 Being an automorphism, the map must preserve the order of elements, but it does not preserve cycle structure.For instance, a 2-cycle maps to a product of three 2-cycles; it is easy to see that a 2-cycle affects all of the 6 graph factorizations in some way, and hence has no fixed points when viewed as a permutation of factorizations.The fact that it is possible to construct this automorphism at all relies on a large number of numerical coincidences which apply only to .