g~:  4
r: 4


 b =  2 



Example  # 1
-- C~ --
G~ Id: SmallGroup <6, 2>
G~ name: C6
GrpPC : G of order 6 = 2 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^3 = Id(G)
generating vector: [ G.2, G.2^2, G.1 * G.2, G.1 * G.2^2 ]
signature: [ 3, 3, 6, 6 ]
genus: 4
decomp H^0(K_C~): [ 0, 0, 1, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.1
branch points: 2
verify (B1): true
verify (B2): false
-- C --
G Id: SmallGroup <3, 1>
G name: C3
GrpPC : H of order 3
PC-Relations:
    H.1^3 = Id(H)
generating vector: [ H.1, H.1^2, H.1, H.1^2 ]
signature: [ 3, 3, 3, 3 ]
genus: 2
decomp H^0(K_C): [ 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 2
-- C~ --
G~ Id: SmallGroup <12, 4>
G~ name: D6
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = Id(G), 
    G.3^3 = Id(G), 
    G.3^G.1 = G.3^2
generating vector: [ G.1 * G.2, G.1, G.3, G.2 * G.3^2 ]
signature: [ 2, 2, 3, 6 ]
genus: 4
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.2
branch points: 2
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <6, 1>
G name: S3
GrpPC : H of order 6 = 2 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^3 = Id(H), 
    H.2^H.1 = H.2^2
generating vector: [ H.1, H.1, H.2, H.2^2 ]
signature: [ 2, 2, 3, 3 ]
genus: 2
decomp H^0(K_C): [ 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1
2


 b =  6 



Example  # 1
-- C~ --
G~ Id: SmallGroup <6, 2>
G~ name: C6
GrpPC : G of order 6 = 2 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^3 = Id(G)
generating vector: [ G.1, G.1 * G.2, G.1 * G.2, G.1 * G.2 ]
signature: [ 2, 6, 6, 6 ]
genus: 4
decomp H^0(K_C~): [ 0, 1, 0, 2, 1, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 1
sigma: G.1
branch points: 6
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <3, 1>
G name: C3
GrpPC : H of order 3
PC-Relations:
    H.1^3 = Id(H)
generating vector: [ Id(H), H.1, H.1, H.1 ]
signature: [ 1, 3, 3, 3 ]
genus: 1
decomp H^0(K_C): [ 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 0

Example  # 2
-- C~ --
G~ Id: SmallGroup <12, 5>
G~ name: C2*C6
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = Id(G), 
    G.3^3 = Id(G)
generating vector: [ G.1, G.2, G.3, G.1 * G.2 * G.3^2 ]
signature: [ 2, 2, 3, 6 ]
genus: 4
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 1
sigma: G.2
branch points: 6
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <6, 2>
G name: C6
GrpPC : H of order 6 = 2 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^3 = Id(H)
generating vector: [ H.1, Id(H), H.2, H.1 * H.2^2 ]
signature: [ 2, 1, 3, 6 ]
genus: 1
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 0

Example  # 3
-- C~ --
G~ Id: SmallGroup <12, 5>
G~ name: C2*C6
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = Id(G), 
    G.3^3 = Id(G)
generating vector: [ G.1, G.2, G.3, G.1 * G.2 * G.3^2 ]
signature: [ 2, 2, 3, 6 ]
genus: 4
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 1
sigma: G.1
branch points: 6
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <6, 2>
G name: C6
GrpPC : H of order 6 = 2 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^3 = Id(H)
generating vector: [ Id(H), H.1, H.2, H.1 * H.2^2 ]
signature: [ 1, 2, 3, 6 ]
genus: 1
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 0
3


 b =  10 



Example  # 1
-- C~ --
G~ Id: SmallGroup <8, 4>
G~ name: Q8
GrpPC : G of order 8 = 2^3
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = G.3, 
    G.2^G.1 = G.2 * G.3
generating vector: [ G.3, G.2 * G.3, G.1, G.1 * G.2 ]
signature: [ 2, 4, 4, 4 ]
genus: 4
decomp H^0(K_C~): [ 0, 0, 0, 0, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 1
sigma: G.3
branch points: 10
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <4, 2>
G name: C2^2
GrpPC : H of order 4 = 2^2
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H)
generating vector: [ Id(H), H.2, H.1, H.1 * H.2 ]
signature: [ 1, 2, 2, 2 ]
genus: 0
decomp H^0(K_C): [ 0, 0, 0, 0 ]
N = dim S^2H^0(K_C)^G = 0
1