Here is the problem as posed by Jerrum:
"The computational complexity of the following problem is investigated: Given a permutation group specified as a set of generators, and a single target permutation which is a member of the group, what is the shortest expression for the target permutation in terms of the generators?"
Jerrum, Mark R. "The complexity of finding minimum-length generator sequences." Theoretical Computer Science 36 (1985): 265-289.
What is the current state of the art algorithm for finding min-length generator sequences over small ( n<13 ) permutations?
The simple approach seems to be caching prefixes and growing them when a length $i+1$ word (composed into a permutation) is not contained in any of the length $1..i$ words. You either reach the target permutation, or at some length your words no longer compose into new permutations so you know the target is unreachable.