If a group $G$ is $(2,3,t)$-generated, where $t$ is an odd divisor of $|G|$ then $G$ is a factor of the modular group which is isomorphic to $\mathbb{Z}_2 * \mathbb{Z}_3$.
From assumptions, let $x$ and $y$ be the only two generators of $G$. Then $|x|=2$, $|y|=3$, $|xy|=t$. I believe the definition of "factor" here is the same as that of a subgroup.
But can you explain intuitively why $G$ is a factor of $PSL(2, \mathbb{Z})?$