During reading a book, I have faced to this problem telling:
$G$ is a group of order $12$ such that $Z(G)$ has no element of order $2$ . Then $Gā A_4$.
Obviously, this group is not abelian and I think some information about $S_4$ is involved here because of the desired deduction. Can we say $|\frac{G}{Z(G)}|ā 3$? And if so, is it useful for the problem?