Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, if $M\subset N$ are two such integral representations and $M$ is of finite prime-to-$p$ index in $N$, I call $M$ and $N$ $p$-equivalent. This is probably the same as to say that $M\otimes_{\mathbb{Z}} {\mathbb{Z}}_{(p)}$ is isomorphic to $N\otimes_{\mathbb{Z}} {\mathbb{Z}}_{(p)}$, where ${\mathbb{Z}}_{(p)}$ is the ring of fractions $a/b$ with $a,b\in{\mathbb{Z}}$, $b$ is prime to $p$. I would like to classify integral representations of $C_p^n$ up to $p$-equivalence.
Question. What is known or can be said on the integral representations, in particular, on the indecomposable integral representations, of $F=C_p^n$ with respect to this equivalence relation?
For example, in the case $F=C_p$, I think there are only three indecomposable integral representations up to $p$-equivalence: the 1-dimensional trivial representation, the permutation $p$-dimensional representation ${\mathbb{Z}}[C_p]$, and the $(p-1)$-dimensional representation ${\mathbb{Z}}[C_p]/{\mathbb{Z}}$ (where ${\mathbb{Z}}$ is embedded into ${\mathbb{Z}}[C_p]$ diagonally). Is that correct? What can be said in the case $F=C_p\times C_p$ ?
