$\DeclareMathOperator\PSL{PSL}$I'm studying the Hurwitz group $(2, 3, 7; 9)$, with presentation: $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9 \rangle$. This group has $\PSL_2(8)$ as a quotient, so I decided to try the Reidemeister-Schreier process to find the presentation of the normal subgroup $H$ with $G/H=\PSL_2(8)$. I did this in three phases, using the permutation representation of $\PSL_2(8)$ to first find an index 9 subgroup $G_1$ of $G$, which led to a clear choice of an index 8 subgroup $G_2$ in $G_1$. Finally $H$ is an index 7 subgroup of $G_2$.

What I ended up with was a very strange presentation that I need help understanding:

$H = \langle h_1, h_2, h_3, h_4, h_5, h_6, h_7\rangle$, with relations:

$h_1h_2h_6h_7h_4h_5h_2h_3h_7h_1h_5h_6h_3h_4=1$

$h_1h_3h_4h_6=h_3h_4h_6h_1$

$h_1h_2h_4h_3=h_3h_4h_1h_2$, and

$h_3h_4h_6h_7 = h_1h_3$

Along with their cyclic permutations ($h_1\to h_2\to\dots\to h_7\to h_1$).

What is this group? The generators seem to very nearly commute, but I'm not sure they quite do. I'd like to understand exactly what this group is, if there are simpler defining relations for it, and exactly how $\PSL_2(8)$ acts on it in $(2, 3, 7; 9)$.

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