On the non-split extension 2 ^ 2n . Sp (2n , 2)

In this paper we give some general results on the non-split extension group overlineGn=2^ 2 (2n,2),n>2. We then focus on the group overlineG4=28cdotSp(8,2). We construct overlineG4 as a permutation group acting on 512 points. The conjugacy classes are determined using the coset analysis technique. Then we determine the inertia factor groups and Fischer matrices, which are required for the computations of the character table of overlineG4 by means of Clifford-Fischer Theory. There are two inertia factor groups namely H1=Sp(8,2) and H2=27:Sp(6,2), the Schur multiplier and hence the character table of the corresponding covering group of H2 were calculated. Using the information on conjugacy classes, Fischer matrices and ordinary and projective tables of H2, we concluded that we only need to use the ordinary character table of H2 to construct the character table of overline G4. The Fischer matrices of overline G4 are all listed in this paper. The character table of overlineG4 is a 195times195 complex valued matrix, it has been supplied in the PhD Thesis of the first author, which could be accessed online.