Cyclic permutation groups

Iterating cycle structures:
We can begin by describing the effects of iteration on a cycle structure:

({1 6} {6 1} {3 2} {2 3} {3 2} {6 1})

Using this we can effectively determine the cycle structure of cyclic subgroups and cyclic supergroups of any given cyclic permutation group.

Cyclic permutation groups:
Given a permutation we can select a canonical generator for the permutation group generated by that permutation by selecting a permutation amongst the available iterates using the lexicographic ordering of permutations.