On permutation representations of polyhedral groups

We answer affirmatively the following question of Derek Holt: Given integers $l,m,n \geq 2$, can one, in a simple manner, find a finite set $\Omega$ and permutations $a,b$ such that $a$ has order $l$, $b$ has order $m$ and $ab$ has order $n$? The method of proof enables us to prove more general results (Theorems 2 and 3).