On finite quotients of surface braid groups having order at most $127$

Let ${\mathrm{\Sigma }}_b$ be a compact Riemann surface of genus $b\ge 2$ and let ${\mathsf{P}}_2({\mathrm{\Sigma }}_b)={\pi }_1({\mathrm{\Sigma }}_b\times {\mathrm{\Sigma }}_b-\mathrm{\Delta })$ be the corresponding pure braid group on two strands. A finite quotient $\phi \text{\hspace{0.17em}⁣}:{\mathsf{P}}_2({\mathrm{\Sigma }}_b)\to G$ is called “admissible” if $\phi$ does not factor through ${\pi }_1({\mathrm{\Sigma }}_b\times {\mathrm{\Sigma }}_b)$. In this work we classify all admissible quotients of ${\mathsf{P}}_2({\mathrm{\Sigma }}_b)$ such that $\mathrm{\mid }G\mathrm{\mid }\le 127$.