See this paper:
http://www.dartmouth.edu/~rah/games-with-algorithms.pdf
Page 22 contains a survey of the problem you're interested in. Apparently, there is a polynomial time algorithm to check if a solution exists (it will also return some implicitly represented solution), but finding the solution with the fewest number of moves is, not surprisingly, NP-hardPSPACE-complete.
The original reference is:
Mark Jerrum: The Complexity of Finding Minimum-Length Generator Sequences. Theor. Comput. Sci. 36: 265-289 (1985)
