I understand that you have to write out all the disjoint cycles and then take the least common multiple which yields the highest order.
But my question is, do I have to write all elements of $S_5$, write them as disjoint cycles, and then find the largest least common multiple, or is there a shortcut?
$S_5$ has $5!$ elements and I would not like writing all of these permutations out...
I have read the other answers on here but I have not seen anything to help me with this question.
For example, here (https://math.stackexchange.com/a/231893/133156) the answer lists six disjoint cycles of $S_5$, how did he get there without writing them all out?