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Let $P$ be a finite $p$-group ($p$ prime) and $C$ a finite cyclic group of order coprime to $p$.

What can we say about the representation ring of a semi-direct product $R_{\mathbb Q}(P \rtimes C )$?

I'm looking for something similar to the case of a direct product, where we have $R_{\mathbb Q}(P \times C )\cong R_{\mathbb Q}(P)\times R_{\mathbb Q}(C)$.

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Try googling for "crossed products" ($R_{\mathbb{Q}(P\rtimes C)$ is a crossed product of $R_{\mathbb{Q}(P)$ by $C$). –  Alain Valette Nov 10 '11 at 20:38

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