g~:  11
r: 4


 b =  0 



Example  # 1
-- C~ --
G~ Id: SmallGroup <16, 5>
G~ name: C2*C8
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.3, 
    G.3^2 = G.4
generating vector: [ G.3 * G.4, G.2 * G.3, G.1 * G.4, G.1 * G.2 * G.3 ]
signature: [ 4, 4, 8, 8 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 4
sigma: G.2
branch points: 0
verify (B1): true
verify (B2): false
-- C --
G Id: SmallGroup <8, 1>
G name: C8
GrpPC : H of order 8 = 2^3
PC-Relations:
    H.1^2 = H.2, 
    H.2^2 = H.3
generating vector: [ H.2 * H.3, H.2, H.1 * H.3, H.1 * H.2 ]
signature: [ 4, 4, 8, 8 ]
genus: 6
decomp H^0(K_C): [ 0, 1, 1, 1, 0, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 3

Example  # 2
-- C~ --
G~ Id: SmallGroup <24, 9>
G~ name: C2*C12
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = Id(G), 
    G.3^3 = Id(G), 
    G.4^2 = Id(G)
generating vector: [ G.2, G.3, G.1, G.1 * G.2 * G.3^2 * G.4 ]
signature: [ 2, 3, 4, 12 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 
1, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.2 * G.4
branch points: 0
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <12, 2>
G name: C12
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^2 = H.3, 
    H.2^3 = Id(H), 
    H.3^2 = Id(H)
generating vector: [ H.3, H.2, H.1, H.1 * H.2^2 ]
signature: [ 2, 3, 4, 12 ]
genus: 6
decomp H^0(K_C): [ 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 3
-- C~ --
G~ Id: SmallGroup <24, 7>
G~ name: C2*C3:C4
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = Id(G), 
    G.3^2 = Id(G), 
    G.4^3 = Id(G), 
    G.4^G.1 = G.4^2
generating vector: [ G.3, G.1 * G.2, G.1 * G.3 * G.4^2, G.2 * G.3 * G.4 ]
signature: [ 2, 4, 4, 6 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 2, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.2
branch points: 0
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <12, 1>
G name: C3:C4
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^2 = H.2, 
    H.2^2 = Id(H), 
    H.3^3 = Id(H), 
    H.3^H.1 = H.3^2
generating vector: [ H.2, H.1, H.1 * H.2 * H.3^2, H.2 * H.3 ]
signature: [ 2, 4, 4, 6 ]
genus: 6
decomp H^0(K_C): [ 0, 0, 1, 1, 2, 0 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 4
-- C~ --
G~ Id: SmallGroup <24, 7>
G~ name: C2*C3:C4
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = Id(G), 
    G.3^2 = Id(G), 
    G.4^3 = Id(G), 
    G.4^G.1 = G.4^2
generating vector: [ G.2, G.1 * G.2 * G.3, G.1 * G.3 * G.4^2, G.3 * G.4 ]
signature: [ 2, 4, 4, 6 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.2 * G.3
branch points: 0
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <12, 1>
G name: C3:C4
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^2 = H.2, 
    H.2^2 = Id(H), 
    H.3^3 = Id(H), 
    H.3^H.1 = H.3^2
generating vector: [ H.2, H.1, H.1 * H.2 * H.3^2, H.2 * H.3 ]
signature: [ 2, 4, 4, 6 ]
genus: 6
decomp H^0(K_C): [ 0, 0, 1, 1, 2, 0 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 5
-- C~ --
G~ Id: SmallGroup <40, 12>
G~ name: C2*F5
GrpPC : G of order 40 = 2^3 * 5
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = Id(G), 
    G.3^2 = Id(G), 
    G.4^5 = Id(G), 
    G.4^G.1 = G.4^2, 
    G.4^G.3 = G.4^4
generating vector: [ G.2 * G.3 * G.4^3, G.3 * G.4^2, G.1 * G.4^3, G.1 * G.2 * 
G.3 * G.4^2 ]
signature: [ 2, 2, 4, 4 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 1, 1, 1, 1, 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 <20, 3>
G name: F5
GrpPC : H of order 20 = 2^2 * 5
PC-Relations:
    H.1^2 = H.2, 
    H.2^2 = Id(H), 
    H.3^5 = Id(H), 
    H.3^H.1 = H.3^2, 
    H.3^H.2 = H.3^4
generating vector: [ H.2 * H.3^3, H.2 * H.3^2, H.1 * H.3^3, H.1 * H.2 * H.3^2 ]
signature: [ 2, 2, 4, 4 ]
genus: 6
decomp H^0(K_C): [ 0, 0, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 6
-- C~ --
G~ Id: SmallGroup <120, 35>
G~ name: C2*A5
Permutation group G acting on a set of cardinality 7
Order = 120 = 2^3 * 3 * 5
    (1, 2, 3, 5, 4)
    (1, 3)(2, 4)(6, 7)
generating vector: [
    (1, 2)(4, 5)(6, 7),
    (1, 3)(4, 5),
    (1, 3)(2, 4)(6, 7),
    (1, 2, 4)
]
signature: [ 2, 2, 2, 3 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 1, 1, 0, 0, 0, 0, 0, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: (6, 7)
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <60, 5>
G name: A5
Permutation group H acting on a set of cardinality 6
Order = 60 = 2^2 * 3 * 5
    (2, 3, 4, 5, 6)
    (1, 2)(4, 5)
generating vector: [
    (1, 6)(2, 5),
    (2, 3)(4, 6),
    (1, 2)(4, 5),
    (1, 3, 5)(2, 6, 4)
]
signature: [ 2, 2, 2, 3 ]
genus: 6
decomp H^0(K_C): [ 0, 1, 1, 0, 0 ]
N = dim S^2H^0(K_C)^G = 2
6


 b =  4 



Example  # 1
-- 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.1 * G.2, G.1^2 * G.3 * G.4, G.1^2 * G.3 * G.4, G.1 * G.3 
* G.4 ]
signature: [ 3, 3, 3, 6 ]
genus: 11
decomp H^0(K_C~): [ 0, 1, 1, 2, 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 <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.1 * H.2, H.1^2 * H.3, H.1^2 * H.3, H.1 * H.3 ]
signature: [ 3, 3, 3, 3 ]
genus: 5
decomp H^0(K_C): [ 0, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 2
-- C~ --
G~ Id: SmallGroup <32, 11>
G~ name: C4wrC2
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.1^2 = G.4, 
    G.3^2 = G.5, 
    G.4^2 = G.5, 
    G.2^G.1 = G.2 * G.3, 
    G.3^G.1 = G.3 * G.5, 
    G.3^G.2 = G.3 * G.5
generating vector: [ G.2 * G.3 * G.5, G.3 * G.4 * G.5, G.1 * G.2 * G.4 * G.5, 
G.1 * G.3 * G.4 * G.5 ]
signature: [ 2, 2, 4, 8 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 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, 3>
G name: C2^2:C4
GrpPC : H of order 16 = 2^4
PC-Relations:
    H.1^2 = H.4, 
    H.2^H.1 = H.2 * H.3
generating vector: [ H.2 * H.3, H.3 * H.4, H.1 * H.2 * H.4, H.1 * H.3 * H.4 ]
signature: [ 2, 2, 4, 4 ]
genus: 5
decomp H^0(K_C): [ 0, 0, 0, 0, 1, 0, 1, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 2
2


 b =  12 



Example  # 1
-- C~ --
G~ Id: SmallGroup <24, 7>
G~ name: C2*C3:C4
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = Id(G), 
    G.3^2 = Id(G), 
    G.4^3 = Id(G), 
    G.4^G.1 = G.4^2
generating vector: [ G.2, G.1 * G.3, G.1 * G.3 * G.4^2, G.2 * G.3 * G.4 ]
signature: [ 2, 4, 4, 6 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.3
branch points: 12
verify (B1): false
verify (B2): true
-- 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.2, H.1, H.1 * H.3^2, H.2 * H.3 ]
signature: [ 2, 2, 2, 6 ]
genus: 3
decomp H^0(K_C): [ 0, 0, 0, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 2
-- C~ --
G~ Id: SmallGroup <32, 42>
G~ name: D8:C2
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.3^2 = G.5, 
    G.4^2 = G.5, 
    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.4 * G.5, G.1, G.3, G.1 * G.2 * G.3 * G.4 ]
signature: [ 2, 2, 4, 8 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.5
branch points: 12
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.4, H.1, H.3, H.1 * H.2 * H.3 * H.4 ]
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

Example  # 3
-- C~ --
G~ Id: SmallGroup <32, 44>
G~ name: SD16:C2
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.2^2 = G.5, 
    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.3 * G.5, G.1, G.2 * G.4, G.1 * G.2 * G.3 ]
signature: [ 2, 2, 4, 8 ]
genus: 11
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.5
branch points: 12
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.3, H.1, H.2 * H.4, 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
3