$(p,q,r)$-GENERATIONS OF THE SPORADIC GROUP $HN$

A finite group G is called $(l,m,n)$-generated, if it is a quotient group of the triangle group T(l,m,n) = x,y,z | xˡ = yᵐ = zⁿ = xyz = 1. In [16], the question of finding all triples $(p,q,r)$ such that non-abelian finite simple group G is $(p,q,r)-$generated was posed. In this paper we partially answer this question for the sporadic group $HN$. In fact, we prove that the sporadic group $HN$ is $(p,q,r)-$generated if and only if (p,q,r) ≠ (2,3,5), where $p, q$ and r are prime divisors of $|HN|$ and p < q < r.