How to describe all non-abelian extension groups of a given $\mathbb{Z}_n$ by $\mathbb{Z}_m$, where $n$ and $m$ are primes?

Question

Consider the short exact sequence $0 \rightarrow \mathbb{Z}_n \rightarrow G \rightarrow \mathbb{Z}_m \rightarrow 0$, where

1. $\mathbb{Z}_n \simeq$ normal subgroup $N$ of $G$
2. $\mathbb{Z}_m \simeq G/N$
3. $n$ and $m$ are primes

What can we say about classification of all non-abelian extension groups of $\mathbb{Z}_n$ by $\mathbb{Z}_m$? Is there a method to find all non-abelian extension groups without cohomology theory?

Answer 1

Any group of order $nm$ is of such type. Let $n\le m$, and take an element of order $m$, by Cauchy's theorem. Then we have that the cyclic subgroup generated by this element is normal, since subgroups of index the smallest prime are normal. The quotient is a group of order $n$, and thus since all groups of prime index are cyclice, we have produced an exact sequence of your type. Thus what you are looking for is a classification of non-abelian groups of order $pq$. This has been done as in the question linked in the comment.