Let $G=C_{2} \times C_{4}$ with $a=(1,0)$ and $b=(0,1)$. Let $1$ and $a$ denote the cosets modulo the normal subgroup generated by $b$. Similarly, let $1,b,b^{2},b^{3}$ be the cosets modulo the subgroup generated by $a$. The group $G$ has an obvious action on these cosets.
We thus obtain $G$-modules generated by the cosets $N = \mathbb{Z}/5\mathbb{Z} \langle1,a \rangle$ and $M = \mathbb{Z}/5\mathbb{Z} \langle 1,b,b^{2},b^{3} \rangle$. I would like to calculate a composition series for the $G$-module $M$. I suppose that it would look something like
$$0 \longrightarrow \langle 1+b+b^{2}+b^{3} \rangle \longrightarrow M_{1} \longrightarrow M_{2} \longrightarrow M \longrightarrow 0$$
but I'm having a hard time obtaining $M_{1}$ and $M_{2}$ explicitly.