The Coxeter quotient of the fundamental group of a Galois cover of T \times T

Let $X$ be the surface $\text{T}\times \text{T}$ where $\text{T}$ is the complex torus. This paper is the third in a series, studying the fundamental group of the Galois cover of $X$ \wrt a generic projection onto $\text{C}{\mathrm{\P }}^2$. Van Kampen Theorem gives a presentation of the fundamental group of the complement of the branch curve, with 54 generators and more than 2000 relations. Here we introduce a certain natural quotient (obtained by identifying pairs of generators), prove it is a quotient of a Coxeter group related to the degeneration of $X$, and show that this quotient is virtually nilpotent.