The set of all permutations of an n-element set is denoted by Sn.
This is a "group"
    (with respect to the composition of maps).
For n > 2, all the groups Sn are non-commutative (thus quite complicated).
Note: Every finite group is subgroup of some group Sn, thus we see:
    the groups Sn are in some sense the most complicated ones.

But we have considered only individual permutations (and their powers).
The powers of a fixed permutation form a commutative subgroup of Sn.
    (This means: The non-commutativity of the group Sn did not play a role.)

What was essential? The possible cycle lengths!

The order 8 of the doubling transformation d8 (Photo booth).

Fixed points (thus cycles of length 1) (Secret Santa, Treize).

Cycles of large length (100 prisoners).

To work with permutations provides many further surprises!
(For example, those of the Catalan combinatorics.)