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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 ?

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  • $\begingroup$ 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. $\endgroup$ – Derek Holt Sep 23 '18 at 14:29

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