I'm reading this paper On the Geometry of Hurwitz Surfaces. They define $T^{\pm}$ as the $(2,3,7)$ triangle group of $\mathbb{H}^2$ isometries generated by reflections and $T$ as the subgroup of $T^{\pm}$ of rotations.
At the top of page 12, it is stated
"Clearly the only possibilities for such points (centre-points of rotations) are the face-centres, midpoints and vertices of triangles."
I can picture why this would be the case, but how would you prove that if the centre point is not one of these points, then the map does not preserve the tiling?
Thank you.