Consider the groups $\mathbb{S_4}$, $\mathbb{D_6 \times C_2}$ and $\mathbb{A_4 \times C_2}$. Are of any of these two isomorphic?

Question

Consider the groups $\mathbb{S_4}$, $\mathbb{D_6 \times C_2}$ and $\mathbb{A_4 \times C_2}$. Are of any of these two isomorphic?

I tried looking at the size of the groups to eliminate possibilities, however, they're all size 24. I would really appreciate any help on this question.

Answer 1

Certainly $S_4$ is not isomorphic to $A_4\times C_2$, since the first group has trivial center, the second however not. Also $D_6\times C_2$ has non-trivial center. Furthermore it is easy to see that $A_4$ is not isomorphic to $D_6$ by considering the element orders. In fact, $D_6$ has an element of order $6$, but $A_4$ doesn't. Hence all three groups are different, but all have order $24$.