Let $S$ be a finite set, and $a_1$, $a_2$, and $a_3$ be three distinct involutions of $S$ such that $a_1 a_2 a_3$ is a cyclic permutation of $S$.

I am researching on whether the group generated by $a_1, a_2, a_3$ is the symmetric group of $S$, will be grateful for any suggestions on approaching the problem or related references.