Quotient group of $D(2,3,7)$ under the Klein Quartic The Klein Quartic is a quotient space of the hyperbolic plane. Let $k$ be its quotient map. Given two isometries $f,g$ of the hyperbolic plane, we say that $f \cong g$ iff $k \circ f = k \circ g$.
The Von Dyck group $D(2,3,7)$ is a group of isometries of the hyperbolic plane. It can be presented by $\langle r,m|r^7 = m^2 = (rm)^3 = 1 \rangle$.
My question is, what is $D(2,3,7)/\cong$ (both the group, and the quotient map)?
Note: The Klein Quartic can be tiled by $(2,3,7)$ triangles, so the group elements will correspond to those (half of those) triangles (and in particular will be a finite group).

The group is isomorphic to $GL(3,2)$. The quotient map $k$ is generated by
$$k(r) = \begin{bmatrix}0&1&1\\0&0&1\\1&0&0\end{bmatrix}$$
$$k(m) = \begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}$$
These matrices where found by a computer program looking for matrices that satisfied the Von dyck group relations and that generated $GL(3,2)$. There are many other options.
