# Group 208.28 downloaded from the LMFDB on 26 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 
# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups
GPC = gap.new('PcGroupCode(32852327950982229139626672485387,208)'); a = GPC.1; b = GPC.3;
GPerm = PermutationGroup(['(2,3,4,6)(5,8,9,13)(7,11,12,10)(14,15,16,18,17,19,20,21)', '(14,16,17,20)(15,18,19,21)', '(2,4)(3,6)(5,9)(7,12)(8,13)(10,11)(14,16,17,20)(15,18,19,21)', '(14,17)(15,19)(16,20)(18,21)', '(1,2,5,8,12,6,10,11,3,7,13,9,4)'])
MS = MatrixSpace(GF(5), 4, 4)
GLFp = MatrixGroup([MS([[4, 0, 0, 0], [0, 4, 0, 0], [0, 0, 4, 0], [0, 0, 0, 4]]), MS([[0, 4, 3, 4], [2, 3, 2, 1], [1, 3, 1, 4], [2, 1, 1, 1]]), MS([[2, 3, 2, 0], [3, 2, 3, 4], [0, 4, 0, 1], [0, 0, 3, 0]]), MS([[3, 1, 1, 0], [0, 3, 3, 2], [1, 0, 1, 4], [0, 4, 3, 3]]), MS([[2, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]])])

 
# Booleans 
booleans_208_28 = {
    "Agroup": True,
    "Zgroup": False,
    "abelian": False,
    "almost_simple": False,
    "cyclic": False,
    "metabelian": True,
    "metacyclic": True,
    "monomial": True,
    "nilpotent": False,
    "perfect": False,
    "quasisimple": False,
    "rational": False,
    "solvable": True,
    "supersolvable": True
}

 
# Character Table 
UCF = UniversalCyclotomicField()
E = UCF.gen  # E(n) is the standard primitive n-th root of unity

chartbl_208_28 = {}
chartbl_208_28["Identifier"] = "208.28"
chartbl_208_28["Size"] = 208
chartbl_208_28["NrConjugacyClasses"] = 28
chartbl_208_28["ClassNames"] = ["1A", "2A", "2B", "2C", "4A1", "4A-1", "4B1", "4B-1", "8A1", "8A-1", "8A3", "8A-3", "8B1", "8B-1", "8B3", "8B-3", "13A1", "13A2", "13A4", "26A1", "26A3", "26A7", "52A1", "52A-1", "52A3", "52A-3", "52A7", "52A-7"]
chartbl_208_28["SizesCentralizers"] = [208, 208, 16, 16, 208, 208, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52]
chartbl_208_28["OrderClassRepresentatives"] = [1, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 26, 26, 26, 52, 52, 52, 52, 52, 52]
chartbl_208_28["ComputedPowerMaps"] = {2: [1, 1, 1, 1, 2, 2, 2, 2, 7, 8, 8, 7, 8, 7, 7, 8, 18, 19, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22], 13: [1, 2, 3, 4, 5, 6, 7, 8, 12, 11, 10, 9, 16, 15, 14, 13, 1, 1, 1, 2, 2, 2, 5, 6, 6, 5, 6, 5]}
chartbl_208_28["UnderlyingGroup"] = GPC
chartbl_208_28["ConjugacyClasses"] = [GPC.Identity(), b^26, a^2*b, a^2*b^3, b^13, b^39, a^2, a^2*b^26, a, a^3*b^2, a^3, a*b^2, a*b, a^3*b, a^3*b^3, a*b^3, b^4, b^8, b^16, b^2, b^6, b^14, b, b^27, b^3, b^29, b^7, b^9]
chartbl_208_28["Indicators"] = [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]
irr_208_28 = []
irr_208_28 += [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
irr_208_28 += [1, 1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1]
irr_208_28 += [1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1]
irr_208_28 += [1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
irr_208_28 += [1, 1, -1, -1, 1, 1, -1, -1, -1*E(4), E(4), -1*E(4), E(4), E(4), E(4), -1*E(4), -1*E(4), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
irr_208_28 += [1, 1, -1, -1, 1, 1, -1, -1, E(4), -1*E(4), E(4), -1*E(4), -1*E(4), -1*E(4), E(4), E(4), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
irr_208_28 += [1, 1, 1, 1, -1, -1, -1, -1, -1*E(4), E(4), E(4), -1*E(4), E(4), -1*E(4), E(4), -1*E(4), 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1]
irr_208_28 += [1, 1, 1, 1, -1, -1, -1, -1, E(4), -1*E(4), -1*E(4), E(4), -1*E(4), E(4), -1*E(4), E(4), 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1]
irr_208_28 += [1, -1, -1, 1, -1*E(8)^2, E(8)^2, -1*E(8)^2, E(8)^2, E(8)^3, E(8), -1*E(8), -1*E(8)^3, -1*E(8), E(8)^3, E(8), -1*E(8)^3, 1, 1, 1, -1, -1, -1, -1*E(8)^2, E(8)^2, E(8)^2, E(8)^2, -1*E(8)^2, -1*E(8)^2]
irr_208_28 += [1, -1, -1, 1, E(8)^2, -1*E(8)^2, E(8)^2, -1*E(8)^2, -1*E(8), -1*E(8)^3, E(8)^3, E(8), E(8)^3, -1*E(8), -1*E(8)^3, E(8), 1, 1, 1, -1, -1, -1, E(8)^2, -1*E(8)^2, -1*E(8)^2, -1*E(8)^2, E(8)^2, E(8)^2]
irr_208_28 += [1, -1, -1, 1, -1*E(8)^2, E(8)^2, -1*E(8)^2, E(8)^2, -1*E(8)^3, -1*E(8), E(8), E(8)^3, E(8), -1*E(8)^3, -1*E(8), E(8)^3, 1, 1, 1, -1, -1, -1, -1*E(8)^2, E(8)^2, E(8)^2, E(8)^2, -1*E(8)^2, -1*E(8)^2]
irr_208_28 += [1, -1, -1, 1, E(8)^2, -1*E(8)^2, E(8)^2, -1*E(8)^2, E(8), E(8)^3, -1*E(8)^3, -1*E(8), -1*E(8)^3, E(8), E(8)^3, -1*E(8), 1, 1, 1, -1, -1, -1, E(8)^2, -1*E(8)^2, -1*E(8)^2, -1*E(8)^2, E(8)^2, E(8)^2]
irr_208_28 += [1, -1, 1, -1, -1*E(8)^2, E(8)^2, E(8)^2, -1*E(8)^2, -1*E(8), -1*E(8)^3, -1*E(8)^3, -1*E(8), E(8)^3, E(8), E(8)^3, E(8), 1, 1, 1, -1, -1, -1, -1*E(8)^2, E(8)^2, E(8)^2, E(8)^2, -1*E(8)^2, -1*E(8)^2]
irr_208_28 += [1, -1, 1, -1, E(8)^2, -1*E(8)^2, -1*E(8)^2, E(8)^2, E(8)^3, E(8), E(8), E(8)^3, -1*E(8), -1*E(8)^3, -1*E(8), -1*E(8)^3, 1, 1, 1, -1, -1, -1, E(8)^2, -1*E(8)^2, -1*E(8)^2, -1*E(8)^2, E(8)^2, E(8)^2]
irr_208_28 += [1, -1, 1, -1, -1*E(8)^2, E(8)^2, E(8)^2, -1*E(8)^2, E(8), E(8)^3, E(8)^3, E(8), -1*E(8)^3, -1*E(8), -1*E(8)^3, -1*E(8), 1, 1, 1, -1, -1, -1, -1*E(8)^2, E(8)^2, E(8)^2, E(8)^2, -1*E(8)^2, -1*E(8)^2]
irr_208_28 += [1, -1, 1, -1, E(8)^2, -1*E(8)^2, -1*E(8)^2, E(8)^2, -1*E(8)^3, -1*E(8), -1*E(8), -1*E(8)^3, E(8), E(8)^3, E(8), E(8)^3, 1, 1, 1, -1, -1, -1, E(8)^2, -1*E(8)^2, -1*E(8)^2, -1*E(8)^2, E(8)^2, E(8)^2]
irr_208_28 += [4, 4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2]
irr_208_28 += [4, 4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4]
irr_208_28 += [4, 4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1]
irr_208_28 += [4, 4, 0, 0, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, -1*E(13)^4-E(13)^6-E(13)^-6-E(13)^-4, -1*E(13)^2-E(13)^3-E(13)^-3-E(13)^-2, -1*E(13)-E(13)^5-E(13)^-5-E(13)^-1, -1*E(13)^4-E(13)^6-E(13)^-6-E(13)^-4, -1*E(13)-E(13)^5-E(13)^-5-E(13)^-1, -1*E(13)^2-E(13)^3-E(13)^-3-E(13)^-2]
irr_208_28 += [4, 4, 0, 0, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, -1*E(13)-E(13)^5-E(13)^-5-E(13)^-1, -1*E(13)^4-E(13)^6-E(13)^-6-E(13)^-4, -1*E(13)^2-E(13)^3-E(13)^-3-E(13)^-2, -1*E(13)-E(13)^5-E(13)^-5-E(13)^-1, -1*E(13)^2-E(13)^3-E(13)^-3-E(13)^-2, -1*E(13)^4-E(13)^6-E(13)^-6-E(13)^-4]
irr_208_28 += [4, 4, 0, 0, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, E(13)^2+E(13)^3+E(13)^-3+E(13)^-2, E(13)^4+E(13)^6+E(13)^-6+E(13)^-4, E(13)+E(13)^5+E(13)^-5+E(13)^-1, -1*E(13)^2-E(13)^3-E(13)^-3-E(13)^-2, -1*E(13)-E(13)^5-E(13)^-5-E(13)^-1, -1*E(13)^4-E(13)^6-E(13)^-6-E(13)^-4, -1*E(13)^2-E(13)^3-E(13)^-3-E(13)^-2, -1*E(13)^4-E(13)^6-E(13)^-6-E(13)^-4, -1*E(13)-E(13)^5-E(13)^-5-E(13)^-1]
irr_208_28 += [4, -4, 0, 0, -4*E(52)^13, 4*E(52)^13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1-E(52)^4+E(52)^6-E(52)^8-E(52)^12+E(52)^14+E(52)^18-E(52)^20+E(52)^22, E(52)^4-E(52)^6+E(52)^20-E(52)^22, E(52)^8+E(52)^12-E(52)^14-E(52)^18, 1+E(52)^4-E(52)^6+E(52)^8+E(52)^12-E(52)^14-E(52)^18+E(52)^20-E(52)^22, -1*E(52)^4+E(52)^6-E(52)^20+E(52)^22, -1*E(52)^8-E(52)^12+E(52)^14+E(52)^18, E(52)^3+E(52)^11+E(52)^15+E(52)^23, E(52)^3+E(52)^7-E(52)^9+E(52)^11-E(52)^13+E(52)^15-E(52)^17+E(52)^19+E(52)^23, -1*E(52)^7+E(52)^9+E(52)^17-E(52)^19, -1*E(52)^3-E(52)^11-E(52)^15-E(52)^23, E(52)^7-E(52)^9-E(52)^17+E(52)^19, -1*E(52)^3-E(52)^7+E(52)^9-E(52)^11+E(52)^13-E(52)^15+E(52)^17-E(52)^19-E(52)^23]
irr_208_28 += [4, -4, 0, 0, 4*E(52)^13, -4*E(52)^13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1-E(52)^4+E(52)^6-E(52)^8-E(52)^12+E(52)^14+E(52)^18-E(52)^20+E(52)^22, E(52)^4-E(52)^6+E(52)^20-E(52)^22, E(52)^8+E(52)^12-E(52)^14-E(52)^18, 1+E(52)^4-E(52)^6+E(52)^8+E(52)^12-E(52)^14-E(52)^18+E(52)^20-E(52)^22, -1*E(52)^4+E(52)^6-E(52)^20+E(52)^22, -1*E(52)^8-E(52)^12+E(52)^14+E(52)^18, -1*E(52)^3-E(52)^11-E(52)^15-E(52)^23, -1*E(52)^3-E(52)^7+E(52)^9-E(52)^11+E(52)^13-E(52)^15+E(52)^17-E(52)^19-E(52)^23, E(52)^7-E(52)^9-E(52)^17+E(52)^19, E(52)^3+E(52)^11+E(52)^15+E(52)^23, -1*E(52)^7+E(52)^9+E(52)^17-E(52)^19, E(52)^3+E(52)^7-E(52)^9+E(52)^11-E(52)^13+E(52)^15-E(52)^17+E(52)^19+E(52)^23]
irr_208_28 += [4, -4, 0, 0, -4*E(52)^13, 4*E(52)^13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(52)^4-E(52)^6+E(52)^20-E(52)^22, E(52)^8+E(52)^12-E(52)^14-E(52)^18, -1-E(52)^4+E(52)^6-E(52)^8-E(52)^12+E(52)^14+E(52)^18-E(52)^20+E(52)^22, -1*E(52)^4+E(52)^6-E(52)^20+E(52)^22, -1*E(52)^8-E(52)^12+E(52)^14+E(52)^18, 1+E(52)^4-E(52)^6+E(52)^8+E(52)^12-E(52)^14-E(52)^18+E(52)^20-E(52)^22, E(52)^7-E(52)^9-E(52)^17+E(52)^19, -1*E(52)^3-E(52)^11-E(52)^15-E(52)^23, E(52)^3+E(52)^7-E(52)^9+E(52)^11-E(52)^13+E(52)^15-E(52)^17+E(52)^19+E(52)^23, -1*E(52)^7+E(52)^9+E(52)^17-E(52)^19, -1*E(52)^3-E(52)^7+E(52)^9-E(52)^11+E(52)^13-E(52)^15+E(52)^17-E(52)^19-E(52)^23, E(52)^3+E(52)^11+E(52)^15+E(52)^23]
irr_208_28 += [4, -4, 0, 0, 4*E(52)^13, -4*E(52)^13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(52)^4-E(52)^6+E(52)^20-E(52)^22, E(52)^8+E(52)^12-E(52)^14-E(52)^18, -1-E(52)^4+E(52)^6-E(52)^8-E(52)^12+E(52)^14+E(52)^18-E(52)^20+E(52)^22, -1*E(52)^4+E(52)^6-E(52)^20+E(52)^22, -1*E(52)^8-E(52)^12+E(52)^14+E(52)^18, 1+E(52)^4-E(52)^6+E(52)^8+E(52)^12-E(52)^14-E(52)^18+E(52)^20-E(52)^22, -1*E(52)^7+E(52)^9+E(52)^17-E(52)^19, E(52)^3+E(52)^11+E(52)^15+E(52)^23, -1*E(52)^3-E(52)^7+E(52)^9-E(52)^11+E(52)^13-E(52)^15+E(52)^17-E(52)^19-E(52)^23, E(52)^7-E(52)^9-E(52)^17+E(52)^19, E(52)^3+E(52)^7-E(52)^9+E(52)^11-E(52)^13+E(52)^15-E(52)^17+E(52)^19+E(52)^23, -1*E(52)^3-E(52)^11-E(52)^15-E(52)^23]
irr_208_28 += [4, -4, 0, 0, -4*E(52)^13, 4*E(52)^13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(52)^8+E(52)^12-E(52)^14-E(52)^18, -1-E(52)^4+E(52)^6-E(52)^8-E(52)^12+E(52)^14+E(52)^18-E(52)^20+E(52)^22, E(52)^4-E(52)^6+E(52)^20-E(52)^22, -1*E(52)^8-E(52)^12+E(52)^14+E(52)^18, 1+E(52)^4-E(52)^6+E(52)^8+E(52)^12-E(52)^14-E(52)^18+E(52)^20-E(52)^22, -1*E(52)^4+E(52)^6-E(52)^20+E(52)^22, -1*E(52)^3-E(52)^7+E(52)^9-E(52)^11+E(52)^13-E(52)^15+E(52)^17-E(52)^19-E(52)^23, -1*E(52)^7+E(52)^9+E(52)^17-E(52)^19, -1*E(52)^3-E(52)^11-E(52)^15-E(52)^23, E(52)^3+E(52)^7-E(52)^9+E(52)^11-E(52)^13+E(52)^15-E(52)^17+E(52)^19+E(52)^23, E(52)^3+E(52)^11+E(52)^15+E(52)^23, E(52)^7-E(52)^9-E(52)^17+E(52)^19]
irr_208_28 += [4, -4, 0, 0, 4*E(52)^13, -4*E(52)^13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(52)^8+E(52)^12-E(52)^14-E(52)^18, -1-E(52)^4+E(52)^6-E(52)^8-E(52)^12+E(52)^14+E(52)^18-E(52)^20+E(52)^22, E(52)^4-E(52)^6+E(52)^20-E(52)^22, -1*E(52)^8-E(52)^12+E(52)^14+E(52)^18, 1+E(52)^4-E(52)^6+E(52)^8+E(52)^12-E(52)^14-E(52)^18+E(52)^20-E(52)^22, -1*E(52)^4+E(52)^6-E(52)^20+E(52)^22, E(52)^3+E(52)^7-E(52)^9+E(52)^11-E(52)^13+E(52)^15-E(52)^17+E(52)^19+E(52)^23, E(52)^7-E(52)^9-E(52)^17+E(52)^19, E(52)^3+E(52)^11+E(52)^15+E(52)^23, -1*E(52)^3-E(52)^7+E(52)^9-E(52)^11+E(52)^13-E(52)^15+E(52)^17-E(52)^19-E(52)^23, -1*E(52)^3-E(52)^11-E(52)^15-E(52)^23, -1*E(52)^7+E(52)^9+E(52)^17-E(52)^19]
chartbl_208_28["Irr"] = Matrix(UCF, 28, 28, irr_208_28)