Conjugacy and the Class Equation · Hard problem

How many elements of $S_6$ have the same ORDER as $\sigma=(1\,2\,3)(4\,5\,6)$? (Its order is $\operatorname{lcm}(3,3)=3$ — find every cycle type in $S_6$ giving order $3$, not just $\sigma$'s own conjugacy class.)