ConjugacyClassRepresentative from Permutation
The canonical permutation with the same cycle type.
FindStat does not fix an ordering for this map. Convention used here: cycles in decreasing length, filled with consecutive integers, each cycle (a a+1 … a+len-1) written as the one-line word a+1, …, a+len-1, a.
Called as CombinatorialMap(x, "ConjugacyClassRepresentative"). The head's page has its examples.