g~:  49
r: 4


 b =  0 



Example  # 1
-- C~ --
G~ Id: SmallGroup <256, 5299>
G~ name: C2.D4^2.C2
GrpPC : G of order 256 = 2^8
PC-Relations:
    G.2^2 = G.8, 
    G.4^2 = G.6, 
    G.5^2 = G.7 * G.8, 
    G.7^2 = G.8, 
    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.4^G.3 = G.4 * G.7, 
    G.5^G.1 = G.5 * G.7, 
    G.5^G.2 = G.5 * G.7, 
    G.5^G.3 = G.5 * G.7, 
    G.6^G.3 = G.6 * G.8, 
    G.7^G.1 = G.7 * G.8, 
    G.7^G.2 = G.7 * G.8, 
    G.7^G.3 = G.7 * G.8
generating vector: [ G.3, G.2 * G.3 * G.6, G.1 * G.5 * G.7, G.1 * G.2 * G.5 * 
G.6 * G.8 ]
signature: [ 2, 2, 2, 8 ]
genus: 49
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 
1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 11
sigma: G.8
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <128, 351>
G name: D4^2.C2
GrpPC : H of order 128 = 2^7
PC-Relations:
    H.4^2 = H.6, 
    H.5^2 = H.7, 
    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, 
    H.4^H.3 = H.4 * H.7, 
    H.5^H.1 = H.5 * H.7, 
    H.5^H.2 = H.5 * H.7, 
    H.5^H.3 = H.5 * H.7
generating vector: [ H.3, H.2 * H.3 * H.6, H.1 * H.5 * H.7, H.1 * H.2 * H.5 * 
H.6 ]
signature: [ 2, 2, 2, 8 ]
genus: 25
decomp H^0(K_C): [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 
0, 0, 0, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 10

Example  # 2
-- C~ --
G~ Id: SmallGroup <256, 6670>
G~ name: C2.D4^2.C2
GrpPC : G of order 256 = 2^8
PC-Relations:
    G.2^2 = G.8, 
    G.4^2 = G.8, 
    G.5^2 = G.8, 
    G.6^2 = G.8, 
    G.2^G.1 = G.2 * G.4, 
    G.3^G.1 = G.3 * G.5, 
    G.4^G.1 = G.4 * G.8, 
    G.4^G.2 = G.4 * G.8, 
    G.4^G.3 = G.4 * G.6, 
    G.5^G.1 = G.5 * G.8, 
    G.5^G.2 = G.5 * G.6 * G.7 * G.8, 
    G.5^G.3 = G.5 * G.8, 
    G.5^G.4 = G.5 * G.7 * G.8, 
    G.6^G.1 = G.6 * G.7, 
    G.6^G.3 = G.6 * G.8, 
    G.6^G.4 = G.6 * G.8, 
    G.6^G.5 = G.6 * G.8, 
    G.7^G.2 = G.7 * G.8, 
    G.7^G.3 = G.7 * G.8
generating vector: [ G.2 * G.7, G.1 * G.8, G.3 * G.5, G.1 * G.2 * G.3 * G.7 * 
G.8 ]
signature: [ 2, 2, 2, 8 ]
genus: 49
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 1, 
0, 0, 1, 1, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 10
sigma: G.8
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <128, 928>
G name: D4^2.C2
GrpPC : H of order 128 = 2^7
PC-Relations:
    H.2^H.1 = H.2 * H.4, 
    H.3^H.1 = H.3 * H.5, 
    H.4^H.3 = H.4 * H.6, 
    H.5^H.2 = H.5 * H.6 * H.7, 
    H.5^H.4 = H.5 * H.7, 
    H.6^H.1 = H.6 * H.7
generating vector: [ H.2 * H.7, H.1, H.3 * H.5, H.1 * H.2 * H.3 * H.7 ]
signature: [ 2, 2, 2, 8 ]
genus: 25
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2 ]
N = dim S^2H^0(K_C)^G = 9
2