g~: 13 r: 4 b = 0 Example # 1 -- C~ -- G~ Id: SmallGroup <16, 5> G~ name: C2*C8 GrpPC : G of order 16 = 2^4 PC-Relations: G.1^2 = G.3, G.3^2 = G.4 generating vector: [ G.1, G.1 * G.2 * G.4, G.1 * G.2 * G.3 * G.4, G.1 * G.3 ] signature: [ 8, 8, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 2, 1, 1, 1, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 5 sigma: G.2 branch points: 0 verify (B1): true verify (B2): false -- C -- G Id: SmallGroup <8, 1> G name: C8 GrpPC : H of order 8 = 2^3 PC-Relations: H.1^2 = H.2, H.2^2 = H.3 generating vector: [ H.1, H.1 * H.3, H.1 * H.2 * H.3, H.1 * H.2 ] signature: [ 8, 8, 8, 8 ] genus: 7 decomp H^0(K_C): [ 0, 1, 1, 1, 1, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 4 Example # 2 -- C~ -- G~ Id: SmallGroup <20, 5> G~ name: C2*C10 GrpPC : G of order 20 = 2^2 * 5 PC-Relations: G.1^2 = Id(G), G.2^2 = Id(G), G.3^5 = Id(G) generating vector: [ G.1, G.1 * G.3^2, G.2 * G.3^4, G.2 * G.3^4 ] signature: [ 2, 10, 10, 10 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 2, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 0, 0, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 3 sigma: G.1 * G.2 branch points: 0 verify (B1): true verify (B2): false -- C -- G Id: SmallGroup <10, 2> G name: C10 GrpPC : H of order 10 = 2 * 5 PC-Relations: H.1^2 = Id(H), H.2^5 = Id(H) generating vector: [ H.1, H.1 * H.2^2, H.1 * H.2^4, H.1 * H.2^4 ] signature: [ 2, 10, 10, 10 ] genus: 7 decomp H^0(K_C): [ 0, 1, 1, 1, 1, 0, 0, 2, 0, 1 ] N = dim S^2H^0(K_C)^G = 2 Example # 3 -- C~ -- G~ Id: SmallGroup <24, 9> G~ name: C2*C12 GrpPC : G of order 24 = 2^3 * 3 PC-Relations: G.1^2 = G.4, G.2^2 = Id(G), G.3^3 = Id(G), G.4^2 = Id(G) generating vector: [ G.2, G.1 * G.2, G.3 * G.4, G.1 * G.3^2 ] signature: [ 2, 4, 6, 12 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0 ] N~ = dim S^2H^0(K_C~)^G~ = 3 sigma: G.2 * G.4 branch points: 0 verify (B1): true verify (B2): true -- C -- G Id: SmallGroup <12, 2> G name: C12 GrpPC : H of order 12 = 2^2 * 3 PC-Relations: H.1^2 = H.3, H.2^3 = Id(H), H.3^2 = Id(H) generating vector: [ H.3, H.1 * H.3, H.2 * H.3, H.1 * H.2^2 ] signature: [ 2, 4, 6, 12 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 2 Example # 4 -- C~ -- G~ Id: SmallGroup <24, 7> G~ name: C2*C3:C4 GrpPC : G of order 24 = 2^3 * 3 PC-Relations: G.1^2 = G.3, G.2^2 = Id(G), G.3^2 = Id(G), G.4^3 = Id(G), G.4^G.1 = G.4^2 generating vector: [ G.1 * G.3, G.1 * G.2 * G.4, G.1 * G.2 * G.4, G.1 * G.3 ] signature: [ 4, 4, 4, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 0 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.2 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <12, 1> G name: C3:C4 GrpPC : H of order 12 = 2^2 * 3 PC-Relations: H.1^2 = H.2, H.2^2 = Id(H), H.3^3 = Id(H), H.3^H.1 = H.3^2 generating vector: [ H.1 * H.2, H.1 * H.3, H.1 * H.3, H.1 * H.2 ] signature: [ 4, 4, 4, 4 ] genus: 7 decomp H^0(K_C): [ 0, 1, 1, 1, 2, 0 ] N = dim S^2H^0(K_C)^G = 3 Example # 5 -- C~ -- G~ Id: SmallGroup <32, 5> G~ name: C2^2:C8 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.4^2 = G.5, G.2^G.1 = G.2 * G.3 generating vector: [ G.2, G.2 * G.5, G.1 * G.2 * G.5, G.1 * G.2 * G.3 * G.4 * G.5 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.3 * G.5 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 6> G name: OD16 GrpPC : H of order 16 = 2^4 PC-Relations: H.1^2 = H.3, H.3^2 = H.4, H.2^H.1 = H.2 * H.4 generating vector: [ H.2, H.2 * H.4, H.1 * H.2 * H.4, H.1 * H.2 * H.3 ] signature: [ 2, 2, 8, 8 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 1, 0, 1, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 6 -- C~ -- G~ Id: SmallGroup <32, 9> G~ name: D4:C4 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.3^2 = G.5, G.2^G.1 = G.2 * G.3, G.3^G.1 = G.3 * G.5, G.3^G.2 = G.3 * G.5 generating vector: [ G.2 * G.4, G.2 * G.3 * G.4 * G.5, G.1 * G.2 * G.4, G.1 * G.2 * G.5 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 5 sigma: G.4 branch points: 0 verify (B1): true verify (B2): true -- C -- G Id: SmallGroup <16, 7> G name: D8 GrpPC : H of order 16 = 2^4 PC-Relations: H.3^2 = H.4, H.2^H.1 = H.2 * H.3, H.3^H.1 = H.3 * H.4, H.3^H.2 = H.3 * H.4 generating vector: [ H.2, H.2 * H.3 * H.4, H.1 * H.2, H.1 * H.2 * H.4 ] signature: [ 2, 2, 8, 8 ] genus: 7 decomp H^0(K_C): [ 0, 0, 1, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 4 Example # 7 -- C~ -- G~ Id: SmallGroup <32, 9> G~ name: D4:C4 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.3^2 = G.5, G.2^G.1 = G.2 * G.3, G.3^G.1 = G.3 * G.5, G.3^G.2 = G.3 * G.5 generating vector: [ G.2 * G.3 * G.5, G.2 * G.4 * G.5, G.1 * G.2 * G.4, G.1 * G.2 * G.4 * G.5 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 5 sigma: G.4 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 7> G name: D8 GrpPC : H of order 16 = 2^4 PC-Relations: H.3^2 = H.4, H.2^H.1 = H.2 * H.3, H.3^H.1 = H.3 * H.4, H.3^H.2 = H.3 * H.4 generating vector: [ H.2 * H.3 * H.4, H.2 * H.4, H.1 * H.2, H.1 * H.2 * H.4 ] signature: [ 2, 2, 8, 8 ] genus: 7 decomp H^0(K_C): [ 0, 0, 1, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 4 Example # 8 -- C~ -- G~ Id: SmallGroup <32, 37> G~ name: C2*OD16 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.4^2 = G.5, G.2^G.1 = G.2 * G.5 generating vector: [ G.2, G.2 * G.3 * G.5, G.1 * G.2 * G.3, G.1 * G.2 * G.4 * G.5 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.3 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 6> G name: OD16 GrpPC : H of order 16 = 2^4 PC-Relations: H.1^2 = H.3, H.3^2 = H.4, H.2^H.1 = H.2 * H.4 generating vector: [ H.2, H.2 * H.4, H.1 * H.2, H.1 * H.2 * H.3 * H.4 ] signature: [ 2, 2, 8, 8 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 1, 0, 1, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 9 -- C~ -- G~ Id: SmallGroup <32, 37> G~ name: C2*OD16 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.4^2 = G.5, G.2^G.1 = G.2 * G.5 generating vector: [ G.2, G.2 * G.3 * G.5, G.1 * G.2 * G.3, G.1 * G.2 * G.4 * G.5 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.3 * G.5 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 6> G name: OD16 GrpPC : H of order 16 = 2^4 PC-Relations: H.1^2 = H.3, H.3^2 = H.4, H.2^H.1 = H.2 * H.4 generating vector: [ H.2, H.2, H.1 * H.2 * H.4, H.1 * H.2 * H.3 * H.4 ] signature: [ 2, 2, 8, 8 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 1, 0, 1, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 10 -- C~ -- G~ Id: SmallGroup <32, 40> G~ name: C2*SD16 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.5, G.4^2 = G.5, G.2^G.1 = G.2 * G.4, G.4^G.1 = G.4 * G.5, G.4^G.2 = G.4 * G.5 generating vector: [ G.2 * G.3 * G.5, G.2, G.1 * G.2 * G.3 * G.4, G.1 * G.2 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.3 * G.5 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 8> G name: SD16 GrpPC : H of order 16 = 2^4 PC-Relations: H.1^2 = H.4, H.3^2 = H.4, H.2^H.1 = H.2 * H.3, H.3^H.1 = H.3 * H.4, H.3^H.2 = H.3 * H.4 generating vector: [ H.2, H.2, H.1 * H.2 * H.3 * H.4, H.1 * H.2 ] signature: [ 2, 2, 8, 8 ] genus: 7 decomp H^0(K_C): [ 0, 0, 1, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 11 -- C~ -- G~ Id: SmallGroup <32, 24> G~ name: (C2*C4):C4 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.5, G.3^2 = G.4, G.2^G.1 = G.2 * G.4 generating vector: [ G.2 * G.5, G.1 * G.2 * G.4, G.3 * G.4 * G.5, G.1 * G.3 * G.5 ] signature: [ 2, 4, 4, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.5 branch points: 0 verify (B1): true verify (B2): true -- C -- G Id: SmallGroup <16, 13> G name: D4:C2 GrpPC : H of order 16 = 2^4 PC-Relations: H.3^2 = H.4, H.2^H.1 = H.2 * H.4 generating vector: [ H.2, H.1 * H.2 * H.4, H.3 * H.4, H.1 * H.3 ] signature: [ 2, 4, 4, 4 ] genus: 7 decomp H^0(K_C): [ 0, 0, 0, 1, 0, 0, 0, 0, 2, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 12 -- C~ -- G~ Id: SmallGroup <32, 29> G~ name: C2^2:Q8 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.2^2 = G.4, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.5 generating vector: [ G.3 * G.4 * G.5, G.2 * G.3, G.1 * G.2 * G.4 * G.5, G.1 * G.4 ] signature: [ 2, 4, 4, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.4 * G.5 branch points: 0 verify (B1): false verify (B2): true -- C -- G Id: SmallGroup <16, 13> G name: D4:C2 GrpPC : H of order 16 = 2^4 PC-Relations: H.1^2 = H.4, H.2^2 = H.4, H.2^H.1 = H.2 * H.4, H.3^H.1 = H.3 * H.4 generating vector: [ H.3, H.2 * H.3, H.1 * H.2, H.1 * H.4 ] signature: [ 2, 4, 4, 4 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 0, 0, 2, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 13 -- C~ -- G~ Id: SmallGroup <32, 33> G~ name: C4^2:C2 GrpPC : G of order 32 = 2^5 PC-Relations: G.2^2 = G.4 * G.5, G.3^2 = G.4, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.5 generating vector: [ G.1, G.3 * G.5, G.1 * G.2 * G.4, G.2 * G.3 * G.4 * G.5 ] signature: [ 2, 4, 4, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.4 * G.5 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 13> G name: D4:C2 GrpPC : H of order 16 = 2^4 PC-Relations: H.3^2 = H.4, H.2^H.1 = H.2 * H.4, H.3^H.1 = H.3 * H.4 generating vector: [ H.1, H.3 * H.4, H.1 * H.2 * H.4, H.2 * H.3 ] signature: [ 2, 4, 4, 4 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 0, 0, 2, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 14 -- C~ -- G~ Id: SmallGroup <36, 12> G~ name: C6*S3 GrpPC : G of order 36 = 2^2 * 3^2 PC-Relations: G.1^2 = Id(G), G.2^2 = Id(G), G.3^3 = Id(G), G.4^3 = Id(G), G.4^G.1 = G.4^2 generating vector: [ G.1 * G.4^2, G.1 * G.2 * G.4, G.1 * G.2 * G.3, G.1 * G.3^2 * G.4 ] signature: [ 2, 2, 6, 6 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.2 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <18, 3> G name: C3*S3 GrpPC : H of order 18 = 2 * 3^2 PC-Relations: H.1^2 = Id(H), H.2^3 = Id(H), H.3^3 = Id(H), H.3^H.1 = H.3^2 generating vector: [ H.1 * H.3^2, H.1 * H.3, H.1 * H.2, H.1 * H.2^2 * H.3 ] signature: [ 2, 2, 6, 6 ] genus: 7 decomp H^0(K_C): [ 0, 1, 1, 1, 0, 0, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 15 -- C~ -- G~ Id: SmallGroup <48, 14> G~ name: D6:C4 GrpPC : G of order 48 = 2^4 * 3 PC-Relations: G.1^2 = Id(G), G.2^2 = G.3, G.3^2 = Id(G), G.4^2 = Id(G), G.5^3 = Id(G), G.2^G.1 = G.2 * G.4, G.5^G.1 = G.5^2 generating vector: [ G.1 * G.3 * G.5, G.1 * G.4 * G.5, G.1 * G.2 * G.4 * G.5^2, G.1 * G.2 * G.4 * G.5^2 ] signature: [ 2, 2, 4, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.3 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <24, 8> G name: C3:D4 GrpPC : H of order 24 = 2^3 * 3 PC-Relations: H.1^2 = Id(H), H.2^2 = Id(H), H.3^2 = Id(H), H.4^3 = Id(H), H.2^H.1 = H.2 * H.3, H.4^H.1 = H.4^2 generating vector: [ H.1 * H.4, H.1 * H.3 * H.4, H.1 * H.2 * H.3 * H.4^2, H.1 * H.2 * H.3 * H.4^2 ] signature: [ 2, 2, 4, 4 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 16 -- C~ -- G~ Id: SmallGroup <48, 35> G~ name: C2*C4*S3 GrpPC : G of order 48 = 2^4 * 3 PC-Relations: G.1^2 = Id(G), G.2^2 = Id(G), G.3^2 = G.4, G.4^2 = Id(G), G.5^3 = Id(G), G.5^G.1 = G.5^2 generating vector: [ G.1 * G.2 * G.4 * G.5, G.1 * G.5^2, G.1 * G.2 * G.3, G.1 * G.3 * G.5^2 ] signature: [ 2, 2, 4, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.2 * G.4 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <24, 5> G name: C4*S3 GrpPC : H of order 24 = 2^3 * 3 PC-Relations: H.1^2 = Id(H), H.2^2 = H.3, H.3^2 = Id(H), H.4^3 = Id(H), H.4^H.1 = H.4^2 generating vector: [ H.1 * H.4, H.1 * H.4^2, H.1 * H.2 * H.3, H.1 * H.2 * H.4^2 ] signature: [ 2, 2, 4, 4 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 17 -- C~ -- G~ Id: SmallGroup <64, 130> G~ name: C4:C4.C2^2 GrpPC : G of order 64 = 2^6 PC-Relations: G.3^2 = G.6, G.4^2 = G.6, 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 generating vector: [ G.1 * G.4 * G.6, G.3 * G.4, G.2 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 ] signature: [ 2, 2, 2, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.5 * G.6 branch points: 0 verify (B1): false verify (B2): true -- C -- G Id: SmallGroup <32, 43> G name: C8:C2^2 GrpPC : H of order 32 = 2^5 PC-Relations: H.3^2 = H.5, H.4^2 = H.5, H.2^H.1 = H.2 * H.4, H.3^H.1 = H.3 * H.5, H.4^H.1 = H.4 * H.5, H.4^H.2 = H.4 * H.5 generating vector: [ H.1 * H.4 * H.5, H.3 * H.4, H.2, H.1 * H.2 * H.3 * H.4 ] signature: [ 2, 2, 2, 8 ] genus: 7 decomp H^0(K_C): [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 18 -- C~ -- G~ Id: SmallGroup <64, 150> G~ name: C2.C8:C2^2 GrpPC : G of order 64 = 2^6 PC-Relations: G.2^2 = G.4 * G.6, G.4^2 = G.6, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.5, G.3^G.2 = G.3 * G.6, G.4^G.1 = G.4 * G.6 generating vector: [ G.1, G.3 * G.5, G.1 * G.2 * G.4, G.2 * G.3 ] signature: [ 2, 2, 2, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.5 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <32, 43> G name: C8:C2^2 GrpPC : H of order 32 = 2^5 PC-Relations: H.2^2 = H.4 * H.5, H.4^2 = H.5, H.2^H.1 = H.2 * H.4, H.3^H.2 = H.3 * H.5, H.4^H.1 = H.4 * H.5 generating vector: [ H.1, H.3, H.1 * H.2 * H.4, H.2 * H.3 ] signature: [ 2, 2, 2, 8 ] genus: 7 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 19 -- C~ -- G~ Id: SmallGroup <64, 150> G~ name: C2.C8:C2^2 GrpPC : G of order 64 = 2^6 PC-Relations: G.2^2 = G.4 * G.6, G.4^2 = G.6, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.5, G.3^G.2 = G.3 * G.6, G.4^G.1 = G.4 * G.6 generating vector: [ G.1, G.3 * G.5, G.1 * G.2 * G.4, G.2 * G.3 ] signature: [ 2, 2, 2, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.5 * G.6 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <32, 43> G name: C8:C2^2 GrpPC : H of order 32 = 2^5 PC-Relations: H.2^2 = H.4 * H.5, H.4^2 = H.5, H.2^H.1 = H.2 * H.4, H.3^H.1 = H.3 * H.5, H.3^H.2 = H.3 * H.5, H.4^H.1 = H.4 * H.5 generating vector: [ H.1, H.3 * H.5, H.1 * H.2 * H.4, H.2 * H.3 ] signature: [ 2, 2, 2, 8 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 20 -- C~ -- G~ Id: SmallGroup <72, 23> G~ name: C3^2:D4 GrpPC : G of order 72 = 2^3 * 3^2 PC-Relations: G.1^2 = Id(G), G.2^2 = G.3, G.3^2 = Id(G), G.4^3 = Id(G), G.5^3 = Id(G), G.2^G.1 = G.2 * G.3, G.4^G.2 = G.4^2, G.5^G.1 = G.5^2 generating vector: [ G.1 * G.2 * G.3, G.1 * G.2 * G.4 * G.5^2, G.1 * G.3 * G.5, G.1 * G.4^2 * G.5^2 ] signature: [ 2, 2, 2, 6 ] genus: 13 decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.3 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <36, 10> G name: S3^2 GrpPC : H of order 36 = 2^2 * 3^2 PC-Relations: H.1^2 = Id(H), H.2^2 = Id(H), H.3^3 = Id(H), H.4^3 = Id(H), H.3^H.2 = H.3^2, H.4^H.1 = H.4^2 generating vector: [ H.1 * H.2, H.1 * H.2 * H.3 * H.4^2, H.1 * H.4, H.1 * H.3^2 * H.4^2 ] signature: [ 2, 2, 2, 6 ] genus: 7 decomp H^0(K_C): [ 0, 0, 1, 0, 0, 0, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 21 -- C~ -- G~ Id: SmallGroup <96, 102> G~ name: C4:D12 GrpPC : G of order 96 = 2^5 * 3 PC-Relations: G.1^2 = Id(G), G.2^2 = G.5, G.3^2 = G.4, 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.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.3 * G.4 * G.6, G.1 * G.5 * G.6^2, G.1 * G.2 * G.6^2 ] signature: [ 2, 2, 2, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 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.1 * H.2 * H.3 * H.5, H.1 * H.3 * H.5, H.1 * H.4 * H.5^2, H.1 * H.2 * H.5^2 ] signature: [ 2, 2, 2, 4 ] genus: 7 decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 22 -- C~ -- G~ Id: SmallGroup <96, 187> G~ name: C4:S4 GrpPC : G of order 96 = 2^5 * 3 PC-Relations: G.1^2 = Id(G), G.2^2 = G.3, G.3^2 = Id(G), G.4^3 = Id(G), G.5^2 = Id(G), G.6^2 = Id(G), G.2^G.1 = G.2 * G.3, G.4^G.1 = G.4^2, G.5^G.1 = G.6, G.5^G.4 = G.6, G.6^G.1 = G.5, G.6^G.4 = G.5 * G.6 generating vector: [ G.1 * G.3 * G.4^2, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.4^2 * G.5, G.1 * G.2 * G.3 * G.5 ] signature: [ 2, 2, 2, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.3 branch points: 0 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <48, 48> G name: C2*S4 GrpPC : H of order 48 = 2^4 * 3 PC-Relations: H.1^2 = Id(H), H.2^2 = Id(H), H.3^3 = Id(H), H.4^2 = Id(H), H.5^2 = Id(H), H.3^H.1 = H.3^2, H.4^H.1 = H.5, H.4^H.3 = H.5, H.5^H.1 = H.4, H.5^H.3 = H.4 * H.5 generating vector: [ H.1 * H.3^2, H.1 * H.2 * H.3 * H.5, H.1 * H.3^2 * H.4, H.1 * H.2 * H.4 ] signature: [ 2, 2, 2, 4 ] genus: 7 decomp H^0(K_C): [ 0, 0, 1, 0, 0, 0, 0, 1, 0, 1 ] N = dim S^2H^0(K_C)^G = 3 22 b = 4 Example # 1 -- C~ -- G~ Id: SmallGroup <18, 2> G~ name: C18 GrpPC : G of order 18 = 2 * 3^2 PC-Relations: G.1^2 = Id(G), G.2^3 = G.3, G.3^3 = Id(G) generating vector: [ G.3^2, G.1 * G.3, G.2^2, G.1 * G.2 * G.3^2 ] signature: [ 3, 6, 9, 18 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 0 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.1 branch points: 4 verify (B1): true verify (B2): false -- C -- G Id: SmallGroup <9, 1> G name: C9 GrpPC : H of order 9 = 3^2 PC-Relations: H.1^3 = H.2 generating vector: [ H.2^2, H.2, H.1^2, H.1 * H.2^2 ] signature: [ 3, 3, 9, 9 ] genus: 6 decomp H^0(K_C): [ 0, 1, 1, 0, 1, 1, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 2 -- C~ -- G~ Id: SmallGroup <24, 9> G~ name: C2*C12 GrpPC : G of order 24 = 2^3 * 3 PC-Relations: G.1^2 = G.4, G.2^2 = Id(G), G.3^3 = Id(G), G.4^2 = Id(G) generating vector: [ G.2 * G.4, G.3, G.1 * G.3, G.1 * G.2 * G.3 ] signature: [ 2, 3, 12, 12 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 3 sigma: G.4 branch points: 4 verify (B1): true verify (B2): false -- C -- G Id: SmallGroup <12, 5> G name: C2*C6 GrpPC : H of order 12 = 2^2 * 3 PC-Relations: H.1^2 = Id(H), H.2^2 = Id(H), H.3^3 = Id(H) generating vector: [ H.2, H.3, H.1 * H.3, H.1 * H.2 * H.3 ] signature: [ 2, 3, 6, 6 ] genus: 6 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1 ] N = dim S^2H^0(K_C)^G = 2 Example # 3 -- C~ -- G~ Id: SmallGroup <24, 9> G~ name: C2*C12 GrpPC : G of order 24 = 2^3 * 3 PC-Relations: G.1^2 = G.4, G.2^2 = Id(G), G.3^3 = Id(G), G.4^2 = Id(G) generating vector: [ G.3^2, G.1 * G.2 * G.4, G.1, G.2 * G.3 ] signature: [ 3, 4, 4, 6 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.2 branch points: 4 verify (B1): true verify (B2): false -- C -- G Id: SmallGroup <12, 2> G name: C12 GrpPC : H of order 12 = 2^2 * 3 PC-Relations: H.1^2 = H.3, H.2^3 = Id(H), H.3^2 = Id(H) generating vector: [ H.2^2, H.1 * H.3, H.1, H.2 ] signature: [ 3, 4, 4, 3 ] genus: 6 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 3 b = 8 Example # 1 -- C~ -- G~ Id: SmallGroup <24, 3> G~ name: SL(2,3) GrpPC : G of order 24 = 2^3 * 3 PC-Relations: G.1^3 = Id(G), G.2^2 = G.4, G.3^2 = G.4, G.4^2 = Id(G), G.2^G.1 = G.3, G.3^G.1 = G.2 * G.3, G.3^G.2 = G.3 * G.4 generating vector: [ G.1^2 * G.2 * G.3 * G.4, G.1^2 * G.2 * G.4, G.1 * G.4, G.1 * G.3 * G.4 ] signature: [ 3, 3, 6, 6 ] genus: 13 decomp H^0(K_C~): [ 0, 1, 1, 2, 0, 2, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 3 sigma: G.4 branch points: 8 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <12, 3> G name: A4 GrpPC : H of order 12 = 2^2 * 3 PC-Relations: H.1^3 = Id(H), H.2^2 = Id(H), H.3^2 = Id(H), H.2^H.1 = H.3, H.3^H.1 = H.2 * H.3 generating vector: [ H.1^2 * H.2 * H.3, H.1^2 * H.2, H.1, H.1 * H.3 ] signature: [ 3, 3, 3, 3 ] genus: 5 decomp H^0(K_C): [ 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 2 Example # 2 -- C~ -- G~ Id: SmallGroup <32, 5> G~ name: C2^2:C8 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.4^2 = G.5, G.2^G.1 = G.2 * G.3 generating vector: [ G.2, G.2 * G.5, G.1 * G.2 * G.5, G.1 * G.2 * G.3 * G.4 * G.5 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.5 branch points: 8 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 3> G name: C2^2:C4 GrpPC : H of order 16 = 2^4 PC-Relations: H.1^2 = H.4, H.2^H.1 = H.2 * H.3 generating vector: [ H.2, H.2, H.1 * H.2, H.1 * H.2 * H.3 * H.4 ] signature: [ 2, 2, 4, 4 ] genus: 5 decomp H^0(K_C): [ 0, 1, 0, 0, 1, 0, 1, 0, 0, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 3 -- C~ -- G~ Id: SmallGroup <32, 38> G~ name: OD16:C2 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.4^2 = G.5, G.3^G.2 = G.3 * G.5 generating vector: [ G.2, G.3 * G.5, G.1 * G.2 * G.3 * G.4 * G.5, G.1 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0 ] N~ = dim S^2H^0(K_C~)^G~ = 3 sigma: G.5 branch points: 8 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 10> G name: C2^2*C4 GrpPC : H of order 16 = 2^4 PC-Relations: H.1^2 = H.4 generating vector: [ H.2, H.3, H.1 * H.2 * H.3 * H.4, H.1 ] signature: [ 2, 2, 4, 4 ] genus: 5 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 2 Example # 4 -- C~ -- G~ Id: SmallGroup <32, 40> G~ name: C2*SD16 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.5, G.4^2 = G.5, G.2^G.1 = G.2 * G.4, G.4^G.1 = G.4 * G.5, G.4^G.2 = G.4 * G.5 generating vector: [ G.2 * G.3 * G.5, G.2, G.1 * G.2 * G.3 * G.4, G.1 * G.2 ] signature: [ 2, 2, 8, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.5 branch points: 8 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <16, 11> G name: C2*D4 GrpPC : H of order 16 = 2^4 PC-Relations: H.2^H.1 = H.2 * H.4 generating vector: [ H.2 * H.3, H.2, H.1 * H.2 * H.3 * H.4, H.1 * H.2 ] signature: [ 2, 2, 4, 4 ] genus: 5 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 5 -- C~ -- G~ Id: SmallGroup <32, 29> G~ name: C2^2:Q8 GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4, G.2^2 = G.4, G.2^G.1 = G.2 * G.4, G.3^G.1 = G.3 * G.5 generating vector: [ G.3 * G.4 * G.5, G.2 * G.3 * G.4 * G.5, G.1 * G.3, G.1 * G.2 * G.3 * G.5 ] signature: [ 2, 4, 4, 4 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.4 branch points: 8 verify (B1): false verify (B2): true -- C -- G Id: SmallGroup <16, 11> G name: C2*D4 GrpPC : H of order 16 = 2^4 PC-Relations: H.3^H.1 = H.3 * H.4 generating vector: [ H.3 * H.4, H.2 * H.3 * H.4, H.1 * H.3, H.1 * H.2 * H.3 * H.4 ] signature: [ 2, 2, 4, 4 ] genus: 5 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1 ] N = dim S^2H^0(K_C)^G = 3 Example # 6 -- C~ -- G~ Id: SmallGroup <48, 33> G~ name: SL(2,3):C2 GrpPC : G of order 48 = 2^4 * 3 PC-Relations: G.1^2 = G.5, G.2^3 = Id(G), G.3^2 = G.5, G.4^2 = G.5, G.5^2 = Id(G), G.3^G.2 = G.4, G.4^G.2 = G.3 * G.4, G.4^G.3 = G.4 * G.5 generating vector: [ G.1 * G.3 * G.4, G.1 * G.4, G.2 * G.4, G.2^2 * G.5 ] signature: [ 2, 2, 3, 6 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 3 sigma: G.5 branch points: 8 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <24, 13> G name: C2*A4 GrpPC : H of order 24 = 2^3 * 3 PC-Relations: H.1^2 = Id(H), H.2^3 = Id(H), H.3^2 = Id(H), H.4^2 = Id(H), H.3^H.2 = H.4, H.4^H.2 = H.3 * H.4 generating vector: [ H.1 * H.3 * H.4, H.1 * H.4, H.2 * H.4, H.2^2 ] signature: [ 2, 2, 3, 3 ] genus: 5 decomp H^0(K_C): [ 0, 0, 0, 1, 1, 0, 1, 0 ] N = dim S^2H^0(K_C)^G = 2 Example # 7 -- C~ -- G~ Id: SmallGroup <64, 130> G~ name: C4:C4.C2^2 GrpPC : G of order 64 = 2^6 PC-Relations: G.3^2 = G.6, G.4^2 = G.6, 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 generating vector: [ G.1 * G.4 * G.6, G.3 * G.4, G.2 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 ] signature: [ 2, 2, 2, 8 ] genus: 13 decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1 ] N~ = dim S^2H^0(K_C~)^G~ = 4 sigma: G.6 branch points: 8 verify (B1): false verify (B2): false -- C -- G Id: SmallGroup <32, 27> G name: C2^2wrC2 GrpPC : H of order 32 = 2^5 PC-Relations: H.2^H.1 = H.2 * H.4, H.3^H.1 = H.3 * H.5 generating vector: [ H.1 * H.4, H.3 * H.4, H.2 * H.5, H.1 * H.2 * H.3 * H.4 ] signature: [ 2, 2, 2, 4 ] genus: 5 decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0 ] N = dim S^2H^0(K_C)^G = 3 7