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For a finite group $G$ the regular action $\rho$ of $G$ on itself (by right multiplication) has the property that the normalizer of $\rho(G)$ in the symmetric group $S_G$ is isomorphic to the holomorph $G\rtimes \mathop{Aut}(G)\;$. [I got reminded to this fact by this recent question at mathoverflow.]

What is the according normalizer in $\mathop{GL}_F(V)$ when taking $G$ as a base of a vector space $V$ over a field $F$ (which can be chosen as you like) with $G$ acting regularly on this base?

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The easiest case to think about is when $F$ is a splitting field for $G$ and $F$ has characteristic coprime to $|G|$. Then, as an $FG$-module, $\rho(G)$ is isomorphic to a the direct sum $\oplus k_iV_i$, where $V_i$ are the simple $FG$-modules, and $k_i = \dim(V_i)$. The endomorphism ring of this module is the direct sum $\oplus M(k_i,F)$ of algebras consisting of the $k_i \times k_i$ matrices over $F$.

So the centralizer of $\rho(G)$ in ${\rm GL}_F(V)$ will be isomorphic to the direct product of the general linear groups $ {\rm GL}_{k_i}(F)$. To get the normalizer, just adjoin the image in ${\rm GL}_F(V)$ of the permutation representation of ${\rm Aut}(G)$ arising in the normalizer of the regular permutation representation of $G$.

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    Right, there can't be more. Thanks, again.2012-07-24