g~:  8
r: 4


 b =  2 



Example  # 1
-- C~ --
G~ Id: SmallGroup <10, 2>
G~ name: C10
GrpPC : G of order 10 = 2 * 5
PC-Relations:
    G.1^2 = Id(G), 
    G.2^5 = Id(G)
generating vector: [ G.2^2, G.2, G.1 * G.2, G.1 * G.2 ]
signature: [ 5, 5, 10, 10 ]
genus: 8
decomp H^0(K_C~): [ 0, 0, 0, 1, 1, 2, 1, 0, 2, 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 <5, 1>
G name: C5
GrpPC : H of order 5
PC-Relations:
    H.1^5 = Id(H)
generating vector: [ H.1^2, H.1, H.1, H.1 ]
signature: [ 5, 5, 5, 5 ]
genus: 4
decomp H^0(K_C): [ 0, 0, 1, 1, 2 ]
N = dim S^2H^0(K_C)^G = 1

Example  # 2
-- C~ --
G~ Id: SmallGroup <10, 2>
G~ name: C10
GrpPC : G of order 10 = 2 * 5
PC-Relations:
    G.1^2 = Id(G), 
    G.2^5 = Id(G)
generating vector: [ G.2^2, G.2^4, G.1 * G.2, G.1 * G.2^3 ]
signature: [ 5, 5, 10, 10 ]
genus: 8
decomp H^0(K_C~): [ 0, 0, 1, 1, 1, 2, 1, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.1
branch points: 2
verify (B1): true
verify (B2): false
-- C --
G Id: SmallGroup <5, 1>
G name: C5
GrpPC : H of order 5
PC-Relations:
    H.1^5 = Id(H)
generating vector: [ H.1^2, H.1^4, H.1, H.1^3 ]
signature: [ 5, 5, 5, 5 ]
genus: 4
decomp H^0(K_C): [ 0, 1, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 2

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.3, G.1 * G.2 * G.3^2, G.2 * G.3^2, G.1 * G.3 ]
signature: [ 3, 6, 6, 6 ]
genus: 8
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 3
sigma: G.2
branch points: 2
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.2^2, H.2^2, H.1 * H.2 ]
signature: [ 3, 6, 3, 6 ]
genus: 4
decomp H^0(K_C): [ 0, 0, 1, 1, 1, 1 ]
N = dim S^2H^0(K_C)^G = 2

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

Example  # 5
-- C~ --
G~ Id: SmallGroup <24, 10>
G~ name: C3*D4
GrpPC : G of order 24 = 2^3 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^2 = Id(G), 
    G.3^3 = Id(G), 
    G.4^2 = Id(G), 
    G.2^G.1 = G.2 * G.4
generating vector: [ G.2 * G.4, G.1, G.3, G.1 * G.2 * G.3^2 * G.4 ]
signature: [ 2, 2, 3, 12 ]
genus: 8
decomp H^0(K_C~): [ 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1 ]
N~ = dim S^2H^0(K_C~)^G~ = 2
sigma: G.4
branch points: 2
verify (B1): false
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.1, H.3, H.1 * H.2 * H.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
5