I will start with example. I have two permutations (of 5 elements) - (1 2) and (1 2)(3 5 4). When the first permutation changes something, the second do the same change (but it can do anything with fixed elements of the first).
What's the best way (terminology) to call or understand such relation between two permutations?
[EDIT] Here is how to check for such relation in GAP:
ForAll(Orbits(Group(p1)), i -> i in Orbits(Group(p2)));
If true, then p2 contains all cycles of p1.