g~:  41
r: 4


 b =  0 



Example  # 1
-- C~ --
G~ Id: SmallGroup <192, 386>
G~ name: (C3*C4wrC2):C2
GrpPC : G of order 192 = 2^6 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.6, 
    G.3^2 = G.4 * G.6, 
    G.4^2 = G.6, 
    G.5^2 = G.6, 
    G.6^2 = Id(G), 
    G.7^3 = Id(G), 
    G.3^G.1 = G.3 * G.4, 
    G.3^G.2 = G.3 * G.5, 
    G.4^G.1 = G.4 * G.6, 
    G.5^G.2 = G.5 * G.6, 
    G.5^G.3 = G.5 * G.6, 
    G.7^G.1 = G.7^2
generating vector: [ G.1 * G.3 * G.4 * G.5 * G.7^2, G.1 * G.4 * G.6 * G.7, G.2 *
G.4 * G.6, G.2 * G.3 * G.4 * G.5 * G.7 ]
signature: [ 2, 2, 2, 12 ]
genus: 41
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 
0, 1, 1, 1, 1, 0, 1, 1, 0, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 10
sigma: G.6
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <96, 89>
G name: C2^2:D12
GrpPC : H of order 96 = 2^5 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^2 = H.4, 
    H.4^2 = Id(H), 
    H.5^2 = Id(H), 
    H.6^3 = Id(H), 
    H.3^H.1 = H.3 * H.4, 
    H.3^H.2 = H.3 * H.5, 
    H.6^H.1 = H.6^2
generating vector: [ H.1 * H.3 * H.4 * H.5 * H.6^2, H.1 * H.4 * H.6, H.2 * H.4, 
H.2 * H.3 * H.4 * H.5 * H.6 ]
signature: [ 2, 2, 2, 12 ]
genus: 21
decomp H^0(K_C): [ 0, 0, 0, 1, 0, 0, 0, 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 = 9

Example  # 2
-- C~ --
G~ Id: SmallGroup <192, 621>
G~ name: (D6.C2^2):C2:C2
GrpPC : G of order 192 = 2^6 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.6, 
    G.3^2 = Id(G), 
    G.4^2 = G.6, 
    G.5^2 = Id(G), 
    G.6^2 = Id(G), 
    G.7^3 = Id(G), 
    G.3^G.1 = G.3 * G.4, 
    G.3^G.2 = G.3 * G.5, 
    G.4^G.1 = G.4 * G.6, 
    G.4^G.2 = G.4 * G.6, 
    G.4^G.3 = G.4 * G.6, 
    G.5^G.1 = G.5 * G.6, 
    G.7^G.1 = G.7^2
generating vector: [ G.1 * G.2 * G.4 * G.7, G.1 * G.6, G.3 * G.5, G.2 * G.3 * 
G.4 * G.6 * G.7 ]
signature: [ 2, 2, 2, 12 ]
genus: 41
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 
1, 0, 1, 1, 1, 0, 1, 1, 0, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 8
sigma: G.6
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <96, 144>
G name: C2^3:D6
GrpPC : H of order 96 = 2^5 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^2 = Id(H), 
    H.4^2 = Id(H), 
    H.5^2 = Id(H), 
    H.6^3 = Id(H), 
    H.3^H.1 = H.3 * H.4, 
    H.3^H.2 = H.3 * H.5, 
    H.6^H.1 = H.6^2
generating vector: [ H.1 * H.2 * H.4 * H.6, H.1, H.3 * H.5, H.2 * H.3 * H.4 * 
H.6 ]
signature: [ 2, 2, 2, 12 ]
genus: 21
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 
0, 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 7
2