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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.2012-05-23
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    @plm I've added some explanation in an edit.2012-05-23
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    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 $H$.2012-05-23
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    @darijgrinberg Thanks. I think that is what I was looking for.2012-05-24
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    @darijgrinberg Please consider posting your comment as an answer, so that the question can be marked as answered.2013-06-18
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    Does anybody knows how the irreducible complex characters can be constructed for groups with united factor group (see B. Huppert, Finite groups I, pages 49-50, here also central products are defined).2014-09-03
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    What is sum of degrees of irreducible complex characters of the central product compared to that for the direct product (which is the product of the sums of the components and and upper bound for that of the central product as decribed above)?2015-01-26

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