Disjoint direct product decompositions of permutation groups

Let H≤Sn be an intransitive group with orbits Ω1,Ω2,…,Ωk. Then certainly H is a subdirect product of the direct product of its projections onto each orbit, H|Ω1×H|Ω2×…×H|Ωk. Here we provide a polynomial time algorithm for computing the finest partition P of the H-orbits such that H=∏c∈PH|c and we demonstrate its usefulness in some applications.