# Representations of Central Products

What is a good reference for learning about representations/characters of central products of groups?

By central product, I mean the following. If $G$ and $H$ are groups, containing isomorphic central subgroups $G_1$ and $H_1$ given by an isomorphism $\theta$, then $$G*H = (G \times H)/\langle (g,\theta(g)^{-1}) \rangle$$ is what I'm calling the central product, which obviously depends on $G_1$, $H_1$, and $\theta$.

Update: I've found some basic information about central products in the book by Gorenstein, but I'm still wondering if anywhere else has more discussion of this.

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What do you call a central product? Is it a central extension? That is, a group $E$ such that $C\subset E$ is central, and $E/C=G$ is your original group. – plm May 23 '12 at 15:48
@plm I've added some explanation in an edit. – Dane May 23 '12 at 18:09
If you are working over an algebraically closed field, then irreducible representations of $G\times H$ are tensor products of irreducible representations of $G$ with irreducible representations of $H$. Now, irreducible representations of $G*H$ are irreducible representations of $G\times H$ on which every $g\in G_1$ acts the same way as $\theta\left(g\right)$. So if you care for irreducible representations, finding those of $G\times H$ should be rather easy once you know those of $G$ and $H$. – darij grinberg May 23 '12 at 18:41
@darijgrinberg Thanks. I think that is what I was looking for. – Dane May 24 '12 at 0:02
@darijgrinberg Please consider posting your comment as an answer, so that the question can be marked as answered. – ˈjuː.zɚ79365 Jun 18 '13 at 14:48