A triangle group is denoted by and has finite presentation We examine a method for composition of permutation representations of a triangle group that was used in Everitt's proof of Higman's 1968 conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations. In particular situations, where the representations being composed are all equivalent copies of an alternating group in the same degree containing at least one handle, then has a quotient . We also prove that if G has a quotient containing at least two handles, then G has a quotient for many . This article contains the main results of the author's PhD thesis.