A standard way to find solutions to a finite set of equations in a finite symmetric group ${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use the low index subgroups algorithm to find all subgroups of index $\leq n$ of that group, and to compute the permutations induced by the generators on the right cosets of these subgroups by multiplication from the right. The drawback of this approach is that the time taken by the low index subgroups algorithm grows in general faster than exponentially with $n$.

Now, finite permutation groups are objects which in many respects can be handled computationally very well, while finitely presented groups tend to be a lot more cumbersome when it comes to computation. So moving to finitely presented groups in order to solve a permutation group problem seems like a strange indirection.

**Question:** Is there an efficient genuine permutation group method for solving
a set of equations in a finite symmetric group, i.e. one which does *not* use finitely
presented groups and coset enumeration? -- Here "efficient" means "practical for degrees
at least in the hundreds", or "with at least similar range of applicability as stabilizer
chain methods".

giganticinput length -- or in other words, the brute force method which simply checks for every $k$-tuple of group elements whether it is a solution would still be polynomial time in the input length(!) -- For my purposes, I assume that the sum of the lengths of the equations which are to be solved in ${\rm S}_n$ isSMALLISHin comparison with $n!$. $\endgroup$ – Stefan Kohl Mar 1 '15 at 17:21