Regarding fundamental domain of 2 genus surface

Question

Let $\mathbb{H}^2$ be the hyperbolic plane with $(2,3,7)$ tiling. Let $\Gamma$ be a subgroup of $(2,3,7)$ triangle group such that $\mathbb{H}^2/\Gamma$ is the compact orientable surface of genus 2 with $(2,3,7)$ tiling on it.

Is there any way to see explicitly what will be a fundamental domain of $\Gamma$ inside the $(2,3,7)$ tiling of the plane?

The fundamental domain must have the following properties:

1. It will contain 168 $(2,3,7)$ triangle.
2. If we take the presentation of $\Gamma$ is $\langle g_1,g_2,g_3,g_4 \mid [g_1,g_2][g_3,g_4] \rangle$ then the fundamental domain will be an hyperbolic octagon.
3. Alternate edges of that octagon has same length.

Will it also be true that the sum of all internal angles of the octagon is $2\pi$?

I want to see a fundamental domain explicitly as part of the tiling.

Thanks in advance.