# Semi-direct products of $\mathbb Z/12\mathbb Z$ by $\mathbb Z / 2\mathbb Z$

I need to prove that there exists at least three non-isomorphic semi-direct products $$\mathbb Z/12\mathbb Z \rtimes \mathbb Z / 2\mathbb Z$$ To find such semi-direct products, we need to understand what the homomorphisms of groups $$\mathbb Z / 2\mathbb Z \rightarrow \operatorname{Aut}(\mathbb Z/12\mathbb Z) \simeq (\mathbb Z/12\mathbb Z)^{\times}$$ are. Such a homomorphism is determined by the image of $$1$$ which must be an element of order $$2$$ in $$(\mathbb Z/12\mathbb Z)^{\times}$$.
But we have $$(\mathbb Z/12\mathbb Z)^{\times}=\{1,5,7,11\}$$ and it turns out that each of these elements has order $$2$$. Hence, we have seemingly four possible semi-direct product structures $$\mathbb Z/12\mathbb Z \rtimes \mathbb Z / 2\mathbb Z$$.

Now, is there a way to easily tell these structures apart ? How would you prove that among these four structures, at least three of them are not isomorphic ?

Of course, the direct product structure induced by $$1$$ is the only one that gives an abelian group, but what about the others ?

• They are all distinct. You can distinguish them by looking at their subgroups. For example two of them have abelian Sylow 2-subgroups. Two of them have a subgroup $S_3$ etc. – Derek Holt Sep 23 '18 at 14:29