16.46 (2006)

Among the finitely-presented groups that act arc-transitively on the (infinite) 3-valent tree with finite vertex-stabilizer are the two groups
$$G_3 = \langle h, a, P, Q \mid h^3, a^2, P^2, Q^2, [P, Q], [h, P], (hQ)^2, a^{-1}PaQ \rangle$$ and
$$G_4 = \langle h, a, p, q, r \mid h^3, a^2, p^2, q^2, r^2, [p, q], [p, r], p(qr)^2, h^{-1}phq, h^{-1}qhpq, (hr)^2, [a, p], a^{-1}qar \rangle,$$ each of which contains the modular group $G_1 = \langle h, a \mid h^3, a^2 \rangle \cong \text{PSL}_2(\mathbb{Z})$ as a subgroup of finite index. The free product of $G_3$ and $G_4$ with subgroup $G_1 = \langle h, a \rangle$ amalgamated has a normal subgroup $K$ of index 8 generated by $A = h$, $B = aha$, $C = p$, $D = PpP$, $E = QpQ$, and $F = PQpQP$, with dihedral complement $\langle a, P, Q \rangle$. The group $K$ has presentation $$\begin{align}
\langle A, B, C, D, E, F \mid &{} A^3, B^3, C^2, D^2, E^2, F^2, (AC)^3, (AD)^3, (AE)^3, (AF)^3, (BC)^3, (BD)^3, (BE)^3, (BF)^3,\\
& (ABA^{-1}C)^2, (ABA^{-1}D)^2,
(A^{-1}BAE)^2, (A^{-1}BAF)^2, (BAB^{-1}C)^2, (B^{-1}ABD)^2,
(BAB^{-1}E)^2, (B^{-1}ABF)^2 \rangle.
\end{align}$$ Does this group have a non-trivial finite quotient?