g~:  31
r: 4


 b =  4 



Example  # 1
-- C~ --
G~ Id: SmallGroup <64, 53>
G~ name: SD64
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.1^2 = G.6, 
    G.3^2 = G.4 * G.5, 
    G.4^2 = G.5 * G.6, 
    G.5^2 = G.6, 
    G.2^G.1 = G.2 * G.3, 
    G.3^G.1 = G.3 * G.4, 
    G.3^G.2 = G.3 * G.4, 
    G.4^G.1 = G.4 * G.5, 
    G.4^G.2 = G.4 * G.5, 
    G.5^G.1 = G.5 * G.6, 
    G.5^G.2 = G.5 * G.6
generating vector: [ G.2 * G.4, G.2 * G.5, G.1 * G.2, G.1 * G.2 * G.3 * G.4 * 
G.5 ]
signature: [ 2, 2, 32, 32 ]
genus: 31
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 1, 0, 0, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 9
sigma: G.6
branch points: 4
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <32, 18>
G name: D16
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.3^2 = H.4 * H.5, 
    H.4^2 = H.5, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.4, 
    H.3^H.2 = H.3 * H.4, 
    H.4^H.1 = H.4 * H.5, 
    H.4^H.2 = H.4 * H.5
generating vector: [ H.2 * H.4, H.2 * H.5, H.1 * H.2, H.1 * H.2 * H.3 * H.4 * 
H.5 ]
signature: [ 2, 2, 16, 16 ]
genus: 15
decomp H^0(K_C): [ 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 8
1


 b =  12 



Example  # 1
-- C~ --
G~ Id: SmallGroup <144, 141>
G~ name: (C12*S3):C2
GrpPC : G of order 144 = 2^4 * 3^2
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.6^3 = Id(G), 
    G.2^G.1 = G.2 * G.4, 
    G.5^G.2 = G.5^2, 
    G.6^G.1 = G.6^2
generating vector: [ G.1 * G.4 * G.6, G.1 * G.2 * G.3 * G.5, G.2 * G.4, G.3 * 
G.4 * G.5 * G.6 ]
signature: [ 2, 2, 2, 12 ]
genus: 31
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 
0, 0, 1, 1, 0, 1, 1, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 7
sigma: G.4
branch points: 12
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <72, 46>
G name: C2*S3^2
GrpPC : H of order 72 = 2^3 * 3^2
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^2 = Id(H), 
    H.4^3 = Id(H), 
    H.5^3 = Id(H), 
    H.4^H.2 = H.4^2, 
    H.5^H.1 = H.5^2
generating vector: [ H.1 * H.5, H.1 * H.2 * H.3 * H.4, H.2, H.3 * H.4 * H.5 ]
signature: [ 2, 2, 2, 6 ]
genus: 13
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0 ]
N = dim S^2H^0(K_C)^G = 6
1