# Group 256.213 downloaded from the LMFDB on 03 February 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(10967439262427047720075888956597762925,256); a := GPC.1; b := GPC.3; c := GPC.6; GPerm := Group( (1,9)(2,10)(3,11)(4,12)(5,14,6,13)(7,16,8,15), (9,15,11,14,10,16,12,13), (1,7,3,6,2,8,4,5)(9,13,12,16,10,14,11,15), (5,6)(7,8)(13,14)(15,16), (9,11,10,12)(13,15,14,16), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,14,15), (9,10)(11,12)(13,14)(15,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) ); GLFp := Group([[[ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^3, Z(5)^3, Z(5)^2, Z(5)^2 ], [ Z(5)^3, Z(5), 0*Z(5), Z(5)^3 ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5) ]], [[ Z(5)^2, Z(5)^2, Z(5)^0, Z(5)^0 ], [ 0*Z(5), Z(5)^0, Z(5)^0, Z(5)^3 ], [ 0*Z(5), Z(5)^0, Z(5)^2, Z(5)^3 ], [ Z(5)^0, Z(5)^2, Z(5)^3, Z(5)^0 ]], [[ Z(5)^2, Z(5)^0, Z(5), Z(5)^0 ], [ Z(5)^3, 0*Z(5), Z(5)^0, Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5), Z(5)^2 ], [ Z(5)^3, Z(5)^2, Z(5)^0, Z(5)^0 ]], [[ Z(5)^2, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^2, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^2, 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^2 ]], [[ Z(5)^3, 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5), Z(5)^3, Z(5), Z(5)^0 ], [ 0*Z(5), Z(5)^2, 0*Z(5), Z(5)^2 ], [ Z(5)^3, Z(5), Z(5)^2, Z(5)^2 ]], [[ Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^3, Z(5)^2, Z(5)^2, Z(5)^2 ], [ Z(5)^3, Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^3 ]], [[ Z(5)^2, 0*Z(5), 0*Z(5), 0*Z(5) ], [ Z(5)^2, Z(5)^0, 0*Z(5), Z(5)^2 ], [ 0*Z(5), Z(5), 0*Z(5), 0*Z(5) ], [ Z(5), Z(5)^3, Z(5)^0, 0*Z(5) ]], [[ Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5) ]]]); # Booleans booleans_256_213 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := true, metacyclic := false, monomial := true, nilpotent := true, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_256_213:=rec(); chartbl_256_213.IsFinite:= true; chartbl_256_213.UnderlyingCharacteristic:= 0; chartbl_256_213.UnderlyingGroup:= GPC; chartbl_256_213.Size:= 256; chartbl_256_213.InfoText:= "Character table for group 256.213 downloaded from the LMFDB."; chartbl_256_213.Identifier:= " C8^2:C4 "; chartbl_256_213.NrConjugacyClasses:= 40; chartbl_256_213.ConjugacyClasses:= [ of ..., f5, f8, f2, f4*f8, f4*f5*f8, f4, f7, f7*f8, f5*f7, f5*f7*f8, f2*f4*f8, f2*f7, f1, f1*f2, f6, f6*f7, f3, f3*f8, f5*f6, f5*f6*f7, f4*f6, f3*f6*f7, f3*f7, f3*f4*f7, f4*f6*f7, f4*f5*f6, f2*f3*f7*f8, f2*f3*f4*f7, f2*f3, f2*f6, f2*f6*f7, f2*f4*f6, f2*f4*f6*f7, f1*f7, f1*f2*f7, f1*f6, f1*f2*f6*f7, f1*f6*f7, f1*f2*f6]; chartbl_256_213.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]; chartbl_256_213.ComputedPowerMaps:= [ , [1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 2, 3, 4, 4, 8, 9, 7, 7, 8, 9, 10, 11, 5, 6, 11, 10, 6, 5, 7, 9, 8, 11, 10, 12, 12, 28, 29, 29, 28]]; chartbl_256_213.SizesCentralizers:= [256, 256, 128, 64, 256, 256, 128, 128, 128, 128, 128, 64, 32, 16, 16, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 32, 16, 16, 16, 16, 16, 16]; chartbl_256_213.ClassNames:= ["1A", "2A", "2B", "2C", "4A1", "4A-1", "4B", "4C1", "4C-1", "4D1", "4D-1", "4E", "4F", "4G1", "4G-1", "8A1", "8A-1", "8B1", "8B-1", "8C1", "8C-1", "8D1", "8D-1", "8E1", "8E-1", "8F1", "8F-1", "8G1", "8G-1", "8H", "8I1", "8I-1", "8J1", "8J-1", "8K1", "8K-1", "16A1", "16A-1", "16B1", "16B-1"]; chartbl_256_213.OrderClassRepresentatives:= [1, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16]; chartbl_256_213.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], [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], [1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1*E(4), E(4), E(4), E(4), 1, -1, -1, -1*E(4), -1*E(4), E(4), 1, -1*E(4), 1, -1*E(4), E(4), 1, E(4), -1, -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, -1, -1, -1, -1, 1, E(4), -1*E(4), -1*E(4), -1*E(4), 1, -1, -1, E(4), E(4), -1*E(4), 1, E(4), 1, E(4), -1*E(4), 1, -1*E(4), -1, E(4), -1*E(4), E(4), -1*E(4), E(4), -1, -1, 1, 1], [1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1*E(4), E(4), -1*E(4), -1*E(4), 1, -1, -1, E(4), E(4), -1*E(4), 1, E(4), 1, E(4), -1*E(4), 1, -1*E(4), -1, E(4), -1*E(4), E(4), E(4), -1*E(4), 1, 1, -1, -1], [1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, E(4), -1*E(4), E(4), E(4), 1, -1, -1, -1*E(4), -1*E(4), E(4), 1, -1*E(4), 1, -1*E(4), E(4), 1, E(4), -1, -1*E(4), E(4), -1*E(4), -1*E(4), E(4), 1, 1, -1, -1], [1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1*E(4), E(4), -1, -1, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, 1, 1, 1, -1*E(4), E(4), E(4), -1*E(4), E(4), -1*E(4)], [1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, E(4), -1*E(4), -1, -1, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, 1, 1, 1, E(4), -1*E(4), -1*E(4), E(4), -1*E(4), E(4)], [1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1*E(4), E(4), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1, -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, -1, E(4), -1*E(4), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, -1, -1, -1, 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, -1, -1, -1, -1*E(4), -1*E(4), 1, -1, -1, E(4), E(4), -1*E(4), 1, E(4), -1, E(4), -1*E(4), -1, E(4), 1, -1*E(4), E(4), -1*E(4), 1, 1, -1*E(4), E(4), E(4), -1*E(4)], [1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, -1, -1, E(4), E(4), 1, -1, -1, -1*E(4), -1*E(4), E(4), 1, -1*E(4), -1, -1*E(4), E(4), -1, -1*E(4), 1, E(4), -1*E(4), E(4), 1, 1, E(4), -1*E(4), -1*E(4), E(4)], [1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, -1*E(4), -1*E(4), 1, -1, -1, E(4), E(4), -1*E(4), 1, E(4), -1, E(4), -1*E(4), -1, E(4), 1, -1*E(4), E(4), -1*E(4), -1, -1, E(4), -1*E(4), -1*E(4), E(4)], [1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, -1, 1, 1, E(4), E(4), 1, -1, -1, -1*E(4), -1*E(4), E(4), 1, -1*E(4), -1, -1*E(4), E(4), -1, -1*E(4), 1, E(4), -1*E(4), E(4), -1, -1, -1*E(4), E(4), E(4), -1*E(4)], [2, 2, 2, -2, 2, 2, 2, 2, 2, 2, 2, -2, -2, 0, 0, 0, 0, -2, -2, -2, 0, 0, 0, -2, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 2, -2, -2, -2, -2, 2, -2, 0, 0, 0, 0, -2, 2, 2, 0, 0, 0, -2, 0, 2, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, -2, -2, -2, 0, 0, 0, -2, 0, -2, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, 0, 0, 0, 0, -2, 2, 2, 0, 0, 0, -2, 0, -2, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 2, 0, 0, 0, -1-E(4), -1-E(4), 0, -2*E(4), 2*E(4), 1-E(4), -1+E(4), 1+E(4), 0, -1+E(4), -2*E(4), 1-E(4), 1+E(4), 2*E(4), -1+E(4), 0, 1+E(4), 1-E(4), -1-E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 2, 0, 0, 0, -1+E(4), -1+E(4), 0, 2*E(4), -2*E(4), 1+E(4), -1-E(4), 1-E(4), 0, -1-E(4), 2*E(4), 1+E(4), 1-E(4), -2*E(4), -1-E(4), 0, 1-E(4), 1+E(4), -1+E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 2, 0, 0, 0, 1-E(4), 1-E(4), 0, 2*E(4), -2*E(4), -1-E(4), 1+E(4), -1+E(4), 0, 1+E(4), 2*E(4), -1-E(4), -1+E(4), -2*E(4), 1+E(4), 0, -1+E(4), -1-E(4), 1-E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 2, 0, 0, 0, 1+E(4), 1+E(4), 0, -2*E(4), 2*E(4), -1+E(4), 1-E(4), -1-E(4), 0, 1-E(4), -2*E(4), -1+E(4), -1-E(4), 2*E(4), 1-E(4), 0, -1-E(4), -1+E(4), 1+E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, -2, 0, 0, 0, -1-E(4), -1-E(4), 0, -2*E(4), 2*E(4), 1-E(4), -1+E(4), 1+E(4), 0, -1+E(4), 2*E(4), 1-E(4), 1+E(4), -2*E(4), 1-E(4), 0, -1-E(4), -1+E(4), 1+E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, -2, 0, 0, 0, -1+E(4), -1+E(4), 0, 2*E(4), -2*E(4), 1+E(4), -1-E(4), 1-E(4), 0, -1-E(4), -2*E(4), 1+E(4), 1-E(4), 2*E(4), 1+E(4), 0, -1+E(4), -1-E(4), 1-E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, -2, 0, 0, 0, 1-E(4), 1-E(4), 0, 2*E(4), -2*E(4), -1-E(4), 1+E(4), -1+E(4), 0, 1+E(4), -2*E(4), -1-E(4), -1+E(4), 2*E(4), -1-E(4), 0, 1-E(4), 1+E(4), -1+E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, -2, 0, 0, 0, 1+E(4), 1+E(4), 0, -2*E(4), 2*E(4), -1+E(4), 1-E(4), -1-E(4), 0, 1-E(4), 2*E(4), -1+E(4), -1-E(4), -2*E(4), -1+E(4), 0, 1+E(4), 1-E(4), -1-E(4), 0, 0, 0, 0, 0, 0], [4, 4, -4, 0, 4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4*E(4), 0, 0, 0, 0, 0, 4*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, -4, 0, 4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4*E(4), 0, 0, 0, 0, 0, -4*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, -4, 0, -4, -4, 4, -4*E(4), 4*E(4), -4*E(4), 4*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, -4, 0, -4, -4, 4, 4*E(4), -4*E(4), 4*E(4), -4*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, -4*E(4), 4*E(4), 0, -2+2*E(4), -2-2*E(4), 2-2*E(4), 2+2*E(4), 0, 0, 0, 0, 2*E(4), -2*E(4), 0, 0, 0, -2, 2*E(4), -2, 0, -2*E(4), 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, 4*E(4), -4*E(4), 0, -2-2*E(4), -2+2*E(4), 2+2*E(4), 2-2*E(4), 0, 0, 0, 0, -2*E(4), 2*E(4), 0, 0, 0, -2, -2*E(4), -2, 0, 2*E(4), 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, -4*E(4), 4*E(4), 0, -2+2*E(4), -2-2*E(4), 2-2*E(4), 2+2*E(4), 0, 0, 0, 0, -2*E(4), 2*E(4), 0, 0, 0, 2, -2*E(4), 2, 0, 2*E(4), 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, 4*E(4), -4*E(4), 0, -2-2*E(4), -2+2*E(4), 2+2*E(4), 2-2*E(4), 0, 0, 0, 0, 2*E(4), -2*E(4), 0, 0, 0, 2, 2*E(4), 2, 0, -2*E(4), 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, -4*E(4), 4*E(4), 0, 2-2*E(4), 2+2*E(4), -2+2*E(4), -2-2*E(4), 0, 0, 0, 0, -2, 2, 0, 0, 0, 2*E(4), 2, -2*E(4), 0, -2, 0, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, 4*E(4), -4*E(4), 0, 2+2*E(4), 2-2*E(4), -2-2*E(4), -2+2*E(4), 0, 0, 0, 0, -2, 2, 0, 0, 0, -2*E(4), 2, 2*E(4), 0, -2, 0, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, -4*E(4), 4*E(4), 0, 2-2*E(4), 2+2*E(4), -2+2*E(4), -2-2*E(4), 0, 0, 0, 0, 2, -2, 0, 0, 0, -2*E(4), -2, 2*E(4), 0, 2, 0, 2*E(4), -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, 4*E(4), -4*E(4), 0, 2+2*E(4), 2-2*E(4), -2-2*E(4), -2+2*E(4), 0, 0, 0, 0, 2, -2, 0, 0, 0, 2*E(4), -2, -2*E(4), 0, 2, 0, -2*E(4), 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; ConvertToLibraryCharacterTableNC(chartbl_256_213);