g~:  21
r: 4


 b =  0 



Example  # 1
-- C~ --
G~ Id: SmallGroup <32, 3>
G~ name: C4*C8
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = G.4, 
    G.3^2 = G.5
generating vector: [ G.2 * G.4 * G.5, G.2 * G.4, G.1 * G.5, G.1 * G.3 * G.4 * 
G.5 ]
signature: [ 4, 4, 8, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 
0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 7
sigma: G.4 * G.5
branch points: 0
verify (B1): true
verify (B2): false
-- C --
G Id: SmallGroup <16, 5>
G name: C2*C8
GrpPC : H of order 16 = 2^4
PC-Relations:
    H.1^2 = H.3, 
    H.2^2 = H.4, 
    H.3^2 = H.4
generating vector: [ H.2, H.2 * H.4, H.1 * H.4, H.1 * H.3 ]
signature: [ 4, 4, 8, 8 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 6

Example  # 2
-- C~ --
G~ Id: SmallGroup <32, 13>
G~ name: C8:C4
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.3, 
    G.3^2 = G.5, 
    G.2^G.1 = G.2 * G.3, 
    G.3^G.1 = G.3 * G.5
generating vector: [ G.1 * G.4, G.1 * G.4 * G.5, G.2 * G.3 * G.5, G.2 * G.4 * 
G.5 ]
signature: [ 4, 4, 8, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 1, 0, 0, 1, 0, 1, 0, 1, 2, 2, 2, 2, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 7
sigma: G.4 * G.5
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <16, 8>
G name: SD16
GrpPC : H of order 16 = 2^4
PC-Relations:
    H.1^2 = H.4, 
    H.2^2 = H.3, 
    H.3^2 = H.4, 
    H.2^H.1 = H.2 * H.3, 
    H.3^H.1 = H.3 * H.4
generating vector: [ H.1 * H.4, H.1, H.2 * H.3 * H.4, H.2 ]
signature: [ 4, 4, 8, 8 ]
genus: 11
decomp H^0(K_C): [ 0, 1, 0, 0, 1, 2, 2 ]
N = dim S^2H^0(K_C)^G = 6

Example  # 3
-- C~ --
G~ Id: SmallGroup <48, 14>
G~ name: D6:C4
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.3, 
    G.3^2 = Id(G), 
    G.4^2 = Id(G), 
    G.5^3 = Id(G), 
    G.2^G.1 = G.2 * G.4, 
    G.5^G.1 = G.5^2
generating vector: [ G.1 * G.3 * G.4 * G.5, G.1 * G.4 * G.5, G.2 * G.5^2, G.2 * 
G.5 ]
signature: [ 2, 2, 12, 12 ]
genus: 21
decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 7
sigma: G.3 * G.4
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <24, 6>
G name: D12
GrpPC : H of order 24 = 2^3 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = H.3, 
    H.3^2 = Id(H), 
    H.4^3 = Id(H), 
    H.2^H.1 = H.2 * H.3, 
    H.4^H.1 = H.4^2
generating vector: [ H.1 * H.4, H.1 * H.3 * H.4, H.2 * H.4^2, H.2 * H.4 ]
signature: [ 2, 2, 12, 12 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 1, 0, 1, 1, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 6

Example  # 4
-- C~ --
G~ Id: SmallGroup <48, 21>
G~ name: C3*C2^2:C4
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = G.5, 
    G.2^2 = Id(G), 
    G.3^3 = Id(G), 
    G.4^2 = Id(G), 
    G.5^2 = Id(G), 
    G.2^G.1 = G.2 * G.4
generating vector: [ G.2, G.2 * G.4 * G.5, G.1 * G.2 * G.3^2 * G.4, G.1 * G.2 * 
G.3 * G.4 ]
signature: [ 2, 2, 12, 12 ]
genus: 21
decomp H^0(K_C~): [ 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 
1, 1, 1, 0, 0, 1, 1, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.5
branch points: 0
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.2 * H.4, H.1 * H.2 * H.3^2 * H.4, H.1 * H.2 * H.3 * 
H.4 ]
signature: [ 2, 2, 12, 12 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 5

Example  # 5
-- C~ --
G~ Id: SmallGroup <48, 19>
G~ name: C6.D4
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = G.3, 
    G.2^2 = Id(G), 
    G.3^2 = Id(G), 
    G.4^2 = Id(G), 
    G.5^3 = Id(G), 
    G.2^G.1 = G.2 * G.4, 
    G.5^G.1 = G.5^2
generating vector: [ G.2, G.1 * G.3 * G.5^2, G.1 * G.3 * G.4 * G.5, G.2 * G.3 * 
G.4 * G.5 ]
signature: [ 2, 4, 4, 6 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 2, 1, 1, 1, 2, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.3 * G.4
branch points: 0
verify (B1): false
verify (B2): true
-- C --
G Id: SmallGroup <24, 8>
G name: C3:D4
GrpPC : H of order 24 = 2^3 * 3
PC-Relations:
    H.1^2 = H.3, 
    H.2^2 = Id(H), 
    H.3^2 = Id(H), 
    H.4^3 = Id(H), 
    H.2^H.1 = H.2 * H.3, 
    H.4^H.1 = H.4^2
generating vector: [ H.2, H.1 * H.3 * H.4^2, H.1 * H.4, H.2 * H.4 ]
signature: [ 2, 4, 4, 6 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 1, 0, 1, 0, 1, 1, 2 ]
N = dim S^2H^0(K_C)^G = 5

Example  # 6
-- C~ --
G~ Id: SmallGroup <64, 118>
G~ name: C4*D8
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.1^2 = G.5, 
    G.3^2 = G.5, 
    G.4^2 = G.6, 
    G.2^G.1 = G.2 * G.4, 
    G.4^G.1 = G.4 * G.6, 
    G.4^G.2 = G.4 * G.6
generating vector: [ G.1 * G.3, G.2 * G.6, G.3 * G.4, G.1 * G.2 * G.6 ]
signature: [ 2, 2, 4, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 
1, 1, 0, 1, 0, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 7
sigma: G.5
branch points: 0
verify (B1): true
verify (B2): true
-- C --
G Id: SmallGroup <32, 39>
G name: C2*D8
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.4^2 = H.5, 
    H.2^H.1 = H.2 * H.4, 
    H.4^H.1 = H.4 * H.5, 
    H.4^H.2 = H.4 * H.5
generating vector: [ H.1 * H.3, H.2 * H.5, H.3 * H.4, H.1 * H.2 * H.5 ]
signature: [ 2, 2, 4, 8 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 6

Example  # 7
-- C~ --
G~ Id: SmallGroup <64, 147>
G~ name: C2^2:D8
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.2^2 = G.4 * G.6, 
    G.4^2 = G.6, 
    G.2^G.1 = G.2 * G.4, 
    G.3^G.1 = G.3 * G.5, 
    G.4^G.1 = G.4 * G.6
generating vector: [ G.1 * G.4 * G.5 * G.6, G.1 * G.2 * G.5, G.3 * G.4 * G.6, 
G.2 * G.3 * G.4 ]
signature: [ 2, 2, 4, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 
0, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 7
sigma: G.5
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <32, 39>
G name: C2*D8
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.2^2 = H.4 * H.5, 
    H.4^2 = H.5, 
    H.2^H.1 = H.2 * H.4, 
    H.4^H.1 = H.4 * H.5
generating vector: [ H.1 * H.4 * H.5, H.1 * H.2, H.3 * H.4 * H.5, H.2 * H.3 * 
H.4 ]
signature: [ 2, 2, 4, 8 ]
genus: 11
decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 6

Example  # 8
-- C~ --
G~ Id: SmallGroup <64, 149>
G~ name: C4.(C2*D4)
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.2^2 = G.4, 
    G.4^2 = G.6, 
    G.2^G.1 = G.2 * G.4, 
    G.3^G.1 = G.3 * G.5, 
    G.3^G.2 = G.3 * G.6, 
    G.4^G.1 = G.4 * G.6
generating vector: [ G.1 * G.4, G.3 * G.5, G.1 * G.2, G.2 * G.3 ]
signature: [ 2, 2, 4, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.5
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <32, 43>
G name: C8:C2^2
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.2^2 = H.4, 
    H.4^2 = H.5, 
    H.2^H.1 = H.2 * H.4, 
    H.3^H.2 = H.3 * H.5, 
    H.4^H.1 = H.4 * H.5
generating vector: [ H.1 * H.4, H.3, H.1 * H.2, H.2 * H.3 ]
signature: [ 2, 2, 4, 8 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2 ]
N = dim S^2H^0(K_C)^G = 5

Example  # 9
-- C~ --
G~ Id: SmallGroup <64, 163>
G~ name: C2.D8:C2
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.2^2 = G.6, 
    G.3^2 = G.4, 
    G.5^2 = G.6, 
    G.2^G.1 = G.2 * G.4, 
    G.3^G.1 = G.3 * G.5, 
    G.5^G.1 = G.5 * G.6, 
    G.5^G.3 = G.5 * G.6
generating vector: [ G.2 * G.4 * G.5 * G.6, G.1 * G.6, G.3 * G.4, G.1 * G.2 * 
G.3 * G.4 * G.6 ]
signature: [ 2, 2, 4, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.4 * G.6
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <32, 43>
G name: C8:C2^2
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.2^2 = H.5, 
    H.3^2 = H.5, 
    H.4^2 = H.5, 
    H.2^H.1 = H.2 * H.5, 
    H.3^H.1 = H.3 * H.4, 
    H.4^H.1 = H.4 * H.5, 
    H.4^H.3 = H.4 * H.5
generating vector: [ H.2 * H.4, H.1 * H.5, H.3 * H.5, H.1 * H.2 * H.3 ]
signature: [ 2, 2, 4, 8 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2 ]
N = dim S^2H^0(K_C)^G = 5

Example  # 10
-- C~ --
G~ Id: SmallGroup <96, 91>
G~ name: C3:C4:D4
GrpPC : G of order 96 = 2^5 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.4, 
    G.3^2 = Id(G), 
    G.4^2 = Id(G), 
    G.5^2 = Id(G), 
    G.6^3 = Id(G), 
    G.3^G.1 = G.3 * G.4 * G.5, 
    G.3^G.2 = G.3 * G.5, 
    G.6^G.1 = G.6^2
generating vector: [ G.3 * G.5, G.1 * G.4 * G.6^2, G.1 * G.2 * G.3 * G.6, G.2 * 
G.6 ]
signature: [ 2, 2, 2, 12 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 
0, 1, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.4 * G.5
branch points: 0
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <48, 38>
G name: S3*D4
GrpPC : H of order 48 = 2^4 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = H.4, 
    H.3^2 = Id(H), 
    H.4^2 = Id(H), 
    H.5^3 = Id(H), 
    H.3^H.2 = H.3 * H.4, 
    H.5^H.1 = H.5^2
generating vector: [ H.3 * H.4, H.1 * H.4 * H.5^2, H.1 * H.2 * H.3 * H.5, H.2 * 
H.5 ]
signature: [ 2, 2, 2, 12 ]
genus: 11
decomp H^0(K_C): [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 5
10


 b =  8 



Example  # 1
-- C~ --
G~ Id: SmallGroup <48, 4>
G~ name: C8*S3
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.3, 
    G.3^2 = G.4, 
    G.4^2 = Id(G), 
    G.5^3 = Id(G), 
    G.5^G.1 = G.5^2
generating vector: [ G.1, G.1 * G.4 * G.5, G.2, G.2 * G.3 * G.5^2 ]
signature: [ 2, 2, 8, 24 ]
genus: 21
decomp H^0(K_C~): [ 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 
1, 1, 0, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 4
sigma: G.4
branch points: 8
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <24, 5>
G name: C4*S3
GrpPC : H of order 24 = 2^3 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = H.3, 
    H.3^2 = Id(H), 
    H.4^3 = Id(H), 
    H.4^H.1 = H.4^2
generating vector: [ H.1, H.1 * H.4, H.2, H.2 * H.3 * H.4^2 ]
signature: [ 2, 2, 4, 12 ]
genus: 9
decomp H^0(K_C): [ 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 2 ]
N = dim S^2H^0(K_C)^G = 3

Example  # 2
-- C~ --
G~ Id: SmallGroup <48, 6>
G~ name: C8:S3
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.3, 
    G.3^2 = G.4, 
    G.4^2 = Id(G), 
    G.5^3 = Id(G), 
    G.2^G.1 = G.2 * G.3, 
    G.3^G.1 = G.3 * G.4, 
    G.5^G.1 = G.5^2
generating vector: [ G.1 * G.3 * G.4 * G.5^2, G.1 * G.4, G.2, G.2 * G.5^2 ]
signature: [ 2, 2, 8, 24 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.4
branch points: 8
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <24, 6>
G name: D12
GrpPC : H of order 24 = 2^3 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = H.3, 
    H.3^2 = Id(H), 
    H.4^3 = Id(H), 
    H.2^H.1 = H.2 * H.3, 
    H.4^H.1 = H.4^2
generating vector: [ H.1 * H.3 * H.4^2, H.1, H.2, H.2 * H.4^2 ]
signature: [ 2, 2, 4, 12 ]
genus: 9
decomp H^0(K_C): [ 0, 0, 1, 0, 0, 1, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 5

Example  # 3
-- C~ --
G~ Id: SmallGroup <64, 124>
G~ name: (C4*C8):C2
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.1^2 = G.5, 
    G.3^2 = G.5, 
    G.4^2 = G.6, 
    G.5^2 = G.6, 
    G.2^G.1 = G.2 * G.4, 
    G.4^G.1 = G.4 * G.6, 
    G.4^G.2 = G.4 * G.6
generating vector: [ G.2 * G.6, G.1 * G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.4 * 
G.5 * G.6, G.3 ]
signature: [ 2, 2, 4, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 
1, 0, 1, 0, 1, 2, 1, 0 ]
N~ = dim S^2H^0(K_C~)^G~ = 4
sigma: G.6
branch points: 8
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <32, 25>
G name: C4*D4
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.1^2 = H.5, 
    H.3^2 = H.5, 
    H.2^H.1 = H.2 * H.4
generating vector: [ H.2, H.1 * H.3 * H.4 * H.5, H.1 * H.2 * H.4 * H.5, H.3 ]
signature: [ 2, 2, 4, 4 ]
genus: 9
decomp H^0(K_C): [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1 ]
N = dim S^2H^0(K_C)^G = 3

Example  # 4
-- C~ --
G~ Id: SmallGroup <64, 133>
G~ name: (C2*Q16):C2
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.1^2 = G.6, 
    G.3^2 = G.6, 
    G.4^2 = G.6, 
    G.2^G.1 = G.2 * G.4, 
    G.3^G.1 = G.3 * G.5, 
    G.4^G.1 = G.4 * G.6, 
    G.4^G.2 = G.4 * G.6
generating vector: [ G.2 * G.6, G.3 * G.4, G.1 * G.3 * G.4 * G.6, G.1 * G.2 * 
G.5 ]
signature: [ 2, 2, 4, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.6
branch points: 8
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <32, 27>
G name: C2^2wrC2
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.2^H.1 = H.2 * H.4, 
    H.3^H.1 = H.3 * H.5
generating vector: [ H.2, H.3 * H.4, H.1 * H.3 * H.4, H.1 * H.2 * H.5 ]
signature: [ 2, 2, 4, 4 ]
genus: 9
decomp H^0(K_C): [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1 ]
N = dim S^2H^0(K_C)^G = 5

Example  # 5
-- C~ --
G~ Id: SmallGroup <64, 176>
G~ name: C8.D4
GrpPC : G of order 64 = 2^6
PC-Relations:
    G.2^2 = G.4, 
    G.3^2 = G.5 * G.6, 
    G.4^2 = G.6, 
    G.2^G.1 = G.2 * G.4, 
    G.3^G.1 = G.3 * G.5, 
    G.4^G.1 = G.4 * G.6
generating vector: [ G.1 * G.2 * G.3, G.1, G.3 * G.4 * G.5, G.2 * G.4 * G.5 * 
G.6 ]
signature: [ 2, 2, 4, 8 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 
0, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 6
sigma: G.6
branch points: 8
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <32, 34>
G name: C4:D4
GrpPC : H of order 32 = 2^5
PC-Relations:
    H.2^2 = H.4, 
    H.3^2 = H.5, 
    H.2^H.1 = H.2 * H.4, 
    H.3^H.1 = H.3 * H.5
generating vector: [ H.1 * H.2 * H.3, H.1, H.3 * H.4 * H.5, H.2 * H.4 * H.5 ]
signature: [ 2, 2, 4, 4 ]
genus: 9
decomp H^0(K_C): [ 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0 ]
N = dim S^2H^0(K_C)^G = 5

Example  # 6
-- C~ --
G~ Id: SmallGroup <96, 192>
G~ name: (C2.S4):C2
GrpPC : G of order 96 = 2^5 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = G.6, 
    G.3^3 = Id(G), 
    G.4^2 = G.6, 
    G.5^2 = G.6, 
    G.6^2 = Id(G), 
    G.3^G.1 = G.3^2, 
    G.4^G.1 = G.5, 
    G.4^G.3 = G.5 * G.6, 
    G.5^G.1 = G.4, 
    G.5^G.3 = G.4 * G.5, 
    G.5^G.4 = G.5 * G.6
generating vector: [ G.2 * G.4 * G.5 * G.6, G.1 * G.3 * G.6, G.1 * G.6, G.2 * 
G.3 * G.4 * G.5 * G.6 ]
signature: [ 2, 2, 2, 12 ]
genus: 21
decomp H^0(K_C~): [ 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 5
sigma: G.6
branch points: 8
verify (B1): false
verify (B2): false
-- C --
G Id: SmallGroup <48, 48>
G name: C2*S4
GrpPC : H of order 48 = 2^4 * 3
PC-Relations:
    H.1^2 = Id(H), 
    H.2^2 = Id(H), 
    H.3^3 = Id(H), 
    H.4^2 = Id(H), 
    H.5^2 = Id(H), 
    H.3^H.1 = H.3^2, 
    H.4^H.1 = H.5, 
    H.4^H.3 = H.5, 
    H.5^H.1 = H.4, 
    H.5^H.3 = H.4 * H.5
generating vector: [ H.2 * H.4 * H.5, H.1 * H.3, H.1, H.2 * H.3 * H.4 * H.5 ]
signature: [ 2, 2, 2, 6 ]
genus: 9
decomp H^0(K_C): [ 0, 0, 0, 1, 1, 0, 0, 1, 0, 1 ]
N = dim S^2H^0(K_C)^G = 4
6