Irreducible real representations of $D_{2k}$ and $(C_{i}\times C_{j})\rtimes D_{2k}$

Question

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.

Answer 1

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?).