# Show that every group of order 1965 is isomorphic to $\mathbb{Z}/393\mathbb{Z} \rtimes \mathbb{Z}/5\mathbb{Z}$

Where do I start in such problem? $$1965=3\times 131\times 5.$$

I think was able to show $$n_3=1$$ so $$S_3$$ is normal and G is not simple. $$n_{131}=1$$ What else can I do there? Should I split in the abelian and non-abelian cases or something like it? Should I use that to show that G is abelian somehow? Would you know where I can find similar exercises?

Thank you for your time and patience.

• Since $S_3$ and $S_{131}$ are both normal, and intersect trivially, the subgroup they generate is isomorphic to $S_3\times S_{131}$. What is that isomorphic to? – Arturo Magidin Nov 5 '19 at 22:05