Okay I am having a lot of trouble with these sorts of problems, I really don't know what general sort of technique I should go about using to classify groups. Here are the questions, any advice is appreciated:

1) Classify the groups which have order 1125

2) Assume that $\displaystyle p^2$ is not congruent to 1 modulo q, and that $\displaystyle q^2$ is not congruent to 1 modulo p, and that p>q (p, q are unique primes). Classify the groups which have order $\displaystyle p^2q^2$

Okay so what I did was used Sylows theorems to prove that there is only 1 sylow q-subgroup, by assuming that S_q=p, and derived a contradiction of the assumption that p>q. Then as a corollary, S_q is normal in G.

Now I'm not sure what to do from this point. I'd imagine I would break the thing down by cases (First case is that S_p is normal, second case is that S_p = q) and see what I can derive about the gropu, but I have no idea how to go about this. Any advice is appreciated