On the non-split extension $2^{2n}{^{cdot}}Sp(2n,2)$

In this paper we give some general results on the non-splitextension group $overline{G}_{n} = 2^{2n}{^{cdot}}Sp(2n,2), ngeq2.$ We then focus on the group $overline{G}_{4} =2^{8}{^{cdot}}Sp(8,2).$ We construct $overline{G}_{4}$ as apermutation group acting on 512 points. The conjugacy classes aredetermined using the coset analysis technique. Then we determine theinertia factor groups and Fischer matrices, which are required forthe computations of the character table of $overline{G}_{4}$ bymeans of Clifford-Fischer Theory. There are two inertia factorgroups namely $H_{1} = Sp(8,2)$ and $H_{2} = 2^{7}{:}Sp(6,2),$ theSchur multiplier and hence the character table of the correspondingcovering group of $H_{2}$ were calculated. Using the information onconjugacy classes, Fischer matrices and ordinary and projectivetables of $H_{2},$ we concluded that we only need to use theordinary character table of $H_{2}$ to construct the character tableof $overline{G}_{4}.$ The Fischer matrices of $overline{G}_{4}$are all listed in this paper. The character table of$overline{G}_{4}$ is a $195 times 195$ complex valued matrix, ithas been supplied in the PhD Thesis of the firstauthor, which could be accessed online.