It is known that the standard Eilenberg-Maclane space $K(G,1)$ is uniquely determined by any given group $G$. However, if $G$ contains an element of finite order, then every $K(G,1)$ CW complex must be infinite dimensional.
We are interested in the situation of $G$ is finite and non-abelian, and we do not ask additional condition on its universal covering space (in other words, $\pi_i(M)=0$, for $i>1$).
The following article by Rubinstein constructs such manifolds, asked in the title:
On $3$-manifolds that have finite fundamental group and contain Klein bottles.
For $4$-manifolds, see this MO-question.