Linear fractional group as Galois group

Abstract: We compute all signatures of $PSL_2(\mathbb{F}7)$, and $PSL_2(\mathbb{F}{11})$ which classify all orientation preserving actions of the groups $PSL_2(\mathbb{F}7)$, and $PSL_2(\mathbb{F}{11})$ on compact, connected, orientable surfaces with orbifold genus $\geq 0$. This classification is well-grounded in the other branches of Mathematics like topology, smooth, and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.