Consider I given finite groups $T, R$ with representations act on the same finite vector space $V$, $\rho_T$ and $\rho_R$. Let's also consider that I given a group homomorphism $\phi: R\to Aut(T)$, $T$ is normal under conjugation with $R$. Then how can I construct matrix representation for $G=T\rtimes_{\phi}R$ acting on $V$?
For direct product it is just product of representations $\rho_T$ and $\rho_R$, I can do as following:
$$ (t,r)(t',r') = (tt', rr') = (tt',1)(1, rr') \\ \rho_G((t,r)(t',r')) = \rho_{G}(tt',1)\rho_{G}(1, rr') $$ So, I can define action on $V\oplus V$.Hence it is enough to define action on each group on $V$, but I given them.
But for semi direct I should some how incorporate $\phi$. Also as example of such construction is Affine group, but I can not figure out how to use it (as inspiration). (So if it is possible I would like to first see how construction works for Affine group and than try to generalise)
What did I try.
One simple example is to consider $r^{-1}tr = r^{-1}rt = t$, than representation $\rho(G)$ is product of representations $\rho_T$ and $\rho_R$.
I have following constrains on representation $\rho_G$ to satisfy:
$$ \rho_{G}(t) = \rho_T(t), \\ \rho_{G}(r) = \rho_R(r), \\ \rho_{G}(r^{-1}tr) = \rho_T(\phi(r)t). $$
Let's start from a simple case: consider both T,R is cyclic group with $t^n=e$ and $r^m=e$, than acting on basis $v_1,\dots,v_K$ I can write:
$$ \rho_T(t) = \sum\limits_{k=1}^{K}|v_{t\cdot k}\rangle\langle v_k|,\, \rho_R(r) = \sum\limits_{k=1}^{K}|v_{r\cdot k}\rangle\langle v_k|, $$ where actions over indexes are permutations that satisfies $t^n=e$ and $r^m=e$.
In order to satisfy $\rho_{G}(r^{-1}tr) = \rho_T(\phi(r)t)$ I can try to construct something like this:
$$ \phi(r)=\sum\limits_{n=1}^{N}|t^{\phi_r(n)}\rangle \langle t|, $$ where $\phi_r(n)$ is a permutation. And here I am stuck.