The image of integer 4 under Cycles[{{2,3,4,6}}] is integer 6:

Under the identity, permutation integers are not moved:

An action of a permutation on another permutation is understood as conjugation:

Images under all elements of a group:

The image of a point in the support of the permutation is the right neighbor of the point:

The image of the last point of a cycle is the first point of that cycle:

A point not present in the permutation support stays invariant:

PermutationReplace on arrays of integers returns the list of respective images:

PermutationReplace on other permutations is understood as conjugation:

On a permutation group, the generators are conjugated:

The second argument is listable:

If both arguments are lists then the second argument is threaded first:

Images under all elements of a group:

PermutationReplace is a right action with respect to PermutationProduct:

PermutationReplace on an identity permutation list coincides with PermutationList:

PermutationReplace on an identity permutation list produces the inverse result of Permute:

The orbit of a point under a permutation group is the union of the images of that point under the elements of the group:

Graphical representation of the elements of groups by sorted lists of images: