I am trying to classify all groups of order $3825=3^2 \cdot 5^2 \cdot 17$. The Sylow theorems indicate that the number of Sylow p-subgroups for each p rime are $n_{17}=1$, and $n_{3}=1,25,85$ and $n_5=1,51$. Moreover, I know that the Sylow $3-$ and $5-$ subgroups $P_3$ and $P_5$ are abelian, as they are of the form $|G|=p^2$. I have classified all abelian groups using the structure theorem/fundamental theorem.
How do I find out the non-abelian groups of this order? In particular how do I realise $G$ as a semi-direct product with $P_{17}$ as one of the factors, as done while classifying groups? As $P_3$ and $P_5$ are not known to be normal, I cannot form something like $P_3 P_5$ as a subgroup. Counting arguments are not strong enough and don't tell me anything about $n_5$ and $n_3$.