# Group 320.1591 downloaded from the LMFDB on 13 November 2025. ## 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(728382543465111153222064060268547,320); a := GPC.1; b := GPC.2; c := GPC.4; d := GPC.5; GPerm := Group( (1,2)(3,4)(5,6)(7,8)(10,11,12,13), (1,3)(2,4)(5,7)(6,8), (1,3)(2,4)(5,8)(6,7), (1,4,3,2)(5,8)(6,7), (10,12)(11,13), (1,3)(2,4), (9,10,11,13,12) ); GLZN := Group([[[ZmodnZObj(9,20), ZmodnZObj(0,20)], [ZmodnZObj(0,20), ZmodnZObj(1,20)]],[[ZmodnZObj(1,20), ZmodnZObj(10,20)], [ZmodnZObj(0,20), ZmodnZObj(1,20)]],[[ZmodnZObj(19,20), ZmodnZObj(10,20)], [ZmodnZObj(10,20), ZmodnZObj(3,20)]],[[ZmodnZObj(9,20), ZmodnZObj(0,20)], [ZmodnZObj(0,20), ZmodnZObj(9,20)]],[[ZmodnZObj(9,20), ZmodnZObj(3,20)], [ZmodnZObj(0,20), ZmodnZObj(3,20)]],[[ZmodnZObj(1,20), ZmodnZObj(4,20)], [ZmodnZObj(0,20), ZmodnZObj(1,20)]],[[ZmodnZObj(19,20), ZmodnZObj(0,20)], [ZmodnZObj(0,20), ZmodnZObj(19,20)]]]); # Booleans booleans_320_1591 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := true, metacyclic := false, monomial := true, nilpotent := false, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_320_1591:=rec(); chartbl_320_1591.IsFinite:= true; chartbl_320_1591.UnderlyingCharacteristic:= 0; chartbl_320_1591.UnderlyingGroup:= GPC; chartbl_320_1591.Size:= 320; chartbl_320_1591.InfoText:= "Character table for group 320.1591 downloaded from the LMFDB."; chartbl_320_1591.Identifier:= " D10.C2^4 "; chartbl_320_1591.NrConjugacyClasses:= 56; chartbl_320_1591.ConjugacyClasses:= [ of ..., f6*f7^2, f4, f4*f6*f7^2, f1, f1*f6*f7^2, f1*f4, f1*f4*f6*f7^2, f3*f6, f3*f4, f1*f3, f3*f4*f6, f1*f3*f6, f1*f3*f4, f1*f3*f4*f6, f3, f5*f7, f4*f5*f7, f1*f5*f7, f1*f4*f5*f7, f3*f5, f3*f4*f5, f1*f3*f5, f1*f3*f4*f5, f2, f2*f3, f2*f5, f2*f3*f5, f2*f4, f2*f3*f4, f1*f2, f1*f2*f3, f2*f4*f5, f2*f3*f4*f5, f1*f2*f5, f1*f2*f3*f5, f1*f2*f4, f1*f2*f3*f4, f1*f2*f4*f5, f1*f2*f3*f4*f5, f7^2, f4*f7, f4*f6, f1*f7, f1*f6, f1*f4*f7, f1*f4*f6, f6, f5, f5*f7^3, f4*f5, f4*f5*f7^3, f1*f5, f1*f5*f7^3, f1*f4*f5, f1*f4*f5*f7^3]; chartbl_320_1591.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, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56]; chartbl_320_1591.ComputedPowerMaps:= [ , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 41, 41, 41, 41, 41, 41, 41, 41, 48, 48, 48, 48, 48, 48, 48, 48], [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, 1, 3, 4, 5, 6, 7, 8, 2, 17, 17, 18, 18, 19, 19, 20, 20]]; chartbl_320_1591.SizesCentralizers:= [320, 320, 320, 320, 320, 320, 320, 320, 64, 64, 64, 64, 64, 64, 64, 64, 160, 160, 160, 160, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80, 80]; chartbl_320_1591.ClassNames:= ["1A", "2A", "2B", "2C", "2D", "2E", "2F", "2G", "2H", "2I", "2J", "2K", "2L", "2M", "2N", "2O", "4A", "4B", "4C", "4D", "4E", "4F", "4G", "4H", "4I1", "4I-1", "4J1", "4J-1", "4K1", "4K-1", "4L1", "4L-1", "4M1", "4M-1", "4N1", "4N-1", "4O1", "4O-1", "4P1", "4P-1", "5A", "10A", "10B", "10C", "10D", "10E", "10F", "10G", "20A1", "20A-1", "20B1", "20B-1", "20C1", "20C-1", "20D1", "20D-1"]; chartbl_320_1591.OrderClassRepresentatives:= [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 10, 10, 10, 10, 10, 10, 10, 20, 20, 20, 20, 20, 20, 20, 20]; chartbl_320_1591.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, 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-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], [1, 1, 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), E(4), 1, E(4), E(4), -1*E(4), 1, 1, E(4), -1*E(4), -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], [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, -1*E(4), -1*E(4), -1*E(4), 1, -1*E(4), -1*E(4), E(4), 1, 1, -1*E(4), E(4), 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, -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), -1, E(4), -1*E(4), -1*E(4), -1, E(4), E(4), -1*E(4), -1, -1, -1*E(4), E(4), E(4), E(4), -1*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, 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, -1*E(4), E(4), E(4), -1, -1*E(4), -1*E(4), E(4), -1, -1, E(4), -1*E(4), -1*E(4), -1*E(4), 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], [2, -2, -2, 2, -2, 2, 2, -2, 2, 2, -2, 2, -2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 2, 2, -2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, -2, 2, 2, -2, -2, 2, -2, -2, 2, 2, -2, -2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, 2, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, -2, -2, 2, -2, 2, 2, -2, 2, 2, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, -2, -2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, -2, 2, -2, 2, -2, -2, 2, -2, 2, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -2, -2, 2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, -2, 2, -2, 2, 2, -2, -2, -2, 2, -2, 2, 2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 2, 2, -2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, -2, 2, 2, -2, -2, 2, 2, 2, -2, -2, 2, 2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, 2, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, -2, -2, 2, -2, 2, -2, 2, -2, -2, -2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, -2, -2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -2, -2, 2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1], [4, 4, -4, -4, -4, -4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, 1, 1], [4, 4, -4, -4, -4, -4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1], [4, 4, -4, -4, 4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, -1, -1, 1, 1, 1, -1, 1, -1, 1, -1, -1, 1], [4, 4, -4, -4, 4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1], [4, 4, 4, 4, -4, -4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, 1], [4, 4, 4, 4, -4, -4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1], [4, 4, 4, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1], [4, -4, -4, 4, -4, 4, 4, -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, 0, 0, 0, -1, 1, -1, -1, 1, 1, -1, 1, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7], [4, -4, -4, 4, -4, 4, 4, -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, 0, 0, 0, -1, 1, -1, -1, 1, 1, -1, 1, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7], [4, -4, -4, 4, 4, -4, -4, 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, 0, 0, 0, -1, 1, 1, -1, -1, 1, 1, -1, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7], [4, -4, -4, 4, 4, -4, -4, 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, 0, 0, 0, -1, 1, 1, -1, -1, 1, 1, -1, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7], [4, -4, 4, -4, -4, 4, -4, 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, 0, 0, 0, -1, -1, -1, 1, 1, 1, 1, -1, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7], [4, -4, 4, -4, -4, 4, -4, 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, 0, 0, 0, -1, -1, -1, 1, 1, 1, 1, -1, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7], [4, -4, 4, -4, 4, -4, 4, -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, 0, 0, 0, -1, -1, 1, 1, -1, 1, -1, 1, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7], [4, -4, 4, -4, 4, -4, 4, -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, 0, 0, 0, -1, -1, 1, 1, -1, 1, -1, 1, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7, 2*E(20)^3-E(20)^5+2*E(20)^7, -2*E(20)^3+E(20)^5-2*E(20)^7]]; ConvertToLibraryCharacterTableNC(chartbl_320_1591);