Riemann surfaces of genus 1 + q2 with 3q2 automorphisms

Groups of automorphisms of Riemann surfaces and maps of genus $p+1$ where $p$ is prime

“…Connections of these results with maps and hypermaps are discussed in Section 11 where, as a corollary to Theorem 1, we obtain the following classification, where the numbering of cases follows and refers to that in Theorem 1(a): Theorem 3. The orientably regular maps or hypermaps of genus g = p + 1 for some prime p ≥ 7, with orientation-preserving automorphism group G of order divisible by p, are as follows (up to duality or triality, permuting the roles of vertices, edges and faces): with G PGL 2 (7), one on each surface;…”

“…(xii) for p = 7 the two surfaces S in Theorem 1(a)(xi) support two regular hypermaps of type (3,3,4) with G PS L 2 (7), one on each surface.…”

“…(ix) ρ = 3, σ = (3 [4] ) and G G p,3 where p ≡ 1 mod (3); (x) ρ = 84, σ = (2, 3, 7) and G PS L 2 (13) where p = 13; (xi) ρ = 48, σ = (2, 3, 8) and G PGL 2 (7) where p = 7;…”

“…(c) The surfaces S in case (iv) and a chiral pair of those in case (v) are the surfaces S 1 , S 1 in (i) with automorphism group A := Aut S G p,6 × C 2 ; when p = 7 two of those in case (v) have A PGL 2 (7) in (xi), and when p = 13 three of those in case (v) have A PS L 2 (13) in (x). The surfaces in case (vi) have A G p,10 in (ii).…”

“…The surfaces in case (ix) are the infinite family in case (v), with G p,3 acting as a subgroup of index 2 in G p,3 × C 2 , and with automorphism groups as described here for case (v). The surfaces in case (xii) have A PGL 2 (7)…”