On a group extension involving the Suzuki group Sz(8)

Abstract

The Suzuki simple group Sz(8) has an automorphism group 3. Using the electronic Atlas [22], the group Sz(8) : 3 has an absolutely irreducible module of dimension 12 over \({\mathbb {F}}_{2}.\) Therefore a split extension group of the form \(2^{12}{:}(Sz(8){:}3):= {\overline{G}}\) exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of \({\overline{G}}\) by analysing the maximal subgroups of Sz(8) : 3 and maximal of the maximal subgroups of Sz(8) : 3 together with various other information. It turns out that the character table of \({\overline{G}}\) is a \(43 \times 43\) complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7.