g~: 29 r: 4 b = 0 Example # 1 -- C~ -- G~ Id: SmallGroup <72, 8> G~ name: C9:D4 GrpPC : G of order 72 = 2^3 * 3^2 PC-Relations: G.1^2 = Id(G), G.2^2 = Id(G), G.3^2 = Id(G), G.4^3 = G.5^2, G.5^3 = Id(G), G.2^G.1 = G.2 * G.3, G.4^G.1 = G.4^2 * G.5, G.5^G.1 = G.5^2 generating vector: [ G.1 * G.4 * G.5^2, G.1 * G.5^2, G.2 * G.3 * G.5^2, G.2 * G.3 * G.4 * G.5 ] signature: [ 2, 2, 6, 18 ] genus: 29 decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 9 sigma: G.3 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <36, 4> G name: D18 GrpPC : H of order 36 = 2^2 * 3^2 PC-Relations: H.1^2 = Id(H), H.2^2 = Id(H), H.3^3 = H.4^2, H.4^3 = Id(H), H.3^H.1 = H.3^2 * H.4, H.4^H.1 = H.4^2 generating vector: [ H.1 * H.3 * H.4^2, H.1 * H.4^2, H.2 * H.4^2, H.2 * H.3 * H.4 ] signature: [ 2, 2, 6, 18 ] genus: 15 decomp H^0(K_C): [ 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 8 Example # 2 -- C~ -- G~ Id: SmallGroup <80, 14> G~ name: D10:C4 GrpPC : G of order 80 = 2^4 * 5 PC-Relations: G.1^2 = Id(G), G.2^2 = G.3, G.3^2 = Id(G), G.4^2 = Id(G), G.5^5 = Id(G), G.2^G.1 = G.2 * G.4, G.5^G.1 = G.5^4 generating vector: [ G.1 * G.5, G.1 * G.3 * G.4 * G.5^3, G.2 * G.3 * G.4, G.2 * G.3 * G.5^3 ] signature: [ 2, 2, 4, 20 ] genus: 29 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 2, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 9 sigma: G.3 * G.4 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <40, 6> G name: D20 GrpPC : H of order 40 = 2^3 * 5 PC-Relations: H.1^2 = Id(H), H.2^2 = H.3, H.3^2 = Id(H), H.4^5 = Id(H), H.2^H.1 = H.2 * H.3, H.4^H.1 = H.4^4 generating vector: [ H.1 * H.4, H.1 * H.4^3, H.2, H.2 * H.3 * H.4^3 ] signature: [ 2, 2, 4, 20 ] genus: 15 decomp H^0(K_C): [ 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 8 Example # 3 -- C~ -- G~ Id: SmallGroup <96, 137> G~ name: (C2*C6):D4 GrpPC : G of order 96 = 2^5 * 3 PC-Relations: G.1^2 = Id(G), G.2^2 = G.5, G.3^2 = Id(G), G.4^2 = Id(G), G.5^2 = Id(G), G.6^3 = Id(G), G.2^G.1 = G.2 * G.5, G.3^G.1 = G.3 * G.4, G.6^G.1 = G.6^2 generating vector: [ G.1 * G.2 * G.4 * G.5 * G.6, G.1, G.2 * G.3 * G.4, G.3 * G.5 * G.6 ] signature: [ 2, 2, 4, 6 ] genus: 29 decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 0 ] N~ = dim S^2H^0(K_C~)^G~ = 9 sigma: G.4 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <48, 36> G name: C2*D12 GrpPC : H of order 48 = 2^4 * 3 PC-Relations: H.1^2 = Id(H), H.2^2 = H.4, H.3^2 = Id(H), H.4^2 = Id(H), H.5^3 = Id(H), H.2^H.1 = H.2 * H.4, H.5^H.1 = H.5^2 generating vector: [ H.1 * H.2 * H.4 * H.5, H.1, H.2 * H.3, H.3 * H.4 * H.5 ] signature: [ 2, 2, 4, 6 ] genus: 15 decomp H^0(K_C): [ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 8 Example # 4 -- C~ -- G~ Id: SmallGroup <96, 145> G~ name: D6:D4 GrpPC : G of order 96 = 2^5 * 3 PC-Relations: G.1^2 = Id(G), G.2^2 = G.5, G.3^2 = Id(G), G.4^2 = Id(G), G.5^2 = Id(G), G.6^3 = Id(G), G.3^G.1 = G.3 * G.4, G.3^G.2 = G.3 * G.5, G.6^G.1 = G.6^2 generating vector: [ G.3 * G.4 * G.5, G.1 * G.4 * G.5 * G.6^2, G.1 * G.2, G.2 * G.3 * G.5 * G.6^2 ] signature: [ 2, 2, 4, 6 ] genus: 29 decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 2, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 8 sigma: G.4 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <48, 38> G name: S3*D4 GrpPC : H of order 48 = 2^4 * 3 PC-Relations: H.1^2 = Id(H), H.2^2 = H.4, H.3^2 = Id(H), H.4^2 = Id(H), H.5^3 = Id(H), H.3^H.2 = H.3 * H.4, H.5^H.1 = H.5^2 generating vector: [ H.3 * H.4, H.1 * H.4 * H.5^2, H.1 * H.2, H.2 * H.3 * H.4 * H.5^2 ] signature: [ 2, 2, 4, 6 ] genus: 15 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 2 ] N = dim S^2H^0(K_C)^G = 7 4 b = 8 Example # 1 -- C~ -- G~ Id: SmallGroup <128, 925> G~ name: D8.C2^2.C2 GrpPC : G of order 128 = 2^7 PC-Relations: G.3^2 = G.7, G.4^2 = G.6 * G.7, G.5^2 = G.7, G.6^2 = G.7, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.5, G.4^G.1 = G.4 * G.6, G.4^G.2 = G.4 * G.6, G.5^G.1 = G.5 * G.7, G.5^G.3 = G.5 * G.7, G.6^G.1 = G.6 * G.7, G.6^G.2 = G.6 * G.7 generating vector: [ G.1 * G.5 * G.6, G.2 * G.4 * G.6, G.3 * G.6, G.1 * G.2 * G.3 * G.6 * G.7 ] signature: [ 2, 2, 2, 16 ] genus: 29 decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 7 sigma: G.7 branch points: 8 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <64, 128> G name: (C2*D8):C2 GrpPC : H of order 64 = 2^6 PC-Relations: H.4^2 = H.6, H.2^H.1 = H.2 * H.4, H.3^H.1 = H.3 * H.5, H.4^H.1 = H.4 * H.6, H.4^H.2 = H.4 * H.6 generating vector: [ H.1 * H.5 * H.6, H.2 * H.4 * H.6, H.3 * H.6, H.1 * H.2 * H.3 * H.6 ] signature: [ 2, 2, 2, 8 ] genus: 13 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ] N = dim S^2H^0(K_C)^G = 6 1