Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?

Let me firstly clarify my question. For example, we obviously have a faithful rep. of Poincare group, $\begin{pmatrix} \Lambda & x \\ 0 & 1 \end{pmatrix}$, where $\Lambda$ is Lorentz transformation and $x$ is translation. But this is a reducible and indecomposable representation. We can always define an irreducible rep. of Poincare group by

$$f:\begin{pmatrix} \Lambda & x \\ 0 & 1 \end{pmatrix}\rightarrow D_{(i,j)}(\Lambda)$$ where $D_{(i,j)}(\Lambda)$ is some irreducible rep. of Lorentz group. But in this case, the translation acts trivially.

Because Euclidean and Poincare group are noncompact, their unitary and irreducible rep. must be infinite dimensional. However it doesn't say we don't have finite dimensional and irreducible but not unitary rep. of these groups. The above is an example. So I want to know whether there is other finite dimensional irreducible rep. of Euclidean and Poincare group in which rotation and translation both act nontrivially? If it doesn't exist, how to prove or tell me the name of theorem.