Let G be a group of order 6. I am able to do the exercise without semidirect products($G \cong Z_6 $ or $S_3$) but I don't know how to use semidirect products to do this.
By Sylow's theorem, there is only subgroup of order 3, say P. As it is unique, P is normal in G. Also, let Q be a 2-Sylow subgroup. Note that $P\cap Q=\{e\}$. That means G is a semidirect product $P \rtimes_{\phi} Q$ for $\phi:Q \to Aut(P)$. So there are two possible automorphisms on P, names $\alpha$ which is the identity on $P$ and $\beta$ which takes $x\in P$ to $x^{-1}$. That gives us two homomorphisms $\phi:Q \to Aut(P)$. How do I proceed after this? (I suspect I don't understand semidirect products well enough and I really don't know how to explicitly show that one gets two nonisomorphic groups)