If permutation p=(148)(25)(396)(7) how to find p^123?

Question

If permutation p=(148)(25)(396)(7) how to find p^123 ?

Answer 1

Hint:

Your permutation is very conveniently written in disjoint cyclic form. A very nice property of permutations written as disjoint cycles is that raising the permutation to a power as a whole is equivalent to raising each cycle to that power individually (seen easily by the fact that disjoint cycles commute with one another and induction).

$p^{123} = (148)^{123}(25)^{123}(396)^{123}(7)^{123}$. Now, consider simplifying each of these cycles individually.