Modular representation indecomposable but not irreducible?

Question

are there any exmaples of finite groups $G$ with a finite dimensional representation, which is indecomposable and not irreducible? (We need $\operatorname{char}(K) \big| |G|$) It would be nice if you have more than one example.

Answer 1

If $G$ is a $p$-group, where $char(k) =p$, then the only irreducibke $G$-module is $k$ with a trivial $G$-action; but there are other indecomposable ones in general.

For instance, let $G=C_2$ be the group with $2$ elements, and $k$ any field of characteristic $2$. Then $k^2$ with the swapping action $(x,y)\mapsto (y,x)$ is indecomposable.

More generally, under the same assumptions, $H^1(G,k)$ (group cohomology) is the same thing as $\mathrm{Ext}^1_{kG}(k,k)$, so it classifies extensions of $kG$-modules of the form $0\to k \to M \to k\to 0$ : whenever $H^1(G,k)\neq 0$, you have at least one nonsplit extension, which must correspond to an indecomposable module $M$ (here there's an additional argument to make, as nonsplit doesn't immediately imply that $M$ is indecomposable, but here it's the case) which obviously isn't irreducible.

But $H^1(G,k) \cong \hom(G,k)$, so you can find tons of examples that way.