Suppose a cycle $\sigma\in S_n$ has decomposition a product of disjoint $p$-cycles, for $p$ a prime. So each cycle has order $p$, and thus $\sigma^p=id$, so $\sigma$ has order dividing $p$, and thus order $p$ when $\sigma\neq id$.
What if $\sigma$ has a decomposition into a product of disjoint $m$-cycles, for $m$ composite. I know $\sigma^m=id$, and thus has order dividing $m$. In every example I've tried, it seems $\sigma$ still has order $m$. Is it possible for $\sigma$ to have order smaller than $m$ (when not the identity of course)? If so, is there an example?