g~:  7
r: 4


 b =  0 



Example  # 1
-- 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.3^2, G.1 * G.3^2, G.2 * G.3^2 ]
signature: [ 2, 6, 6, 6 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.1 * G.2
branch points: 0
verify (B1): true
verify (B2): false
-- 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, H.1 * H.2^2, H.1 * H.2^2, H.1 * H.2^2 ]
signature: [ 2, 6, 6, 6 ]
genus: 4
decomp H^0(K_C): [ 0, 1, 1, 0, 0, 2 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 2
-- C~ --
G~ Id: SmallGroup <16, 4>
G~ name: C4:C4
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.3, 
    G.2^G.1 = G.2 * G.3
generating vector: [ G.4, G.1 * G.3, G.2 * G.3 * G.4, G.1 * G.2 * G.4 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 1, 0, 1, 2, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3 * G.4
branch points: 0
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <8, 4>
G name: Q8
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3, 
    H.2^2 = H.3, 
    H.2^H.1 = H.2 * H.3
generating vector: [ H.3, H.1 * H.3, H.2, H.1 * H.2 * H.3 ]
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

Example  # 3
-- C~ --
G~ Id: SmallGroup <16, 4>
G~ name: C4:C4
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.3, 
    G.2^G.1 = G.2 * G.3
generating vector: [ G.3, G.1 * G.4, G.2 * G.3, G.1 * G.2 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3 * G.4
branch points: 0
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <8, 4>
G name: Q8
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3, 
    H.2^2 = H.3, 
    H.2^H.1 = H.2 * H.3
generating vector: [ H.3, H.1 * H.3, H.2 * H.3, H.1 * H.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

Example  # 4
-- C~ --
G~ Id: SmallGroup <16, 12>
G~ name: C2*Q8
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.4, 
    G.2^G.1 = G.2 * G.4
generating vector: [ G.3 * G.4, G.2 * G.3 * G.4, G.1 * G.2 * G.3 * G.4, G.1 * 
G.3 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3
branch points: 0
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <8, 4>
G name: Q8
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3, 
    H.2^2 = H.3, 
    H.2^H.1 = H.2 * H.3
generating vector: [ H.3, H.2 * H.3, H.1 * H.2 * H.3, H.1 ]
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

Example  # 5
-- C~ --
G~ Id: SmallGroup <48, 48>
G~ name: C2*S4
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 = Id(G), 
    G.5^2 = Id(G), 
    G.3^G.1 = G.3^2, 
    G.4^G.1 = G.5, 
    G.4^G.3 = G.5, 
    G.5^G.1 = G.4, 
    G.5^G.3 = G.4 * G.5
generating vector: [ G.1, G.1 * G.2 * G.4 * G.5, G.1 * G.3^2 * G.4, G.1 * G.2 * 
G.3^2 * G.5 ]
signature: [ 2, 2, 2, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 1, 0, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.2
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <24, 12>
G name: S4
GrpPC : H of order 24 = 2^3 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^3 = Id(H), 
    H.3^2 = Id(H), 
    H.4^2 = Id(H), 
    H.2^H.1 = H.2^2, 
    H.3^H.1 = H.4, 
    H.3^H.2 = H.4, 
    H.4^H.1 = H.3, 
    H.4^H.2 = H.3 * H.4
generating vector: [ H.1, H.1 * H.3 * H.4, H.1 * H.2^2 * H.3, H.1 * H.2^2 * H.4 
]
signature: [ 2, 2, 2, 4 ]
genus: 4
decomp H^0(K_C): [ 0, 1, 0, 1, 0 ]
N = dim S^2H^0(K_C)^G = 2
5


 b =  4 



Example  # 1
-- C~ --
G~ Id: SmallGroup <8, 1>
G~ name: C8
GrpPC : G of order 8 = 2^3
PC-Relations:
    G.1^2 = G.2, 
    G.2^2 = G.3
generating vector: [ G.1, G.1, G.1, G.1 * G.3 ]
signature: [ 8, 8, 8, 8 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 1, 2, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3
branch points: 4
verify (B1): true
verify (B2): false
-- C --
G Id: SmallGroup <4, 1>
G name: C4
GrpPC : H of order 4 = 2^2
PC-Relations:
    H.1^2 = H.2
generating vector: [ H.1, H.1, H.1, H.1 ]
signature: [ 4, 4, 4, 4 ]
genus: 3
decomp H^0(K_C): [ 0, 2, 1, 0 ]
N = dim S^2H^0(K_C)^G = 1

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.3^2, G.1 * G.3^2, G.2 * G.3^2 ]
signature: [ 2, 6, 6, 6 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.2
branch points: 4
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, H.2^2, H.1 * H.2^2, H.2^2 ]
signature: [ 2, 3, 6, 3 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 1, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1

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

Example  # 4
-- C~ --
G~ Id: SmallGroup <16, 6>
G~ name: OD16
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.3, 
    G.3^2 = G.4, 
    G.2^G.1 = G.2 * G.4
generating vector: [ G.2, G.2 * G.4, G.1 * G.4, G.1 * G.3 * G.4 ]
signature: [ 2, 2, 8, 8 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 1, 0, 1, 0, 1, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.4
branch points: 4
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <8, 2>
G name: C2*C4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3
generating vector: [ H.2, H.2, H.1, H.1 * H.3 ]
signature: [ 2, 2, 4, 4 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 0, 1, 0, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 5
-- C~ --
G~ Id: SmallGroup <16, 8>
G~ name: SD16
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.3^2 = G.4, 
    G.2^G.1 = G.2 * G.3, 
    G.3^G.1 = G.3 * G.4, 
    G.3^G.2 = G.3 * G.4
generating vector: [ G.2 * G.3, G.2 * G.3, G.1 * G.2 * G.3 * G.4, G.1 * G.2 ]
signature: [ 2, 2, 8, 8 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 1, 0, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.4
branch points: 4
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <8, 3>
G name: D4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.2^H.1 = H.2 * H.3
generating vector: [ H.2 * H.3, H.2 * H.3, H.1 * H.2 * H.3, H.1 * H.2 ]
signature: [ 2, 2, 4, 4 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 6
-- C~ --
G~ Id: SmallGroup <16, 2>
G~ name: C4^2
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = G.4
generating vector: [ G.3, G.1 * G.4, G.2 * G.3, G.1 * G.2 * G.3 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.4
branch points: 4
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <8, 2>
G name: C2*C4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3
generating vector: [ H.3, H.1, H.2 * H.3, H.1 * H.2 * H.3 ]
signature: [ 2, 4, 2, 4 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 1, 0, 0, 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 7
-- C~ --
G~ Id: SmallGroup <16, 2>
G~ name: C4^2
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = G.4
generating vector: [ G.3, G.1 * G.4, G.2 * G.3, G.1 * G.2 * G.3 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3 * G.4
branch points: 4
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <8, 2>
G name: C2*C4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3, 
    H.2^2 = H.3
generating vector: [ H.3, H.1 * H.3, H.2 * H.3, H.1 * H.2 * H.3 ]
signature: [ 2, 4, 4, 2 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 1, 1, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 8
-- C~ --
G~ Id: SmallGroup <16, 4>
G~ name: C4:C4
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.3, 
    G.2^G.1 = G.2 * G.3
generating vector: [ G.4, G.1 * G.3, G.2 * G.3 * G.4, G.1 * G.2 * G.4 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 1, 0, 1, 2, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3
branch points: 4
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <8, 2>
G name: C2*C4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3
generating vector: [ H.3, H.1, H.2 * H.3, H.1 * H.2 * H.3 ]
signature: [ 2, 4, 2, 4 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 1, 0, 0, 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 9
-- C~ --
G~ Id: SmallGroup <16, 4>
G~ name: C4:C4
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.3, 
    G.2^G.1 = G.2 * G.3
generating vector: [ G.3 * G.4, G.1 * G.2 * G.3 * G.4, G.2 * G.3, G.1 * G.4 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 1, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3
branch points: 4
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <8, 2>
G name: C2*C4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.3
generating vector: [ H.3, H.1 * H.2 * H.3, H.2, H.1 * H.3 ]
signature: [ 2, 4, 2, 4 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 1, 1, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 10
-- C~ --
G~ Id: SmallGroup <24, 13>
G~ name: C2*A4
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^3 = Id(G), 
    G.3^2 = Id(G), 
    G.4^2 = Id(G), 
    G.3^G.2 = G.4, 
    G.4^G.2 = G.3 * G.4
generating vector: [ G.1 * G.4, G.4, G.2^2 * G.3 * G.4, G.1 * G.2 * G.3 ]
signature: [ 2, 2, 3, 6 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.1
branch points: 4
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <12, 3>
G name: A4
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^3 = Id(H), 
    H.2^2 = Id(H), 
    H.3^2 = Id(H), 
    H.2^H.1 = H.3, 
    H.3^H.1 = H.2 * H.3
generating vector: [ H.3, H.3, H.1^2 * H.2 * H.3, H.1 * H.2 ]
signature: [ 2, 2, 3, 3 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 11
-- C~ --
G~ Id: SmallGroup <32, 43>
G~ name: C8:C2^2
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.4^2 = G.5, 
    G.2^G.1 = G.2 * G.4, 
    G.3^G.1 = G.3 * G.5, 
    G.4^G.1 = G.4 * G.5, 
    G.4^G.2 = G.4 * G.5
generating vector: [ G.2, G.1 * G.5, G.3, G.1 * G.2 * G.3 ]
signature: [ 2, 2, 2, 8 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.5
branch points: 4
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <16, 11>
G name: C2*D4
GrpPC : H of order 16 = 2^4
PC-Relations:
    H.2^H.1 = H.2 * H.4
generating vector: [ H.2, H.1, H.3, H.1 * H.2 * H.3 ]
signature: [ 2, 2, 2, 4 ]
genus: 3
decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 2
11


 b =  8 



Example  # 1
-- C~ --
G~ Id: SmallGroup <12, 2>
G~ name: C12
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
    G.1^2 = G.3, 
    G.2^3 = Id(G), 
    G.3^2 = Id(G)
generating vector: [ G.3, G.2^2, G.1 * G.2^2 * G.3, G.1 * G.2^2 * G.3 ]
signature: [ 2, 3, 12, 12 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 1
sigma: G.3
branch points: 8
verify (B1): true
verify (B2): false
-- 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.2^2, H.1 * H.2^2, H.1 * H.2^2 ]
signature: [ 1, 3, 6, 6 ]
genus: 2
decomp H^0(K_C): [ 0, 0, 1, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 0

Example  # 2
-- C~ --
G~ Id: SmallGroup <12, 1>
G~ name: C3:C4
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
    G.1^2 = G.2, 
    G.2^2 = Id(G), 
    G.3^3 = Id(G), 
    G.3^G.1 = G.3^2
generating vector: [ G.3^2, G.1 * G.2 * G.3^2, G.1 * G.2 * G.3, G.2 * G.3^2 ]
signature: [ 3, 4, 4, 6 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 1, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.2
branch points: 8
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.2^2, H.1 * H.2^2, H.1 * H.2, H.2^2 ]
signature: [ 3, 2, 2, 3 ]
genus: 2
decomp H^0(K_C): [ 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 3
-- C~ --
G~ Id: SmallGroup <12, 2>
G~ name: C12
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
    G.1^2 = G.3, 
    G.2^3 = Id(G), 
    G.3^2 = Id(G)
generating vector: [ G.2, G.1, G.1, G.2^2 * G.3 ]
signature: [ 3, 4, 4, 6 ]
genus: 7
decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3
branch points: 8
verify (B1): true
verify (B2): false
-- 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.2, H.1, H.1, H.2^2 ]
signature: [ 3, 2, 2, 3 ]
genus: 2
decomp H^0(K_C): [ 0, 0, 0, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 4
-- C~ --
G~ Id: SmallGroup <16, 4>
G~ name: C4:C4
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.3, 
    G.2^G.1 = G.2 * G.3
generating vector: [ G.3, G.1 * G.4, G.2 * G.3, G.1 * G.2 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.4
branch points: 8
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <8, 3>
G name: D4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.2^2 = H.3, 
    H.2^H.1 = H.2 * H.3
generating vector: [ H.3, H.1, H.2 * H.3, H.1 * H.2 ]
signature: [ 2, 2, 4, 2 ]
genus: 2
decomp H^0(K_C): [ 0, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 5
-- C~ --
G~ Id: SmallGroup <16, 4>
G~ name: C4:C4
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.3, 
    G.2^G.1 = G.2 * G.3
generating vector: [ G.3 * G.4, G.1 * G.2 * G.3 * G.4, G.2 * G.3, G.1 * G.4 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 1, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.4
branch points: 8
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <8, 3>
G name: D4
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.2^2 = H.3, 
    H.2^H.1 = H.2 * H.3
generating vector: [ H.3, H.1 * H.2 * H.3, H.2 * H.3, H.1 ]
signature: [ 2, 2, 4, 2 ]
genus: 2
decomp H^0(K_C): [ 0, 0, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 1
5


 b =  12 



Example  # 1
-- C~ --
G~ Id: SmallGroup <12, 1>
G~ name: C3:C4
GrpPC : G of order 12 = 2^2 * 3
PC-Relations:
    G.1^2 = G.2, 
    G.2^2 = Id(G), 
    G.3^3 = Id(G), 
    G.3^G.1 = G.3^2
generating vector: [ G.1 * G.3^2, G.1 * G.3, G.1, G.1 * G.3 ]
signature: [ 4, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 1, 2, 0, 2, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.2
branch points: 12
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.2^2, H.1 * H.2, H.1, H.1 * H.2 ]
signature: [ 2, 2, 2, 2 ]
genus: 1
decomp H^0(K_C): [ 0, 1, 0 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 2
-- C~ --
G~ Id: SmallGroup <16, 12>
G~ name: C2*Q8
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.4, 
    G.2^G.1 = G.2 * G.4
generating vector: [ G.3 * G.4, G.2 * G.3 * G.4, G.1 * G.2 * G.3 * G.4, G.1 * 
G.3 ]
signature: [ 2, 4, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.4
branch points: 12
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <8, 5>
G name: C2^3
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^2 = Id(H)
generating vector: [ H.3, H.2 * H.3, H.1 * H.2 * H.3, H.1 * H.3 ]
signature: [ 2, 2, 2, 2 ]
genus: 1
decomp H^0(K_C): [ 0, 0, 0, 0, 1, 0, 0, 0 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 3
-- C~ --
G~ Id: SmallGroup <24, 5>
G~ name: C4*S3
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.3, 
    G.3^2 = Id(G), 
    G.4^3 = Id(G), 
    G.4^G.1 = G.4^2
generating vector: [ G.1 * G.3, G.1 * G.4, G.1 * G.2 * G.4, G.1 * G.2 ]
signature: [ 2, 2, 4, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.3
branch points: 12
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <12, 4>
G name: D6
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^3 = Id(H), 
    H.3^H.1 = H.3^2
generating vector: [ H.1, H.1 * H.3, H.1 * H.2 * H.3, H.1 * H.2 ]
signature: [ 2, 2, 2, 2 ]
genus: 1
decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 4
-- C~ --
G~ Id: SmallGroup <24, 3>
G~ name: SL(2,3)
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^3 = Id(G), 
    G.2^2 = G.4, 
    G.3^2 = G.4, 
    G.4^2 = Id(G), 
    G.2^G.1 = G.3, 
    G.3^G.1 = G.2 * G.3, 
    G.3^G.2 = G.3 * G.4
generating vector: [ G.4, G.1 * G.3, G.1 * G.2 * G.3, G.1 ]
signature: [ 2, 3, 3, 3 ]
genus: 7
decomp H^0(K_C~): [ 0, 1, 0, 2, 1, 0, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 1
sigma: G.4
branch points: 12
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <12, 3>
G name: A4
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^3 = Id(H), 
    H.2^2 = Id(H), 
    H.3^2 = Id(H), 
    H.2^H.1 = H.3, 
    H.3^H.1 = H.2 * H.3
generating vector: [ Id(H), H.1 * H.3, H.1 * H.2 * H.3, H.1 ]
signature: [ 1, 3, 3, 3 ]
genus: 1
decomp H^0(K_C): [ 0, 0, 1, 0 ]
N = dim S^2H^0(K_C)^G = 0

Example  # 5
-- C~ --
G~ Id: SmallGroup <48, 41>
G~ name: D6.C2^2
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.4, 
    G.3^2 = G.4, 
    G.4^2 = Id(G), 
    G.5^3 = Id(G), 
    G.3^G.1 = G.3 * G.4, 
    G.3^G.2 = G.3 * G.4, 
    G.5^G.1 = G.5^2
generating vector: [ G.1 * G.4, G.1 * G.2 * G.3 * G.5^2, G.1 * G.3 * G.5^2, G.1 
* G.2 ]
signature: [ 2, 2, 2, 4 ]
genus: 7
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.4
branch points: 12
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <24, 14>
G name: C2^2*S3
GrpPC : H of order 24 = 2^3 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^2 = Id(H), 
    H.4^3 = Id(H), 
    H.4^H.1 = H.4^2
generating vector: [ H.1, H.1 * H.2 * H.3 * H.4^2, H.1 * H.3 * H.4^2, H.1 * H.2 
]
signature: [ 2, 2, 2, 2 ]
genus: 1
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
N = dim S^2H^0(K_C)^G = 1
5