\overline{G}_{2}$ are $A_{9},\, PSL(2,8){:}3$ and $2^{3}{:}GL(3,2).$ We then determine the Fischer matrices for these two groups and apply the Clifford–Fischer theory to compute the ordinary character tables of $\overline{G}_{1}$ and $\overline{G}_{2}.$ The Fischer matrices of $\overline{G}_{1}$ and $\overline{G}_{2}$ are all integer valued, with sizes ranging from 1 to 9 and from 1 to 4 respectively. The full character tables of $\overline{G}_{1}$ and $\overline{G}_{2}$ are $84 \times 84$ and $40 \times 40$ complex valued matrices respectively.