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Let $G = \mathbb{Z}_2 \wr \mathbb{Z}_n$ be the finite lamplighter group. What are the irreducible representations of $G$ - can anyone provide a clear reference?

Austin, Naor and Valette list representations of $G$ here: http://arxiv.org/abs/0705.4662 (page 4), however, it seems to me that they are not all irreducible (dimension count gives $\sum_{\rho} d_{\rho}^2> |G|$).

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One can explicitly describe all the irreducible characters of a semi-direct products by an abelian group. This is done in Serre's rep theory book, Chapter II, Section 8.2, or here in the edit.

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