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Let $G$ be central extension of an abelian group $A$ by some group $H$. Is it possible to characterize all irreducible representions of $G$ in terms of irreducible representations of $A$ and $H$?

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The question is somewhat unprecise. I assume you mean irreps over $\mathbb{C}$ and finite groups. Then the answer is "no" as long as you aren't more specific about the central extension in question. For example, four of the five nonisomorphic groups of order $p^3$ can be obtained as central extensions of $C_p$ by $C_p\times C_p$, and the irreps of the abelian and the nonabelian ones are quite different. (By the way, it seems more usual to call $G$ a central extension of $H$ by $A$, assuming you want to have $A\subseteq \operatorname{\textbf{Z}}(G)$.)
However, the projective representations of $H$ tell you something about representations of central extensions, and here Schur's theory of the Schur multiplier and the "Darstellungsgruppe" of $H$ is helpful. You should look for these keywords in books on representation theory (e. g., the appropriate sections in Huppert, Endliche Gruppen I, Kapitel V, contain much information).

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Can you please recommend some good books in English were I can read about it? – Klim Efremenko Feb 26 '11 at 18:54
@Klim Efremenko: Huppert, Character Theory of Finite Groups, or Curtis and Reiner, Representation Theory of Finite Groups and Associative Algebras – Frieder Ladisch Feb 27 '11 at 21:12

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