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I am considering finite groups of types $D_{2k}$ or $(C_{i}\times C_{j})\rtimes D_{2k}$. I would like to find the irreps of these groups over $\mathbb{R}$ on vector spaces of dimensions $N \lesssim 20$, for which I have bases that that, apart from the trivial irrep, carry the natural representation of $S_N$ in $N-1$ dimensions. Is there a theoretical approach I can follow, or barring that, a practical computational approach, preferably implementable in Mathematica, with a focus on ease of implementation rather than optimality.

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    $\begingroup$ There are existing functions in GAP and Magma for finding the complex representations. This problem is difficult and the algorithms involved are complicated. $\endgroup$
    – Derek Holt
    Sep 2, 2021 at 7:25
  • $\begingroup$ I have significantly updated my question, as I had overlooked that much more is in fact known about the structure of the groups involved. $\endgroup$
    – BeMuSeD
    Sep 2, 2021 at 14:54
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    $\begingroup$ The representations of the dihedral groups are easy and well understood. They all have dimension $1$ or $2$, and the $2$-dimensionals are as rotation and reflection matrices. I don't know about the other examples. You haven't specified an action for the semidirect product. $\endgroup$
    – Derek Holt
    Sep 3, 2021 at 7:47
  • $\begingroup$ Given the, indeed straightforward answer to the question on $D_{2k}$, I have now sharpened the question considerably. I am mulling on your question about the action in the semidirect product. $\endgroup$
    – BeMuSeD
    Sep 3, 2021 at 8:44
  • $\begingroup$ Please don't do that, @BeMuSeD. Ask a new question instead. $\endgroup$
    – Shaun
    Sep 3, 2021 at 14:07

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Leaving aside the requirement of reality, the question I asked is actually an example of a more general fully solved case, namely that of the irreps over $\mathbb{C}$ of the semidirect product $G = A \rtimes_{\varphi} H$, where $A$ abelian, $H$ a group with a known full set of irreps, and $\varphi:H\rightarrow \operatorname{Aut}(A)$ a homomorphism in $H$ under composition. This case is e.g. treated by Serre, Linear representations of Finite Groups, in section 8.2. Let $X=\{\chi_1,\chi_2,\ldots,\chi_{|A|}\}$ be the group of characters of $A$. Define the action of $H$ on this group through $h\chi_i(a)=\chi_i\left(\varphi^{-1}(h)a\right)$. Denote the orbits of $H$ on $X$ as $X_\alpha$, and choose a representative $\chi_\alpha \in X_\alpha$. Let $H_\alpha$ be the stabilizer in $H$ of $\chi_\alpha$, and $\rho_\alpha:H_\alpha\rightarrow V_\alpha$ and irrep of $H_\alpha$. Then $\chi_\alpha \otimes V_\alpha$ carries an irrep of $G_\alpha=A \rtimes_{\varphi} H_\alpha$. Finally, consider the induced representation $\operatorname{Ind}^{G}_{G_\alpha}(\chi_\alpha \otimes V_\alpha)$. The statement is that the latter is irreducible, and that by this procedure all irreps are found. This has been discussed in more detail e.g. in Irreducible representations of a semidirect product, proof in Serre. I am confident, but admittedly lack proof, that the requested real irreps can be derived from the complex ones, through standard procedures (see e.g. Easy way to get real irreducible characters (reps) from complex irreducible characters?).

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