Let $G$ be a Fuchsian group with normalizer $N(G)$ inside $PSL(2,13)$

Due to the Hurwitz formula, it suffices to find a presentation of $G$ of the form: $$\langle x_1,\ldots,x_r,a_1,b_1,\ldots,a_\gamma,b_\gamma: x_1^{m_1} = \cdots = x_r^{m_r} = [a_1\;b_1]\cdots[a_\gamma\;b_\gamma]x_1\cdots x_r = 1\rangle$$

In that case, the genus (of the compactification of the riemann surface $G\backslash\mathcal{H}$) is given by $$2g - 2 = [N(G):G]\left(2\gamma - 2 + \sum_{i=1}^r\left(1-\frac{1}{m_i}\right)\right)$$

Thus, my question reduces to - Is there in general a nice way to find such a presentation of the group?

In particular, suppose $G$ is the Fuchsian group derived from the Hurwitz Quaternion Order $\mathcal{Q}_\text{Hur}$ (defined here: http://en.wikipedia.org/wiki/First_Hurwitz_triplet) - namely $G$ is the subgroup of $\mathcal{Q}_\text{Hur}$ consisting of elements of norm 1 that are also 1 mod $\mathfrak{p}\mathcal{Q}_\text{Hur}$, where $\mathfrak{p}$ is one of the three primes lying above 13 in the ring of integers of $\mathbb{Q}(2\cos(2\pi/7)))$.

Is there a nice way to find the genus(or equiv. a presentation in terms of elliptic and hyperbolic elements) of this group?