# Irreducible representations of a semidirect product

I have two finite groups. The irreducible representations of their product are given by tensor products of the irreducible of representations of the groups.

Is there a way to build the irreducible representations of a semidirect product from the irreducible representations of the groups?

Any references are very welcome. I couldn't find this in Serre, so I'm guessing it isn't straightforward like the product case. So, any tips would also be great.

Just in case it is completely known and available in the literature, I am interested in $SL_2(\mathbb{F_q})\rtimes H$, where $H$ is the Heisenberg group.

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## 1 Answer

I am not aware of a general procedure that would work for any semi-direct product. Serre treats semi-direct products by abelian groups in Part II, Section 8.2. See also my answer to another question. In your particular case, apart from lifting the representations of $H$ from the quotient, I would also try to induce the irreducible representations of $SL_2(\mathbb{F}_q)$ to $G$ and see which ones remain irreducible or where you can split off summands that you already know about. Mackey's irreducibility criterion (see e.g. Serre, Part II, section 7.4) should be quite useful for this. Generally, taking inner products of two such induced characters is easy using Frobenius reciprocity and Mackey's formula.

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