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Zbl 0852.20003
Glasby, S.P.
On the faithful representations, of degree $2\sp n$, of certain extensions of $2$-groups by orthogonal and symplectic groups. (English)
[J] J. Aust. Math. Soc., Ser. A 58, No.2, 232-247 (1995). ISSN 0263-6115

Let $R$ be one of the groups $E=2^{1+2n}_\in$ or $H=4\circ 2^{1+2n}$ where $E$ is the extraspecial group of order $2^{1+2n}$ and of type $E=+$ and $H$ is the central product of a cyclic group of order 4 with $E$. It is known that if $\rho$ is a faithful absolutely irreducible representation of $R$ over the field $K$, then the degree of $\rho$ is $2^n$ and $K$ has characteristic different from 2, moreover in the case of $H$ the field $K$ should contain a square root of $-1$.\par In this paper the author starts with the above unique faithful irreducible representation of $R$ with degree $2^n$ and obtains projective representations of the automorphism group $\text{Aut}(E)$ and of $A$ where $A$ is a subgroup of index 2 in $\text{Aut}(H)$. The groups $\text{Aut}(E)$ and $A$ are of the forms $2^{2n}.O^\in_{2n}(2)$ and $2^{2n}.SP_{2n}(2)$ respectively where $2^{2n}$ is the elementary abelian 2-group of order $2^{2n}$ and the main result of the paper is to use the ideas of Schur to obtain faithful representations of central extensions of $\text{Aut}(E)$ and $A$ which are isomorphic to groups of the forms $2^{1+2n}.O^\in_{2n}(2)$ and $4\circ 2^{1+2n}.SP_{2n}(2)$ respectively.
[M.R.Darafsheh (Tehran)]

MSC 2000:: *20C15 Ordinary representations and characters of groups
20G05 Representation theory of linear algebraic groups

Keywords: extraspecial groups; central products; faithful absolutely irreducible representations; projective representations; automorphism groups

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