  A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$
\Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method
for composition of permutation representations of a triangle group
$\Delta(p,q,r)$ 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, and in particular situations, where the
representations being composed are all equivalent copies of an alternating
group in the same degree, we can give a complete description of the structure
of the composition of the representations. This article contains the main
results of the author's PhD thesis.
