is no overlap between the two. To see that all the $2n$ dimensional representations of item $2(a)$ are distinct, it is enough to observe that the rational canonical forms of the linear transformations on the co-invariant submodules, $$b^{-1}a : M/UM {\ensuremath{\rightarrow}}M/UM$$ are distinct, where $U$ is the ideal generated by $a$ and $b$. To see that the $2n$ dimensional representation of item $2(b)$ does not occur in item $2(a)$ observe that kernel of the $b$-action in both cases have different dimensions: $n$ for the module in $2(a)$ and $n+1$ for that in 2$(b)$. The two $2n+1$ dimensional modules in items $3(b)$ and $3(c)$ are non-isomorphic because it is clear from the diagrams that the dimensions of the invariant submodules in both cases are different: $n$ for those in item $3(b)$, and $n+1$ for those in item $3(c)$.

Of course the hard thing is to show that every indecomposable representation of $V_4$ is isomorphic to one in the above list. Since projective modules over $p$-groups are free, there is only one indecomposable projective $V_4$-module, namely $kV_4$ which occurs as item (1) in the list. Therefore we only consider indecomposable projective-free (modules which do not have projective summands) $V_4$-modules.

One can get 