g~:  15
r: 4


 b =  0 



Example  # 1
-- 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.4, G.2 * G.3, G.1 * G.3 * G.4, G.1 * G.2 * G.3 * G.4 ]
signature: [ 2, 6, 12, 12 ]
genus: 15
decomp H^0(K_C~): [ 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 
0, 0, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 4
sigma: G.2 * G.4
branch points: 0
verify (B1): true
verify (B2): false
-- 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.3, H.1 * H.2 * H.3, H.1 * H.2 ]
signature: [ 2, 6, 12, 12 ]
genus: 8
decomp H^0(K_C): [ 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 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.1, G.1 * G.2 * G.4, G.2 * G.3^2 * G.4, G.3 * G.4 ]
signature: [ 4, 4, 6, 6 ]
genus: 15
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 0, 1, 1, 0, 
1, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 5
sigma: G.2
branch points: 0
verify (B1): true
verify (B2): false
-- 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.1, H.1 * H.3, H.2^2 * H.3, H.2 * H.3 ]
signature: [ 4, 4, 6, 6 ]
genus: 8
decomp H^0(K_C): [ 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 4
2


 b =  4 



Example  # 1
-- C~ --
G~ Id: SmallGroup <16, 1>
G~ name: C16
GrpPC : G of order 16 = 2^4
PC-Relations:
    G.1^2 = G.2, 
    G.2^2 = G.3, 
    G.3^2 = G.4
generating vector: [ G.1 * G.4, G.1 * G.4, G.1 * G.3, G.1 * G.4 ]
signature: [ 16, 16, 16, 16 ]
genus: 15
decomp H^0(K_C~): [ 0, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 0, 2, 0, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.4
branch points: 4
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.1, H.1, H.1 * H.3, H.1 ]
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

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.2 * G.3^2 * G.4, G.1 * G.3^2 * G.4, G.1 * G.3^2 * 
G.4 ]
signature: [ 2, 6, 12, 12 ]
genus: 15
decomp H^0(K_C~): [ 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 
1, 2, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.4
branch points: 4
verify (B1): true
verify (B2): false
-- C --
G Id: SmallGroup <12, 5>
G name: C2*C6
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)
generating vector: [ H.2, H.2 * H.3^2, H.1 * H.3^2, H.1 * H.3^2 ]
signature: [ 2, 6, 6, 6 ]
genus: 7
decomp H^0(K_C): [ 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2 ]
N = dim S^2H^0(K_C)^G = 2

Example  # 3
-- C~ --
G~ Id: SmallGroup <32, 19>
G~ name: SD32
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.1^2 = G.5, 
    G.3^2 = G.4 * G.5, 
    G.4^2 = G.5, 
    G.2^G.1 = G.2 * G.3, 
    G.3^G.1 = G.3 * G.4, 
    G.3^G.2 = G.3 * 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.2 * G.3, G.1 * G.2, G.1 * G.2 * G.4 * G.5 ]
signature: [ 2, 2, 16, 16 ]
genus: 15
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 5
sigma: G.5
branch points: 4
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <16, 7>
G name: D8
GrpPC : H of order 16 = 2^4
PC-Relations:
    H.3^2 = H.4, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.4, 
    H.3^H.2 = H.3 * H.4
generating vector: [ H.2 * H.4, H.2 * H.3, H.1 * H.2, H.1 * H.2 * H.4 ]
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 = 4
3


 b =  12 



Example  # 1
-- C~ --
G~ Id: SmallGroup <40, 5>
G~ name: C4*D5
GrpPC : G of order 40 = 2^3 * 5
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.3, 
    G.3^2 = Id(G), 
    G.4^5 = Id(G), 
    G.4^G.1 = G.4^4
generating vector: [ G.1 * G.3, G.1 * G.4^4, G.2 * G.3, G.2 * G.3 * G.4 ]
signature: [ 2, 2, 4, 20 ]
genus: 15
decomp H^0(K_C~): [ 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 2, 1, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 4
sigma: G.3
branch points: 12
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <20, 4>
G name: D10
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), 
    H.3^H.1 = H.3^4
generating vector: [ H.1, H.1 * H.3^4, H.2, H.2 * H.3 ]
signature: [ 2, 2, 2, 10 ]
genus: 5
decomp H^0(K_C): [ 0, 0, 0, 1, 1, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 3

Example  # 2
-- C~ --
G~ Id: SmallGroup <48, 37>
G~ name: D12:C2
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.4, 
    G.3^2 = Id(G), 
    G.4^2 = Id(G), 
    G.5^3 = Id(G), 
    G.3^G.1 = G.3 * G.4, 
    G.5^G.1 = G.5^2
generating vector: [ G.1, G.1 * G.2 * G.3 * G.5, G.2, G.3 * G.4 * G.5^2 ]
signature: [ 2, 2, 4, 6 ]
genus: 15
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 4
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, H.2, H.3 * H.4^2 ]
signature: [ 2, 2, 2, 6 ]
genus: 5
decomp H^0(K_C): [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 3

Example  # 3
-- C~ --
G~ Id: SmallGroup <48, 39>
G~ name: D4:S3
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.4, 
    G.3^2 = Id(G), 
    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.5^2, G.2 * G.3 * G.4, G.1 * G.2, G.3 * G.4 * 
G.5^2 ]
signature: [ 2, 2, 4, 6 ]
genus: 15
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 4
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.4^2, H.2 * H.3, H.1 * H.2, H.3 * H.4^2 ]
signature: [ 2, 2, 2, 6 ]
genus: 5
decomp H^0(K_C): [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0 ]
N = dim S^2H^0(K_C)^G = 3
3