Let $G$ be the modular group. We know this can be described by the relations (in terms of the $S$ and $T$ transformations) given by $S^4 = I, (ST)^3 = S^2$.

In my work matrix representations of $G$ have arisen where $T^N = I$ for some $N \geq 2$. I am interested to know whether the modular group $G$ with an additional relation $T^N = I$ gives a finite group? And regardless of this, what is known about $G$ with the additional relation $T^N = I$?