The groups generated by maximal sets of symmetries of Riemann surfaces and extremal quantities of their ovals

Given $g\geq 2$, there are formulas for the maximal number of non-conjugate symmetries of a Riemann surface of genus $g$ and the maximal number of ovals for a given number of symmetries. Here we describe the algebraic structure of the automorphism groups of Riemann surfaces, supporting such extremal configurations of symmetries, showing that they are direct products of a dihedral group and some number of cyclic groups of order $2$. This allows us to establish a deeper relation between the mentioned above quantitative (the number of symmetries) and qualitative (configurations of ovals) cases.