Is the symmetric group $S_4$ cyclic

Question

Is the symmetric group $S_4$ cyclic?

By writing all $24$ elements we can write the tabular form of $S_4$. Then choosing each element of $S_4$, we can find its order and thus, we can show that that there is no element of $S_4$ of order 24. Then $S_4$ will be non-cyclic.

But this is a laborious work as $S_4$ has $24$ elements. Is there any other way to show this?

Answer 1

Another approach, if $S^4$ is abelian, then every subgroup is abelian, but $S^3\leq S^4$ and $S^3$ is the very first and unique (up to isomorphism) non-abelian group. $S^3$ in $S^4$ is $\langle(1234),(12)\rangle$.