g~:  16
r: 4


 b =  2 



Example  # 1
-- C~ --
G~ Id: SmallGroup <40, 10>
G~ name: C5*D4
GrpPC : G of order 40 = 2^3 * 5
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = Id(G), 
    G.3^5 = Id(G), 
    G.4^2 = Id(G), 
    G.2^G.1 = G.2 * G.4
generating vector: [ G.2, G.1, G.3^4, G.1 * G.2 * G.3 ]
signature: [ 2, 2, 5, 20 ]
genus: 16
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 
0, 1, 1, 2, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.4
branch points: 2
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <20, 5>
G name: C2*C10
GrpPC : H of order 20 = 2^2 * 5
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^5 = Id(H)
generating vector: [ H.2, H.1, H.3^4, H.1 * H.2 * H.3 ]
signature: [ 2, 2, 5, 10 ]
genus: 8
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 2
-- C~ --
G~ Id: SmallGroup <48, 25>
G~ name: C3*D8
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = Id(G), 
    G.3^3 = Id(G), 
    G.4^2 = G.5, 
    G.5^2 = Id(G), 
    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.5, G.1 * G.4 * G.5, G.3, G.1 * G.2 * G.3^2 * G.4 * 
G.5 ]
signature: [ 2, 2, 3, 24 ]
genus: 16
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 2, 
1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.5
branch points: 2
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <24, 10>
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.3^3 = Id(H), 
    H.4^2 = Id(H), 
    H.2^H.1 = H.2 * H.4
generating vector: [ H.2, H.1 * H.4, H.3, H.1 * H.2 * H.3^2 * H.4 ]
signature: [ 2, 2, 3, 12 ]
genus: 8
decomp H^0(K_C): [ 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 2
2