# Group 504.156 downloaded from the LMFDB on 16 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 dict: 
#	 Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, 
#	 metacyclic, monomial, nilpotent, perfect, quasisimple, rational, 
#	 solvable, supersolvable 
 
# The character table is stored in the dict 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 universal cyclotomic 
# field whose rows are the irreducible characters and whose columns are 
# the conjugacy classes, in the order given by 'ClassNames'. 

 
# Constructions 
GPerm = PermutationGroup(['(1,2,8,9,7,5,3)', '(1,9,8,6,5,4,2)'])
F = GF(8); al = F.0; MS = MatrixSpace(F, 2, 2)
GLFq = MatrixGroup([MS([[1, al^1], [al^6, 0]]), MS([[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 
UCF = UniversalCyclotomicField()
E = UCF.gen  # E(n) is the standard primitive n-th root of unity

chartbl_504_156 = {}
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"] = {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"] = [GPerm.one(), GPerm('(1,7)(2,3)(4,5)(8,9)'), GPerm('(1,6,2)(3,8,9)(4,5,7)'), GPerm('(1,3,5,7,9,8,2)'), GPerm('(1,5,9,2,3,7,8)'), GPerm('(1,7,2,5,8,3,9)'), GPerm('(1,8,4,6,9,5,2,3,7)'), GPerm('(1,4,9,2,7,8,6,5,3)'), GPerm('(1,9,7,6,3,4,2,8,5)')]
chartbl_504_156["Indicators"] = [1, 1, 1, 1, 1, 1, 1, 1, 1]
irr_504_156 = []
irr_504_156 += [1, 1, 1, 1, 1, 1, 1, 1, 1]
irr_504_156 += [7, -1, -2, 0, 0, 0, 1, 1, 1]
irr_504_156 += [7, -1, 1, 0, 0, 0, -1*E(9)-E(9)^-1, -1*E(9)^2-E(9)^-2, -1*E(9)^4-E(9)^-4]
irr_504_156 += [7, -1, 1, 0, 0, 0, -1*E(9)^2-E(9)^-2, -1*E(9)^4-E(9)^-4, -1*E(9)-E(9)^-1]
irr_504_156 += [7, -1, 1, 0, 0, 0, -1*E(9)^4-E(9)^-4, -1*E(9)-E(9)^-1, -1*E(9)^2-E(9)^-2]
irr_504_156 += [8, 0, -1, 1, 1, 1, -1, -1, -1]
irr_504_156 += [9, 1, 0, E(7)^3+E(7)^-3, E(7)+E(7)^-1, E(7)^2+E(7)^-2, 0, 0, 0]
irr_504_156 += [9, 1, 0, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, E(7)+E(7)^-1, 0, 0, 0]
irr_504_156 += [9, 1, 0, E(7)+E(7)^-1, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, 0, 0, 0]
chartbl_504_156["Irr"] = Matrix(UCF, 9, 9, irr_504_156)