# Group 500.27 downloaded from the LMFDB on 04 May 2026. ## Various presentations of this group are stored in this file: # GPC is polycyclic presentation GPerm is permutation group # GLZ, GLFp, GLZA, GLZq, GLFq if they exist are matrix groups # Many characteristics of the group are stored as booleans in a record: # Agroup, Zgroup, abelian, almost_simple,cyclic, metabelian, # metacyclic, monomial, nilpotent, perfect, quasisimple, rational, # solvable, supersolvable # The character table is stored as a record chartbl_n_i where n is the order # of the group and i is which group of that order it is. The record is # converted to a character table using ConvertToLibraryCharacterTableNC # Constructions GPC := PcGroupCode(788021647822868084082751,500); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.5; GPerm := Group( (2,5)(3,4)(6,21)(7,25)(8,24)(9,23)(10,22)(11,16)(12,20)(13,19)(14,18)(15,17), (2,5)(3,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24), (6,7,8,9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22) ); GLFp := Group([[[ Z(5)^2, Z(5)^2, Z(5)^2 ], [ Z(5)^0, Z(5)^2, Z(5)^3 ], [ Z(5)^3, Z(5)^2, 0*Z(5) ]], [[ Z(5)^2, Z(5)^2, Z(5)^2 ], [ 0*Z(5), Z(5), Z(5)^3 ], [ 0*Z(5), Z(5)^2, Z(5)^3 ]], [[ Z(5), 0*Z(5), Z(5)^2 ], [ Z(5), Z(5)^0, Z(5)^3 ], [ Z(5)^0, 0*Z(5), 0*Z(5) ]], [[ Z(5)^2, Z(5)^0, 0*Z(5) ], [ Z(5)^2, Z(5)^0, Z(5)^0 ], [ 0*Z(5), Z(5)^0, Z(5)^2 ]], [[ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ Z(5), 0*Z(5), Z(5)^2 ]]]); # Booleans booleans_500_27 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := false, metacyclic := false, monomial := true, nilpotent := false, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_500_27:=rec(); chartbl_500_27.IsFinite:= true; chartbl_500_27.UnderlyingCharacteristic:= 0; chartbl_500_27.UnderlyingGroup:= GLFp; chartbl_500_27.Size:= 500; chartbl_500_27.InfoText:= "Character table for group 500.27 downloaded from the LMFDB."; chartbl_500_27.Identifier:= " C5^2:D10 "; chartbl_500_27.NrConjugacyClasses:= 20; chartbl_500_27.ConjugacyClasses:= [[1, 0, 0, 0, 1, 0, 0, 0, 1], [1, 3, 4, 0, 0, 2, 0, 3, 0], [1, 0, 0, 4, 1, 1, 2, 0, 4], [4, 4, 4, 0, 2, 3, 0, 4, 3], [3, 0, 3, 4, 1, 1, 2, 0, 4], [0, 0, 1, 3, 1, 2, 4, 0, 2], [1, 4, 2, 0, 4, 4, 0, 4, 3], [1, 3, 4, 0, 2, 3, 0, 3, 0], [4, 0, 2, 2, 1, 3, 3, 0, 3], [2, 0, 4, 4, 1, 1, 1, 0, 0], [0, 1, 4, 0, 3, 1, 4, 1, 0], [0, 2, 2, 2, 0, 0, 4, 2, 3], [0, 2, 2, 0, 0, 2, 4, 2, 3], [0, 1, 4, 2, 3, 4, 4, 1, 0], [2, 3, 3, 2, 0, 0, 1, 3, 4], [4, 3, 1, 1, 0, 1, 3, 3, 2], [1, 2, 1, 4, 0, 3, 2, 2, 0], [1, 1, 3, 4, 3, 2, 2, 1, 2], [0, 4, 3, 4, 2, 4, 1, 4, 2], [2, 4, 1, 2, 2, 1, 3, 4, 0]]; chartbl_500_27.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]; chartbl_500_27.ComputedPowerMaps:= [ , [1, 1, 1, 1, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4]]; chartbl_500_27.SizesCentralizers:= [500, 20, 20, 20, 250, 250, 50, 50, 50, 50, 25, 25, 25, 25, 10, 10, 10, 10, 10, 10]; chartbl_500_27.ClassNames:= ["1A", "2A", "2B", "2C", "5A1", "5A2", "5B1", "5B2", "5C1", "5C2", "5D1", "5D2", "5E1", "5E2", "10A1", "10A3", "10B1", "10B3", "10C1", "10C3"]; chartbl_500_27.OrderClassRepresentatives:= [1, 2, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 10]; chartbl_500_27.Irr:= [[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, 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, 1, -1, -1, -1], [1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1], [2, 0, 0, 2, 2, 2, 2, 2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 0, 0, 0, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 0], [2, 0, 0, 2, 2, 2, 2, 2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 0, 0, 0, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 0], [2, 0, 2, 0, 2, 2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 2, 2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 0, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 0, 0, 0], [2, 0, 2, 0, 2, 2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 2, 2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 0, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 0, 0, 0], [2, 0, -2, 0, 2, 2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 2, 2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 0, -1*E(5)-E(5)^-1, -1*E(5)^2-E(5)^-2, 0, 0, 0], [2, 0, -2, 0, 2, 2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 2, 2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 0, -1*E(5)^2-E(5)^-2, -1*E(5)-E(5)^-1, 0, 0, 0], [2, 0, 0, -2, 2, 2, 2, 2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, 0, 0, 0, -1*E(5)-E(5)^-1, -1*E(5)^2-E(5)^-2, 0], [2, 0, 0, -2, 2, 2, 2, 2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)^2+E(5)^-2, 0, 0, 0, -1*E(5)^2-E(5)^-2, -1*E(5)-E(5)^-1, 0], [4, 0, 0, 0, 4, 4, 2*E(5)^2+2*E(5)^-2, 2*E(5)+2*E(5)^-1, 2*E(5)^2+2*E(5)^-2, 2*E(5)+2*E(5)^-1, -1, -1, 1-E(5)^2-E(5)^-2, 2+E(5)^2+E(5)^-2, 0, 0, 0, 0, 0, 0], [4, 0, 0, 0, 4, 4, 2*E(5)+2*E(5)^-1, 2*E(5)^2+2*E(5)^-2, 2*E(5)+2*E(5)^-1, 2*E(5)^2+2*E(5)^-2, -1, -1, 2+E(5)^2+E(5)^-2, 1-E(5)^2-E(5)^-2, 0, 0, 0, 0, 0, 0], [4, 0, 0, 0, 4, 4, 2*E(5)^2+2*E(5)^-2, 2*E(5)+2*E(5)^-1, 2*E(5)+2*E(5)^-1, 2*E(5)^2+2*E(5)^-2, 2+E(5)^2+E(5)^-2, 1-E(5)^2-E(5)^-2, -1, -1, 0, 0, 0, 0, 0, 0], [4, 0, 0, 0, 4, 4, 2*E(5)+2*E(5)^-1, 2*E(5)^2+2*E(5)^-2, 2*E(5)^2+2*E(5)^-2, 2*E(5)+2*E(5)^-1, 1-E(5)^2-E(5)^-2, 2+E(5)^2+E(5)^-2, -1, -1, 0, 0, 0, 0, 0, 0], [10, 2, 0, 0, 5*E(5)^2+5*E(5)^-2, 5*E(5)+5*E(5)^-1, 0, 0, 0, 0, 0, 0, 0, 0, E(5)+E(5)^-1, 0, 0, 0, 0, E(5)^2+E(5)^-2], [10, 2, 0, 0, 5*E(5)+5*E(5)^-1, 5*E(5)^2+5*E(5)^-2, 0, 0, 0, 0, 0, 0, 0, 0, E(5)^2+E(5)^-2, 0, 0, 0, 0, E(5)+E(5)^-1], [10, -2, 0, 0, 5*E(5)^2+5*E(5)^-2, 5*E(5)+5*E(5)^-1, 0, 0, 0, 0, 0, 0, 0, 0, -1*E(5)-E(5)^-1, 0, 0, 0, 0, -1*E(5)^2-E(5)^-2], [10, -2, 0, 0, 5*E(5)+5*E(5)^-1, 5*E(5)^2+5*E(5)^-2, 0, 0, 0, 0, 0, 0, 0, 0, -1*E(5)^2-E(5)^-2, 0, 0, 0, 0, -1*E(5)-E(5)^-1]]; ConvertToLibraryCharacterTableNC(chartbl_500_27);