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 $${\mathbb {F}}_{2}.$$ F 2 . Therefore a split extension group of the form $$2^{12}{:}(Sz(8){:}3):= {\overline{G}}$$ 2 12 : ( S z ( 8 ) : 3 ) : = 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 $${\overline{G}}$$ 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 $${\overline{G}}$$ G ¯ is a $$43 \times 43$$ 43 × 43 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7.