#= Group 504.156 downloaded from the LMFDB on 06 October 2026. =#
 
#= Various presentations of this group are stored in this file: 
	 GPC is polycyclic presentation, GPerm is permutation group 
	 GLZ, GLFp, GLZN, GLZq, GLFq if they exist are matrix groups 
 
 Many characteristics of the group are stored as booleans in a NamedTuple: 
	 Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, 
	 metacyclic, monomial, nilpotent, perfect, quasisimple, rational, 
	 solvable, supersolvable 
 
 The character table is stored in the dictionary chartbl_n_i, where n is 
 the order of the group and i is which group of that order it is. The 
 key "Irr" holds the table itself, as a matrix over the abelian closure 
 of Q whose rows are the irreducible characters and whose columns are 
 the conjugacy classes, in the order given by "ClassNames". 
=#
 
#= Constructions =#
GPerm = @permutation_group(9, (1,2,8,9,7,5,3), (1,9,8,6,5,4,2))
F = GF(8); al = gen(F); GLFq = matrix_group([matrix(GF(8), [[1, al^1], [al^6, 0]]), matrix(GF(8), [[1, 1], [1, 0]])])

 
#= Booleans =#
booleans_504_156 = (
    Agroup = true,
    Zgroup = false,
    abelian = false,
    almost_simple = true,
    cyclic = false,
    metabelian = false,
    metacyclic = false,
    monomial = false,
    nilpotent = false,
    perfect = true,
    quasisimple = true,
    rational = false,
    solvable = false,
    supersolvable = false,
)

 
#= Character Table =#
K, z = abelian_closure(QQ)  # z(n) is the standard primitive n-th root of unity

chartbl_504_156 = Dict{String, Any}()
chartbl_504_156["Identifier"] = "504.156"
chartbl_504_156["Size"] = 504
chartbl_504_156["NrConjugacyClasses"] = 9
chartbl_504_156["ClassNames"] = ["1A", "2A", "3A", "7A1", "7A2", "7A3", "9A1", "9A2", "9A4"]
chartbl_504_156["SizesCentralizers"] = [504, 8, 9, 7, 7, 7, 9, 9, 9]
chartbl_504_156["OrderClassRepresentatives"] = [1, 2, 3, 7, 7, 7, 9, 9, 9]
chartbl_504_156["ComputedPowerMaps"] = Dict(2 => [1, 1, 3, 5, 6, 4, 8, 9, 7], 3 => [1, 2, 1, 6, 4, 5, 3, 3, 3], 7 => [1, 2, 3, 1, 1, 1, 8, 9, 7])
chartbl_504_156["UnderlyingGroup"] = GPerm
chartbl_504_156["ConjugacyClasses"] = [perm(GPerm, [1, 2, 3, 4, 5, 6, 7, 8, 9]), perm(GPerm, [7, 3, 2, 5, 4, 6, 1, 9, 8]), perm(GPerm, [6, 1, 8, 5, 7, 2, 4, 9, 3]), perm(GPerm, [3, 1, 5, 4, 7, 6, 9, 2, 8]), perm(GPerm, [5, 3, 7, 4, 9, 6, 8, 1, 2]), perm(GPerm, [7, 5, 9, 4, 8, 6, 2, 3, 1]), perm(GPerm, [8, 3, 7, 6, 2, 9, 1, 4, 5]), perm(GPerm, [4, 7, 1, 9, 3, 5, 8, 6, 2]), perm(GPerm, [9, 8, 4, 2, 1, 3, 6, 5, 7])]
chartbl_504_156["Indicators"] = [1, 1, 1, 1, 1, 1, 1, 1, 1]
irr_504_156 = elem_type(K)[]
append!(irr_504_156, [1, 1, 1, 1, 1, 1, 1, 1, 1])
append!(irr_504_156, [7, -1, -2, 0, 0, 0, 1, 1, 1])
append!(irr_504_156, [7, -1, 1, 0, 0, 0, -1*z(9)-z(9)^-1, -1*z(9)^2-z(9)^-2, -1*z(9)^4-z(9)^-4])
append!(irr_504_156, [7, -1, 1, 0, 0, 0, -1*z(9)^2-z(9)^-2, -1*z(9)^4-z(9)^-4, -1*z(9)-z(9)^-1])
append!(irr_504_156, [7, -1, 1, 0, 0, 0, -1*z(9)^4-z(9)^-4, -1*z(9)-z(9)^-1, -1*z(9)^2-z(9)^-2])
append!(irr_504_156, [8, 0, -1, 1, 1, 1, -1, -1, -1])
append!(irr_504_156, [9, 1, 0, z(7)^3+z(7)^-3, z(7)+z(7)^-1, z(7)^2+z(7)^-2, 0, 0, 0])
append!(irr_504_156, [9, 1, 0, z(7)^2+z(7)^-2, z(7)^3+z(7)^-3, z(7)+z(7)^-1, 0, 0, 0])
append!(irr_504_156, [9, 1, 0, z(7)+z(7)^-1, z(7)^2+z(7)^-2, z(7)^3+z(7)^-3, 0, 0, 0])
chartbl_504_156["Irr"] = matrix(K, 9, 9, irr_504_156)