I have just shown that $\rho$, where $\rho$ is a product of disjoint cycles of lengths $m\_i$ in $S\_n$, has order $\operatorname{lcm}(m\_1,m\_2,\ldots,m\_k)$.

Now I have to show that there are no elements of order $6$ in $S\_4$. I know that $S\_4$ has order $24$, but I can't really get anywhere from this. Could someone give me a hint on how to do this. Thanks.