To find order of permutation

Question

Let $\sigma$ be the permutation given by

Is their a short way to do this.Thanks

Answer 1

Hint: Every permutation can be written as a product of disjoint cycles. The order of the permutation is the least common multiple of the order of each cycle.

Answer 2

We have a cycle of length 4, a cycle of length 3, and a cycle of length 2. Raising to a power that is a multiple of 4 makes the length 4 cycle go away, as well as the length 2. Raising to a multiple of 3 makes the 3-cycle go away. But we want them all to go away. What can we do?