Are any 2 groups of order 6 isomorphic?

Question

So my question is pretty simple : Are any groups of order 6 isomorphic ?

I would say no, but I know that if the groups are cyclic then yes. If the answer is indeed no, could I please have a counter-example ?

Thank you.

Answer 1

If you know about permutation groups, then $S_3$ is a non-Abelian group (so not cyclic) of order $6$. There is also the obvious cyclic subgroup of order $6$. These are the only two groups (up to isomorphism) of order $6$.

Answer 2

It is known that $S_3\not\cong \mathbb{Z}_6.$ This follows from the fact that $\mathbb{Z}_6$ is cyclic, whereas $S_3$ is not.