############################################################################# ## #W multfree.dat database of mult.-free perm. characters Thomas Breuer #W Klaus Lux ## #H @(#)$Id: multfree.dat,v 1.5 2003/11/19 09:56:05 gap Exp $ ## #Y Copyright (C) 2000, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany ## ## This file contains the following {\GAP} objects. ## ## `MULTFREEINFO' ## is the global variable that encodes the faithful multiplicity-free ## permutation characters of the sporadic simple groups and their ## automorphism groups, as classified in~\cite{BL96} (modulo bug fixes). ## ## `MultFreePermChars' ## is a function that can be used for computing more detailed data ## about the permutation characters from the compact information ## stored in `MULTFREEINFO'. ## Revision.multfree_dat := "@(#)$Id: multfree.dat,v 1.5 2003/11/19 09:56:05 gap Exp $"; ############################################################################# ## ## Print the banner if wanted. ## if not GAPInfo.CommandLineOptions.q and not GAPInfo.CommandLineOptions.b then Print( "--------------------------------------------------------------------\n", "Loading the Database of Multiplicity-Free Permutation Characters\n", "of the Sporadic Simple Groups and Their Automorphism Groups,\n", "by T. Breuer and K. Lux;\n", "call `MultFreePermChars( )' for accessing the data\n", "for the group whose character table has identifier .\n", "--------------------------------------------------------------------\n" ); fi; ############################################################################# ## #V MULTFREEINFO ## ## `MULTFREEINFO' is an immutable record. ## Its components are the `Identifier' values of the {\GAP} character ## tables of the sporadic simple groups and their automorphism groups. ## The value of the component corresponding to the group $G$, say, ## is a list containing in the first position a string denoting the name of ## $G$ in {\LaTeX} format, ## and in each of the remaining positions a pair `[,]' ## where is a list of positive integers and is a string ## that denotes the name of a subgroup $H$ of $G$, in {\LaTeX} format; ## the sum of irreducible characters of $G$ at the positions in ## is a multiplicity-free permutation character of $G$ that is induced from ## the trivial character of $H$. ## BindGlobal( "MULTFREEINFO", rec() ); MULTFREEINFO.("M11"):= ["$M_{11}$", [[1,2],"$A_6.2_3$"], [[1,2,5],"$A_6 \\leq A_6.2_3$"], [[1,5],"$L_2(11)$"], [[1,5,6,7,9,10],"$11:5 \\leq L_2(11)$"], [[1,2,8],"$3^2:Q_8.2$"], [[1,2,8,10],"$3^2:8 \\leq 3^2:Q_8.2$"], [[1,2,5,8],"$A_5.2$"], ]; MULTFREEINFO.("M12"):= ["$M_{12}$", [[1,2],"$M_{11}$"], [[1,3],"$M_{11}$"], [[1,2,3,8,11],"$L_2(11) \\leq M_{11}$"], [[1,2,7],"$A_6.2^2$"], [[1,2,3,7,8],"$A_6.2_1 \\leq A_6.2^2$"], [[1,2,7,11],"$A_6.2_2 \\leq A_6.2^2$"], [[1,3,7],"$A_6.2^2$"], [[1,2,3,7,8],"$A_6.2_1 \\leq A_6.2^2$"], [[1,3,7,11],"$A_6.2_2 \\leq A_6.2^2$"], [[1,4,5,6,11],"$L_2(11)$"], [[1,2,7,8,12],"$3^2.2.S_4$"], [[1,2,6,7,8,10,12,13],"$3^2:2.A_4 \\leq 3^2.2.S_4$"], [[1,3,7,8,12],"$3^2.2.S_4$"], [[1,3,6,7,8,9,12,13],"$3^2:2.A_4 \\leq 3^2.2.S_4$"], ]; MULTFREEINFO.("M12.2"):= ["$M_{12}.2$", [[1,2,3],"$M_{11}$"], [[1,3,9,12],"$L_2(11).2$"], [[1,2,3,7,8],"$A_6.2^2$"], [[1,2,3,7,8,12,13],"$A_6.2_2 \\leq A_6.2^2$"], [[1,4,5,12],"$L_2(11).2$"], [[1,2,3,7,8,9,10,14,15],"$3^2.2.S_4$"], [[1,2,3,5,6,7,8,9,10,11,14,15,16,17],"$3^2:2.A_4 \\leq 3^2.2.S_4$"], [[1,4,5,7,8,12,18],"$(2^2 \\times A_5).2$"], [[1,4,5,7,8,10,12,13,15,18,21],"$(2 \\times A_5).2 \\leq (2^2 \\times A_5).2$"\ ], [[1,3,7,8,9,12,15,18],"$M_8.(S_4 \\times 2)$"], [[1,3,6,7,8,9,12,13,15,16,17,18,19],"$M_8.(A_4 \\times 2) \\leq M_8.(S_4 \\tim\ es 2)$"], [[1,4,6,7,8,12,15,18],"$4^2:D_{12}.2$"], [[1,4,6,7,8,9,10,11,12,14,15,18,20],"$4^2:(6 \\times 2) \\leq 4^2:D_{12}.2$"], ]; MULTFREEINFO.("J1"):= ["$J_1$", [[1,2,3,4,6],"$L_2(11)$"], [[1,2,3,4,7,8,9,10,11,12,15],"$2^3.7.3$"], ]; MULTFREEINFO.("M22"):= ["$M_{22}$", [[1,2],"$L_3(4)$"], [[1,2,5],"$2^4:A_6$"], [[1,2,5,7,9],"$2^4:A_5 \\leq 2^4:A_6$"], [[1,2,7],"$A_7$"], [[1,2,7],"$A_7$"], [[1,2,5,7],"$2^4:S_5$"], [[1,2,5,6,7],"$2^3:L_3(2)$"], [[1,2,5,7,12],"$A_6.2_3$"], [[1,2,5,7,8,9],"$L_2(11)$"], ]; MULTFREEINFO.("M22.2"):= ["$M_{22}.2$", [[1,3],"$L_3(4).2_2$"], [[1,2,3,4],"$L_3(4) \\leq L_3(4).2_2$"], [[1,3,9],"$2^4:S_6$"], [[1,2,3,4,9,10,13,14,17,18],"$2^4:A_5 \\leq 2^4:S_6$"], [[1,2,3,4,9,10],"$2^4:A_6 \\leq 2^4:S_6$"], [[1,3,9,13,18],"$2^4:S_5 \\leq 2^4:S_6$"], [[1,2,3,4,13,14],"$A_7$"], [[1,3,9,13],"$2^5:S_5$"], [[1,2,3,4,9,10,13,14],"$2^4:S_5 \\leq 2^5:S_5$"], [[1,3,4,9,13,16],"$2^4:(A_5 \\times 2) \\leq 2^5:S_5$"], [[1,3,9,11,13],"$2^3:L_3(2) \\times 2$"], [[1,2,3,4,9,10,11,12,13,14],"$2^3:L_3(2) \\leq 2^3:L_3(2) \\times 2$"], [[1,3,9,13,20],"$A_6.2^2$"], [[1,2,3,4,9,10,13,14,20,21],"$A_6.2_3 \\leq A_6.2^2$"], [[1,3,4,9,10,12,13,16,18,20],"$A_6.2_2 \\leq A_6.2^2$"], [[1,4,9,13,16,18],"$L_2(11).2$"], [[1,2,3,4,9,10,13,14,15,16,17,18],"$L_2(11) \\leq L_2(11).2$"], ]; MULTFREEINFO.("J2"):= ["$J_2$", [[1,6,7],"$U_3(3)$"], [[1,7,10,11],"$3.A_6.2_2$"], [[1,2,3,6,10,12],"$2^{1+4}_{-}:A_5$"], [[1,6,7,10,12,13],"$2^{2+4}.(3 \\times S_3)$"], [[1,7,10,11,12,13,18],"$A_4 \\times A_5$"], ]; MULTFREEINFO.("J2.2"):= ["$J_2.2$", [[1,5,7],"$U_3(3).2$"], [[1,2,5,6,7,8],"$U_3(3) \\leq U_3(3).2$"], [[1,7,10,12],"$3.A_6.2^2$"], [[1,2,7,8,10,11,12,13],"$3.A_6.2_2 \\leq 3.A_6.2^2$"], [[1,4,7,8,10,12,17],"$3.A_6.2_3 \\leq 3.A_6.2^2$"], [[1,3,5,10,14],"$2^{1+4}_{-}:S_5$"], [[1,3,5,8,10,12,13,14,23,25,26,27],"$2^{1+4}_-:5:4 \\leq 2^{1+4}_{-}:S_5$"], [[1,5,7,10,14,16],"$2^{2+4}.(S_3 \\times S_3)$"], [[1,2,5,6,7,8,10,11,14,15,16,17],"$2^{2+4}.(3 \\times S_3) \\leq 2^{2+4}.(S_3 \ \\times S_3)$"], [[1,5,7,9,10,14,15,16,20],"$2^{2+4}.(S_3 \\times 3) \\leq 2^{2+4}.(S_3 \\times\ S_3)$"], [[1,7,10,12,14,16,21],"$(A_4 \\times A_5).2$"], [[1,2,7,8,10,11,12,13,14,15,16,17,20,21],"$A_4 \\times A_5 \\leq (A_4 \\times \ A_5).2$"], [[1,3,10,11,12,14,21,23],"$(A_5 \\times D_{10}).2$"], [[1,8,10,11,12,13,14,16,21,23,26,27],"$5^2:(4 \\times S_3)$"], ]; MULTFREEINFO.("M23"):= ["$M_{23}$", [[1,2],"$M_{22}$"], [[1,2,5],"$L_3(4).2_2$"], [[1,2,5],"$2^4:A_7$"], [[1,2,5,9],"$A_8$"], [[1,2,5,16],"$M_{11}$"], ]; MULTFREEINFO.("HS"):= ["$HS$", [[1,2,3],"$M_{22}$"], [[1,7],"$U_3(5).2$"], [[1,2,5,7],"$U_3(5) \\leq U_3(5).2$"], [[1,7],"$U_3(5).2$"], [[1,2,6,7],"$U_3(5) \\leq U_3(5).2$"], [[1,2,3,7,13],"$L_3(4).2_1$"], [[1,3,4,7,9],"$A_8.2$"], [[1,2,3,4,5,6,7,9,10],"$A_8 \\leq A_8.2$"], [[1,2,3,4,7,9,10,13,18],"$4^3:L_3(2)$"], [[1,2,3,5,7,10,13,16,22],"$M_{11}$"], [[1,2,3,6,7,10,13,16,22],"$M_{11}$"], [[1,3,4,7,9,13,16,17,18],"$4.2^4:S_5$"], ]; MULTFREEINFO.("HS.2"):= ["$HS.2$", [[1,3,5],"$M_{22}.2$"], [[1,2,3,4,5,6],"$M_{22} \\leq M_{22}.2$"], [[1,2,10,11],"$U_3(5).2$"], [[1,2,3,4,9,10,11],"$U_3(5) \\leq U_3(5).2$"], [[1,10,11,14,19,21,22,25,26,29,31,34,35,37,39],"$5^{1+2}_+:[2^5]$"], [[1,3,5,10,19],"$L_3(4).2^2$"], [[1,3,4,5,6,10,13,17,19],"$L_3(4).2_3 \\leq L_3(4).2^2$"], [[1,2,3,4,5,6,10,11,19,20],"$L_3(4).2_1 \\leq L_3(4).2^2$"], [[1,5,7,10,14],"$A_8.2 \\times 2$"], [[1,3,5,7,9,10,14,16],"$A_8 \\times 2 \\leq A_8.2 \\times 2$"], [[1,4,5,7,9,10,14,17],"$A_8.2 \\leq A_8.2 \\times 2$"], [[1,2,5,6,7,8,10,11,14,15],"$A_8.2 \\leq A_8.2 \\times 2$"], [[1,3,5,7,10,14,16,19,26],"$4^3:(L_3(2) \\times 2)$"], [[1,2,3,4,5,6,7,8,10,11,14,15,16,17,19,20,26,27],"$4^3:L_3(2) \\leq 4^3:(L_3(2\ ) \\times 2)$"], [[1,2,3,4,5,6,9,10,11,16,17,19,20,22,23,34,35],"$M_{11}$"], [[1,5,7,10,14,19,22,25,26],"$2^{1+6}_+:S_5$"], [[1,2,5,6,7,8,10,11,14,15,19,20,22,23,24,25,26,27],"$4.2^4:S_5 \\leq 2^{1+6}_+\ :S_5$"], ]; MULTFREEINFO.("J3"):= ["$J_3$", [[1,4,5,6,10,11,12,13],"$L_2(16).2$"], ]; MULTFREEINFO.("J3.2"):= ["$J_3.2$", [[1,4,6,10,13,15,16],"$L_2(16).4$"], [[1,5,10,12,13,14,15,16,18,20,22,24,25,27,29],"$3^2.3^{1+2}:8.2$"], ]; MULTFREEINFO.("M24"):= ["$M_{24}$", [[1,2],"$M_{23}$"], [[1,2,7],"$M_{22}.2$"], [[1,2,7,9],"$2^4:A_8$"], [[1,7,14],"$M_{12}.2$"], [[1,2,7,14,17],"$M_{12} \\leq M_{12}.2$"], [[1,7,9,14],"$2^6:3.S_6$"], [[1,7,9,14,18],"$2^6:3.A_6 \\leq 2^6:3.S_6$"], [[1,2,7,9,17],"$L_3(4).3.2_2$"], [[1,2,7,8,9,17,18],"$L_3(4).3 \\leq L_3(4).3.2_2$"], [[1,7,9,14,19],"$2^6:(L_3(2) \\times S_3)$"], [[1,7,8,9,14,17,19,20],"$2^6:(L_3(2) \\times 3) \\leq 2^6:(L_3(2) \\times S_3)\ $"], [[1,7,9,14,17,18,19,20,23,24,26],"$2^6:(7:3 \\times S_3) \\leq 2^6:(L_3(2) \\t\ imes S_3)$"], ]; MULTFREEINFO.("McL"):= ["$McL$", [[1,2,4],"$U_4(3)$"], [[1,2,4,9],"$M_{22}$"], [[1,2,4,9],"$M_{22}$"], [[1,2,4,9,14],"$U_3(5)$"], [[1,4,12,14,15],"$3^{1+4}:2S_5$"], [[1,4,9,14,15,20],"$2.A_8$"], ]; MULTFREEINFO.("McL.2"):= ["$McL.2$", [[1,3,7],"$U_4(3).2_3$"], [[1,2,3,4,7,8],"$U_4(3) \\leq U_4(3).2_3$"], [[1,2,3,4,7,8,14,15],"$M_{22}$"], [[1,4,7,14,24],"$U_3(5).2$"], [[1,2,3,4,7,8,14,15,24,25],"$U_3(5) \\leq U_3(5).2$"], [[1,7,20,24,26],"$3^{1+4}:4S_5$"], [[1,2,7,8,20,21,24,25,26,27],"$3^{1+4}:2S_5 \\leq 3^{1+4}:4S_5$"], [[1,6,7,20,24,26,27,31],"$3^{1+4}:2S_5 \\leq 3^{1+4}:4S_5$"], [[1,7,14,24,26,30],"$2.S_8$"], [[1,2,7,8,14,15,24,25,26,27,30,31],"$2.A_8 \\leq 2.S_8$"], ]; MULTFREEINFO.("He"):= ["$He$", [[1,2,3,6,9],"$S_4(4).2$"], [[1,2,3,6,7,8,9],"$S_4(4) \\leq S_4(4).2$"], [[1,2,3,6,9,12,14],"$2^2.L_3(4).S_3$"], [[1,2,3,6,7,8,9,12,14,15],"$2^2.L_3(4).3 \\leq 2^2.L_3(4).S_3$"], ]; MULTFREEINFO.("He.2"):= ["$He.2$", [[1,3,5,9],"$S_4(4).4$"], [[1,3,5,8,11,15],"$2^2.L_3(4).D_{12}$"], [[1,3,5,7,8,11,15,17],"$2^2.L_3(4).S_3 \\leq 2^2.L_3(4).D_{12}$"], [[1,3,5,7,8,11,15,18],"$2^2.L_3(4).6 \\leq 2^2.L_3(4).D_{12}$"], ]; MULTFREEINFO.("Ru"):= ["$Ru$", [[1,5,6],"${^2F_4(2)^{\\prime}}.2$"], [[1,4,5,6,7],"${^2F_4(2)^{\\prime}} \\leq {^2F_4(2)^{\\prime}}.2$"], [[1,6,8,14,15,16,21,23,25,32],"$(2^2 \\times Sz(8)):3$"], ]; MULTFREEINFO.("Suz"):= ["$Suz$", [[1,4,5],"$G_2(4)$"], [[1,3,4,9,15],"$3.U_4(3):2$"], [[1,2,3,4,9,10,11,15],"$3.U_4(3) \\leq 3.U_4(3):2$"], [[1,2,3,9,11,12],"$U_5(2)$"], [[1,2,4,6,9,12,16,17,27],"$2^{1+6}_-.U_4(2)$"], [[1,3,4,5,9,11,12,15,17,27,28,30,33],"$2^{4+6}:3A_6$"], ]; MULTFREEINFO.("Suz.2"):= ["$Suz.2$", [[1,7,9],"$G_2(4).2$"], [[1,2,7,8,9,10],"$G_2(4) \\leq G_2(4).2$"], [[1,5,7,14,23],"$3.U_4(3).2^2$"], [[1,2,5,6,7,8,14,15,23,24],"$3.U_4(3).2_3^{\\prime} \\leq 3.U_4(3).2^2$"], [[1,3,5,7,14,16,18,23],"$3.U_4(3).2_3 \\leq 3.U_4(3).2^2$"], [[1,4,5,7,14,17,19,23],"$3.U_4(3).2_1 \\leq 3.U_4(3).2^2$"], [[1,2,3,4,5,6,7,8,14,15,16,17,18,19,23,24],"$3.U_4(3) \\leq 3.U_4(3).2^2$"], [[1,4,5,14,19,21],"$U_5(2).2$"], [[1,2,3,4,5,6,14,15,18,19,20,21],"$U_5(2) \\leq U_5(2).2$"], [[1,4,7,11,14,21,26,27,38],"$2^{1+6}_-.U_4(2).2$"], [[1,2,3,4,7,8,11,12,14,15,20,21,25,26,27,28,38,39],"$2^{1+6}_-.U_4(2) \\leq 2^\ {1+6}_-.U_4(2).2$"], [[1,5,6,14,19,21,22,23,33,48],"$3^5:(M_{11} \\times 2)$"], [[1,5,7,9,14,19,21,23,27,38,40,44,47],"$2^{4+6}:3S_6$"], [[1,2,5,6,7,8,9,10,14,15,18,19,20,21,23,24,27,28,38,39,40,41,44,45,47,48],"$2^\ {4+6}:3A_6 \\leq 2^{4+6}:3S_6$"], ]; MULTFREEINFO.("ON"):= ["$ON$", [[1,2,7,8,11],"$L_3(7).2$"], [[1,2,7,8,10,11,18],"$L_3(7) \\leq L_3(7).2$"], [[1,2,7,9,11],"$L_3(7).2$"], [[1,2,7,9,10,11,18],"$L_3(7) \\leq L_3(7).2$"], ]; MULTFREEINFO.("ON.2"):= ["$ON.2$", [[1,2,3,4,7,8,9,12,13],"$L_3(7).2$"], [[1,2,3,4,7,8,9,10,11,12,13,20,21],"$L_3(7) \\leq L_3(7).2$"], ]; MULTFREEINFO.("Co3"):= ["$Co_3$", [[1,5],"$McL.2$"], [[1,2,4,5],"$McL \\leq McL.2$"], [[1,2,5,9,15],"$HS$"], [[1,2,4,5,9,13,15,24],"$M_{23}$"], [[1,5,13,15,20,31],"$3^5:(2 \\times M_{11})$"], [[1,2,4,5,9,13,15,20,22,24,28,31],"$3^5:M_{11} \\leq 3^5:(2 \\times M_{11})$"]\ , [[1,5,14,15,20,27,29],"$2.S_6(2)$"], ]; MULTFREEINFO.("Co2"):= ["$Co_2$", [[1,4,6],"$U_6(2).2$"], [[1,2,4,6,7],"$U_6(2) \\leq U_6(2).2$"], [[1,4,5,6,8,15,17,28,36,39,44],"$U_5(2).2 \\leq U_6(2).2$"], [[1,2,3,4,5,6,7,8,15,16,17,19,21,28,35,36,39,42,44,48],"$U_5(2) \\leq U_6(2).2\ $"], [[1,4,6,14,17],"$2^{10}:M_{22}:2$"], [[1,2,4,6,7,14,17,20],"$2^{10}:M_{22} \\leq 2^{10}:M_{22}:2$"], [[1,2,4,7,14,18],"$McL$"], [[1,4,6,15,17],"$2^{1+8}:S_6(2)$"], [[1,4,6,14,17,27,33],"$HS.2$"], [[1,2,4,6,7,14,17,18,20,27,33,38],"$HS \\leq HS.2$"], ]; MULTFREEINFO.("Fi22"):= ["$Fi_{22}$", [[1,3,7],"$2.U_6(2)$"], [[1,3,9],"$O_7(3)$"], [[1,3,9],"$O_7(3)$"], [[1,7,9,13],"$O_8^+(2).3.2$"], [[1,4,7,8,9,13,15],"$O_8^+(2).3 \\leq O_8^+(2).3.2$"], [[1,3,7,9,13,14,17],"$O_8^+(2).2 \\leq O_8^+(2).3.2$"], [[1,2,3,5,7,10,11,17],"$2^{10}:M_{22}$"], [[1,3,5,7,9,10,13,17,25,28],"$2^6:S_6(2)$"], [[1,4,5,9,10,26,31,32,39,45,53],"${^2F_4(2)^{\\prime}}$"], ]; MULTFREEINFO.("Fi22.2"):= ["$Fi_{22}.2$", [[1,5,13],"$2.U_6(2).2$"], [[1,2,5,6,13,14],"$2.U_6(2) \\leq 2.U_6(2).2$"], [[1,2,5,6,17,18],"$O_7(3)$"], [[1,13,17,25],"$O_8^+(2).3.2 \\times 2$"], [[1,2,13,14,17,18,25,26],"$O_8^+(2).3.2 \\leq O_8^+(2).3.2 \\times 2$"], [[1,8,13,15,17,25,29],"$O_8^+(2).3 \\times 2 \\leq O_8^+(2).3.2 \\times 2$"], [[1,7,13,16,17,25,30],"$O_8^+(2).S_3 \\leq O_8^+(2).3.2 \\times 2$"], [[1,5,13,17,25,27,33],"$O_8^+(2).2 \\times 2 \\leq O_8^+(2).3.2 \\times 2$"], [[1,2,7,8,13,14,15,16,17,18,25,26,29,30],"$O_8^+(2).3 \\leq O_8^+(2).3.2 \\tim\ es 2$"], [[1,2,5,6,13,14,17,18,25,26,27,28,33,34],"$O_8^+(2).2 \\leq O_8^+(2).3.2 \\tim\ es 2$"], [[1,5,6,7,13,16,17,25,27,28,30,33,34],"$O_8^+(2).2 \\leq O_8^+(2).3.2 \\times \ 2$"], [[1,3,5,9,13,19,21,33],"$2^{10}:M_{22}.2$"], [[1,2,3,4,5,6,9,10,13,14,19,20,21,22,33,34],"$2^{10}:M_{22} \\leq 2^{10}:M_{22\ }.2$"], [[1,5,9,13,17,19,25,33,46,52],"$2^7:S_6(2)$"], [[1,2,5,6,9,10,13,14,17,18,19,20,25,26,33,34,46,47,52,53],"$2^6:S_6(2) \\leq 2\ ^7:S_6(2)$"], [[1,7,9,17,19,49,58,68,75,88],"${^2F_4(2)}$"], ]; MULTFREEINFO.("HN"):= ["$HN$", [[1,2,3,4,5,8,10,11,12,18,20,23],"$A_{12}$"], [[1,2,3,4,5,8,9,10,11,12,18,20,23,24,32,39,40,41,47],"$A_{11} \\leq A_{12}$"], [[1,5,8,9,10,17,18,20,24],"$2.HS.2$"], [[1,4,5,9,11,12,18,19,21,22,25,26,32,34,35,36,37,41,49],"$U_3(8).3_1$"], ]; MULTFREEINFO.("HN.2"):= ["$HN.2$", [[1,3,4,6,9,13,15,20,24,27],"$S_{12}$"], [[1,3,4,6,9,11,13,15,20,24,27,29,36,47,49,51,63],"$S_{11} \\leq S_{12}$"], [[1,6,9,11,13,18,20,24,29],"$4.HS.2$"], [[1,2,6,7,9,10,11,12,13,14,18,19,20,21,24,25,29,30],"$2.HS.2 \\leq 4.HS.2$"], [[1,4,6,9,11,13,18,19,20,22,24,29,33],"$2.HS.2 \\leq 4.HS.2$"], [[1,4,6,11,15,20,22,26,31,36,40,42,43,51,67],"$U_3(8).6$"], ]; MULTFREEINFO.("Ly"):= ["$Ly$", [[1,4,11,12,14],"$G_2(5)$"], [[1,4,11,12,15],"$3.McL.2$"], [[1,4,10,11,12,13,15,16],"$3.McL \\leq 3.McL.2$"], ]; MULTFREEINFO.("Th"):= ["$Th$", [[1,3,7,8,19,21,25,32,37,39,41],"${^3D_4(2)}.3$"], [[1,8,17,18,25,32,37,38,39,42,46],"$2^5.L_5(2)$"], ]; MULTFREEINFO.("Fi23"):= ["$Fi_{23}$", [[1,2,6],"$2.Fi_{22}$"], [[1,6,8],"$O_8^+(3).3.2$"], [[1,5,6,8,9],"$O_8^+(3).3 \\leq O_8^+(3).3.2$"], [[1,2,6,8,10],"$O_8^+(3).2_2 \\leq O_8^+(3).3.2$"], [[1,2,3,6,7,8,10,14,20,24,38,40,42],"$S_8(2)$"], [[1,2,3,6,7,10,13,14,19,20,24,26,38,41,42,60],"$2^{11}.M_{23}$"], ]; MULTFREEINFO.("Co1"):= ["$Co_1$", [[1,3,6,10],"$Co_2$"], [[1,4,7,16,20],"$3.Suz.2$"], [[1,2,4,7,11,16,20,22],"$3.Suz \\leq 3.Suz.2$"], [[1,6,10,16,25,32],"$2^{11}:M_{24}$"], [[1,3,6,10,14,26,32],"$Co_3$"], [[1,3,6,7,10,12,16,29,32,37,46],"$2^{1+8}_+.O_8^+(2)$"], ]; MULTFREEINFO.("J4"):= ["$J_4$", [[1,8,11,14,19,20,21],"$2^{11}:M_{24}$"], [[1,8,11,14,19,20,21,29,30,45,51],"$2^{11}:M_{23} \\leq 2^{11}:M_{24}$"], ]; MULTFREEINFO.("F3+"):= ["$F_{3+}$", [[1,3,4],"$Fi_{23}$"], [[1,2,3,4,5,8,11,13,16,20,24,26,27,29,38,41,45],"$O_{10}^-(2)$"], [[1,3,4,11,12,16,17,19,22,24,29,30,32,39,40,44,45,59],"$3^7.O_7(3)$"], ]; MULTFREEINFO.("F3+.2"):= ["$F_{3+}.2$", [[1,5,7],"$Fi_{23} \\times 2$"], [[1,2,5,6,7,8],"$Fi_{23} \\leq Fi_{23} \\times 2$"], [[1,4,5,7,10,15,21,26,28,37,44,48,51,54,73,75,83],"$O_{10}^-(2).2$"], [[1,2,3,4,5,6,7,8,9,10,15,16,21,22,25,26,28,29,36,37,44,45,48,49,50,51,54,55,7\ 2,73,75,76,83,84],"$O_{10}^-(2) \\leq O_{10}^-(2).2$"], [[1,5,7,21,23,28,30,34,41,44,54,56,60,74,81,83,108],"$3^7.O_7(3).2$"], ]; MULTFREEINFO.("B"):= ["$BM$", [[1,3,5,13,15],"$2.{}^2E_6(2).2$"], [[1,2,3,5,7,13,15,17],"$2.{}^2E_6(2) \\leq 2.{}^2E_6(2).2$"], [[1,3,5,8,13,15,28,30,37,40],"$2^{1+22}.Co_2$"], [[1,2,3,5,7,8,9,12,13,15,17,23,27,30,32,40,41,54,63,68,77,81,83],"$Fi_{23}$"], ]; MULTFREEINFO.("M"):= ["$M$", [[1,2,4,5,9,14,21,34,35],"$2.BM$"], ]; MakeImmutable( MULTFREEINFO ); ############################################################################# ## #F MultFreePermChars( ) ## ## For a string that is the name of a sporadic simple group or of the ## automorphism group of a sporadic simple group, ## `MultFreePermChars' returns a list of records that describe the faithful ## multiplicity-free permutation characters of this group, ## in a format that is similar to the classification shown in~\cite{BL96}. ## ## If is the string `\"all\"' then `MultFreePermChars' returns the ## list of faithful multiplicity-free permutation characters of all sporadic ## simple groups and their automorphism groups. ## ## Each entry in the result list has the following components. ## \beginitems ## group & ## {\LaTeX} format of , ## ## character & ## the permutation character, ## ## rank & ## the rank of the character, ## ## subgroup & ## a string that is a name (in {\LaTeX} format) of the subgroup ## from whose trivial character the permutation character is induced, and ## ## ATLAS & ## a string that describes (in {\LaTeX} format) the constituents of the ## permutation character, relative to the simple group involved; ## the format is described in the section~"ref:PermCharInfoRelative" ## in the {\GAP} Reference Manual. ## \enditems ## DeclareGlobalFunction( "MultFreePermChars" ); InstallGlobalFunction( MultFreePermChars, function( name ) local result, # the result list tbl, # character table with `Identifier' value `name' group, # value of the `group' component of each result entry len, # length of the list stored for `name' chars, # list of the permutation characters for `name' tblsimp, # character table of the derived subgroup of `tbl' info, # list of `ATLAS' values i, # loop over the permutation characters entry; # one entry in the record for `name' result:= []; if IsBound( MULTFREEINFO.( name ) ) then tbl:= CharacterTable( name ); group:= MULTFREEINFO.( name )[1]; len:= Length( MULTFREEINFO.( name ) ); chars:= List( MULTFREEINFO.( name ){ [ 2 .. len ] }, x -> Sum( Irr( tbl ){ x[1] } ) ); if '.' in name then tblsimp:= CharacterTable( name{ [ 1 .. Position( name, '.' )-1 ] } ); info:= PermCharInfoRelative( tblsimp, tbl, chars ).ATLAS; else info:= PermCharInfo( tbl, chars ).ATLAS; fi; for i in [ 2 .. len ] do entry:= MULTFREEINFO.( name )[i]; Add( result, rec( group := group, character := chars[ i-1 ], rank := Length( entry[1] ), subgroup := entry[2], ATLAS := info[ i-1 ] ) ); od; elif name = "all" then for name in RecNames( MULTFREEINFO ) do Append( result, MultFreePermChars( name ) ); od; else Error( " must be the name of a sporadic simple group\n", "or of the automorphism group of a sporadic simple group,\n", "or the string \"all\"" ); fi; return result; end ); ############################################################################# ## #E