[Math] Which of the following group is isomorphic to the group $S_3\times \mathbb Z_2$

Question

The group $S_3\times \mathbb Z_2$ is isomorphic to one of the following groups:

$\mathbb Z_{12}$, $\mathbb Z_6\times \mathbb Z_2$, $A_4$, $D_6$…?

I know that $\mathbb Z_{12}$, $\mathbb Z_6\times \mathbb Z_2$ will not be the answer. So, which one between $A_4$, $D_6$…?

Answer 1

$$G=S_3\times\mathbb{Z_2}$$ is not abelian and not cyclic hence it's not isomorphic to $\mathbb{Z_{12}}$ and $\mathbb{Z_6}\times\mathbb{Z_2}$. Also the group $A_4$ doesn't have an element of order $6$ but $G$ has, say $\{(1,2,3),1\}$, so correct option is $D_6$