We give a fast polynomial-time algorithm for computing a composition series in a primitive permutation group given by a list of generators. Permutation representations for the compilation factors are also obtained.

The algorithm will be a key procedure in an O(sm³ log^c n) algorithm that will the solve the same problems for an arbitrary subgroup of S_n given by s generators. The general algorithm will be described in a forthcoming paper.

The first polynomial-time algorithm for this problem was given by Luks. Our procedure follows the overall architecture of that original algorithm, while replacing the subroutines by much faster ones. New combinatorial estimates of suborbit sizes in primitive gtoupa guarantee the improved performance.