Exceptional Spin groups on hyperelliptic Riemann surfaces

We find all exceptional spin groups attached to the vertices of any exceptional spin graph on any hyperbolic Riemann surface S of genus g>1. In particular, we show that when the order r of a graph is r>2 (i.e.the genus of S must be g>3) then the spin group attached to an exceptional point Q is either isomorphic to the symmetry group S(r) (when the degree of Q is equal to r) or to the symmetry group S(r+1)(when the degree of Q is equal to r+1).