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

Document Type: Research Paper

Authors

1 School of Mathematical Sciences, North-West University (Mafikeng)

2 North-West University

Abstract

In this paper we give some general results on the non-split
extension group $overline{G}_{n} = 2^{2n}{^{cdot}}Sp(2n,2), ngeq
2.$ We then focus on the group $overline{G}_{4} =
2^{8}{^{cdot}}Sp(8,2).$ We construct $overline{G}_{4}$ 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 $overline{G}_{4}$ by
means of Clifford-Fischer Theory. There are two inertia factor
groups namely $H_{1} = Sp(8,2)$ and $H_{2} = 2^{7}{:}Sp(6,2),$ the
Schur multiplier and hence the character table of the corresponding
covering group of $H_{2}$ were calculated. Using the information on
conjugacy classes, Fischer matrices and ordinary and projective
tables of $H_{2},$ we concluded that we only need to use the
ordinary character table of $H_{2}$ to construct the character table
of $overline{G}_{4}.$ 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, it
has been supplied in the PhD Thesis of the first
author, which could be accessed online.

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