How to find order of permutation for this example?

Question

The question asks to compute the orders of the permutations. I think this is a non-disjoint cycle, so I'm not sure exactly how to find the order. Can someone please show me the steps on how to do this or an explanation? Do I just find the product? Thank you. (2 1 4 6 3)(1 2)(3 4 5)

Answer 1

You could write it as a product of disjoint cycles: $(21463)(12)(345)=(245)(36)$. Thus the order is $6$, the lcm of the orders of the disjoint cycles.

Answer 2

The original cycles are not written in disjoint cycles. You should convert it into the disjoint cycles. Once you get the disjoint cycles, the order is the least common multiple of each order of cycle.