# Group 112.17 downloaded from the LMFDB on 28 October 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(11370390178593506981,112); a := GPC.1; b := GPC.2; c := GPC.5; GPerm := Group( (1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,19)(20,21)(22,23), (1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9), (1,4,5,14)(2,7,8,12)(3,10,11,13)(6,9,15,16), (1,5)(2,8)(3,11)(4,14)(6,15)(7,12)(9,16)(10,13), (17,18,20,22,23,21,19) ); GLZN := Group([[[ZmodnZObj(48,91), ZmodnZObj(44,91)], [ZmodnZObj(36,91), ZmodnZObj(4,91)]],[[ZmodnZObj(66,91), ZmodnZObj(13,91)], [ZmodnZObj(52,91), ZmodnZObj(53,91)]],[[ZmodnZObj(64,91), ZmodnZObj(14,91)], [ZmodnZObj(42,91), ZmodnZObj(22,91)]],[[ZmodnZObj(25,91), ZmodnZObj(39,91)], [ZmodnZObj(78,91), ZmodnZObj(38,91)]],[[ZmodnZObj(27,91), ZmodnZObj(0,91)], [ZmodnZObj(0,91), ZmodnZObj(27,91)]]]); # Booleans booleans_112_17 := 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_112_17:=rec(); chartbl_112_17.IsFinite:= true; chartbl_112_17.UnderlyingCharacteristic:= 0; chartbl_112_17.UnderlyingGroup:= GPC; chartbl_112_17.Size:= 112; chartbl_112_17.InfoText:= "Character table for group 112.17 downloaded from the LMFDB."; chartbl_112_17.Identifier:= " C7:Q16 "; chartbl_112_17.NrConjugacyClasses:= 22; chartbl_112_17.ConjugacyClasses:= [ of ..., f4, f3*f4, f1*f4, f1*f2*f5^2, f5^4, f5, f5^5, f2*f3*f4*f5^2, f2*f4*f5^2, f4*f5^2, f4*f5^6, f4*f5^3, f3*f5, f3*f5^3, f3*f5^2, f1*f5, f1*f5^6, f1*f5^3, f1*f5^4, f1*f5^5, f1*f5^2]; chartbl_112_17.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]; chartbl_112_17.ComputedPowerMaps:= [ , [1, 1, 2, 2, 2, 7, 8, 6, 3, 3, 6, 8, 7, 11, 12, 13, 11, 11, 12, 12, 13, 13], [1, 2, 3, 4, 5, 8, 6, 7, 10, 9, 12, 13, 11, 15, 16, 14, 19, 20, 22, 21, 17, 18]]; chartbl_112_17.SizesCentralizers:= [112, 112, 56, 28, 4, 56, 56, 56, 8, 8, 56, 56, 56, 28, 28, 28, 28, 28, 28, 28, 28, 28]; chartbl_112_17.ClassNames:= ["1A", "2A", "4A", "4B", "4C", "7A1", "7A2", "7A3", "8A1", "8A3", "14A1", "14A3", "14A5", "28A1", "28A3", "28A5", "28B1", "28B-1", "28B3", "28B-3", "28B5", "28B-5"]; chartbl_112_17.OrderClassRepresentatives:= [1, 2, 4, 4, 4, 7, 7, 7, 8, 8, 14, 14, 14, 28, 28, 28, 28, 28, 28, 28, 28, 28]; chartbl_112_17.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], [2, 2, -2, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2, 0, -2, 0, -2, 0, 0, 0, 0, -2], [2, -2, 0, 0, 0, 2, 2, 2, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -2, 0, 0, 0, 2, 2, 2, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 0, E(7)^3+E(7)^-3, E(7)+E(7)^-1, E(7)^2+E(7)^-2, 0, 0, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3], [2, 2, 2, 2, 0, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, E(7)+E(7)^-1, 0, 0, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2], [2, 2, 2, 2, 0, E(7)+E(7)^-1, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, 0, 0, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1], [2, 2, 2, -2, 0, E(7)^3+E(7)^-3, E(7)+E(7)^-1, E(7)^2+E(7)^-2, 0, 0, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, -1*E(7)^2-E(7)^-2, E(7)+E(7)^-1, -1*E(7)^3-E(7)^-3, E(7)^2+E(7)^-2, -1*E(7)-E(7)^-1, -1*E(7)^3-E(7)^-3, -1*E(7)^2-E(7)^-2, -1*E(7)-E(7)^-1, E(7)^3+E(7)^-3], [2, 2, 2, -2, 0, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, E(7)+E(7)^-1, 0, 0, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, -1*E(7)-E(7)^-1, E(7)^3+E(7)^-3, -1*E(7)^2-E(7)^-2, E(7)+E(7)^-1, -1*E(7)^3-E(7)^-3, -1*E(7)^2-E(7)^-2, -1*E(7)-E(7)^-1, -1*E(7)^3-E(7)^-3, E(7)^2+E(7)^-2], [2, 2, 2, -2, 0, E(7)+E(7)^-1, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, 0, 0, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, -1*E(7)^3-E(7)^-3, E(7)^2+E(7)^-2, -1*E(7)-E(7)^-1, E(7)^3+E(7)^-3, -1*E(7)^2-E(7)^-2, -1*E(7)-E(7)^-1, -1*E(7)^3-E(7)^-3, -1*E(7)^2-E(7)^-2, E(7)+E(7)^-1], [2, 2, -2, 0, 0, E(7)^3+E(7)^-3, E(7)+E(7)^-1, E(7)^2+E(7)^-2, 0, 0, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, -1*E(7)^2+E(7)^-2, -1*E(7)-E(7)^-1, E(7)^3-E(7)^-3, -1*E(7)^2-E(7)^-2, -1*E(7)+E(7)^-1, -1*E(7)^3+E(7)^-3, E(7)^2-E(7)^-2, E(7)-E(7)^-1, -1*E(7)^3-E(7)^-3], [2, 2, -2, 0, 0, E(7)^3+E(7)^-3, E(7)+E(7)^-1, E(7)^2+E(7)^-2, 0, 0, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^2-E(7)^-2, -1*E(7)-E(7)^-1, -1*E(7)^3+E(7)^-3, -1*E(7)^2-E(7)^-2, E(7)-E(7)^-1, E(7)^3-E(7)^-3, -1*E(7)^2+E(7)^-2, -1*E(7)+E(7)^-1, -1*E(7)^3-E(7)^-3], [2, 2, -2, 0, 0, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, E(7)+E(7)^-1, 0, 0, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, -1*E(7)+E(7)^-1, -1*E(7)^3-E(7)^-3, -1*E(7)^2+E(7)^-2, -1*E(7)-E(7)^-1, E(7)^3-E(7)^-3, E(7)^2-E(7)^-2, E(7)-E(7)^-1, -1*E(7)^3+E(7)^-3, -1*E(7)^2-E(7)^-2], [2, 2, -2, 0, 0, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, E(7)+E(7)^-1, 0, 0, E(7)^2+E(7)^-2, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)-E(7)^-1, -1*E(7)^3-E(7)^-3, E(7)^2-E(7)^-2, -1*E(7)-E(7)^-1, -1*E(7)^3+E(7)^-3, -1*E(7)^2+E(7)^-2, -1*E(7)+E(7)^-1, E(7)^3-E(7)^-3, -1*E(7)^2-E(7)^-2], [2, 2, -2, 0, 0, E(7)+E(7)^-1, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, 0, 0, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, -1*E(7)^3+E(7)^-3, -1*E(7)^2-E(7)^-2, E(7)-E(7)^-1, -1*E(7)^3-E(7)^-3, E(7)^2-E(7)^-2, -1*E(7)+E(7)^-1, E(7)^3-E(7)^-3, -1*E(7)^2+E(7)^-2, -1*E(7)-E(7)^-1], [2, 2, -2, 0, 0, E(7)+E(7)^-1, E(7)^2+E(7)^-2, E(7)^3+E(7)^-3, 0, 0, E(7)+E(7)^-1, E(7)^3+E(7)^-3, E(7)^2+E(7)^-2, E(7)^3-E(7)^-3, -1*E(7)^2-E(7)^-2, -1*E(7)+E(7)^-1, -1*E(7)^3-E(7)^-3, -1*E(7)^2+E(7)^-2, E(7)-E(7)^-1, -1*E(7)^3+E(7)^-3, E(7)^2-E(7)^-2, -1*E(7)-E(7)^-1], [4, -4, 0, 0, 0, 2*E(7)^3+2*E(7)^-3, 2*E(7)+2*E(7)^-1, 2*E(7)^2+2*E(7)^-2, 0, 0, -2*E(7)^3-2*E(7)^-3, -2*E(7)^2-2*E(7)^-2, -2*E(7)-2*E(7)^-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, 0, 2*E(7)^2+2*E(7)^-2, 2*E(7)^3+2*E(7)^-3, 2*E(7)+2*E(7)^-1, 0, 0, -2*E(7)^2-2*E(7)^-2, -2*E(7)-2*E(7)^-1, -2*E(7)^3-2*E(7)^-3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -4, 0, 0, 0, 2*E(7)+2*E(7)^-1, 2*E(7)^2+2*E(7)^-2, 2*E(7)^3+2*E(7)^-3, 0, 0, -2*E(7)-2*E(7)^-1, -2*E(7)^3-2*E(7)^-3, -2*E(7)^2-2*E(7)^-2, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; ConvertToLibraryCharacterTableNC(chartbl_112_17);