$S_3\oplus S_3$ has an element of order $4,6,9$ or $18$. $\oplus$ is the direct sum.
Its order is $36$ so by sylow theorem it has subgroups of order $4$ and $9$. Also order of a permutation is the lcm of the order of disjoint cycles. But when I consider elements of $S_3\oplus S_3$ the cycles might not be disjoint eg. $(12)(123)$. It says the answer is 6 but I'm not convinced because of this reason.
That $\oplus$ symbol? It's the direct sum; anything from one piece commutes with anything from the other. We can represent the elements as ordered pairs $(a,b)$, with group operation $(a,b)\cdot (c,d) = (a\cdot c, b\cdot d)$. It doesn't matter if the cycles aren't disjoint between the two objects - we're not going to multiply across that line anyway.