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

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