[hnhdzvzv] permutation groups

permutation groups are easy to describe, e.g., Rubik's cube.

can every permutation be reached?  that is, is the group isomorphic to symmetric group?  if not, what is the "factorization" of the group into simple finite groups?

find some sequence of operations to reach a given permutation.  find the shortest sequence.  what pair of permutations are separated by the longest shortest sequence (God's number, diameter of the graph, all pairs shortest path)?