Automorphism groups of Riemann surfaces of genus p+1, where p is prime

We show that if $\mathcal{S}$ is a compact Riemann surface of genus $g=p+1$, where $p$ is prime, with a group of automorphisms $G$ such that $\mathrm{\mid }G\mathrm{\mid }\ge \lambda (g-1)$ for some real number $\lambda >6$, then for all sufficiently large $p$ (depending on $\lambda$), $\mathcal{S}$ and $G$ lie in one of six infinite sequences of examples. In particular, if $\lambda =8$ then this holds for all $p\ge 17$ and we obtain the largest groups of automorphisms of Riemann surfaces of genenera $g=p+1$.