Is there a construction that will give a non-abelian group of order $p^mr$ where $p$ is a prime, $r$ and $p$ are relatively prime and $m$ is an arbitrary non-negative integer? I suspect in this generality there is no simple construction so feel free to restrict $m$ and $r$.

I'm reading some notes on group theory and so far I've only seen the group $G=SL(2,p)$ which has order $p(p+1)(p-1)$. This is great because it gives me a class of groups to play with and test out the various theorems. It is a little annoying to have all these theorems and no concrete non-trival examples to test them out on to see all the subtleties since for the abelian groups all these theorems reduce to saying something trivial.

Edit: Mariano makes a good point and I'm not sure how to rule out silly examples like $G\times\mathbb{Z}_p^{m-3}\times\mathbb{Z}_r$. These aren't bad per se but what matters for me is an explicit description of $G$, the non-commutative part, so I have some hope of carrying out some calculations. In essence what I would really like is a construction that parametrizes the non-commutative part depending on all the parameters. In Mariano's example the non-commutative part has no dependence on $m$ and this simplifies the structure of the resulting group.

Thanks for the examples and references. This gives me a lot more concrete stuff to work with. Now hopefully I can work out some of the reasons for the various assumptions used in the proofs.