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## #W exam.gi POLENTA package Bjoern Assmann ## ## examples for polycyclic rational matrix groups ## #Y 2003 ## ############################################################################# ## #F POL_KroneckerProduct ## ## Examples built with the Kroneckerproduct. ## POL_KroneckerProduct := function( G, e ) local newGens, gens; gens := GeneratorsOfGroup( G ); if e = 0 then e := RandomList( [1,-1] ); fi; newGens := List( gens, x->KroneckerProduct(x,x^e) ); return Group( newGens ); end; ############################################################################# ## #F POL_AlmostCrystallographicGroup ## ## Examples coming from aclib ## if not IsPackageMarkedForLoading( "aclib" , "1.0" ) then POL_AlmostCrystallographicGroup := false; else POL_AlmostCrystallographicGroup := function( a,b,c ) local G, mats; G := AlmostCrystallographicGroup(a,b,c); mats := GeneratorsOfGroup( G ); G := Group( mats ); return G; end; fi; ############################################################################# ## #F PolExamples( n ) .............. .. Examples for polycyc rat. matrix groups ## # FIXME: This function is documented and should be turned into a GlobalFunction PolExamples := function( n ) local i,M,P, nat, G, gens, d, l,r,s; # check if aclib is needed l := Concatenation( [9..19],[21..24]); if POL_AlmostCrystallographicGroup = false then if n in l then Print( "package 'aclib' is needed for this example.\n" ); return fail; fi; fi; # some unipotent groups if n=1 then return Group( [ [ [ 1, -4, 1, 2 ], [ 0, 1, 5, 1 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ] , [ [ 1, -1, -2, 4 ], [ 0, 1, -1, 1 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ] , [ [ 1, -4, -2, 1 ], [ 0, 1, -2, -3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ] ] ]); fi; if n=2 then return Group( [ [ [ 1, 5, -3, 1, -3 ], [ 0, 1, 2, 0, 1 ], [ 0, 0, 1, 2, -2 ], [ 0, 0, 0, 1, -2 ], [ 0, 0, 0, 0, 1 ] ] , [ [ 1, 3, -3, -1, 1 ], [ 0, 1, 0, -2, 2 ], [ 0, 0, 1, 4, 1 ], [ 0, 0, 0, 1, -1 ], [ 0, 0, 0, 0, 1 ] ] ]); fi; if n=3 then return Group( [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ] , [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ]); fi; if n=4 then return Group( [ [ [ 5, 2, -8, 17, -1 ], [ -69/4, -15/4, 449/20, -163/5, 53/20 ], [ -2, 4, 9/5, 63/5, 3/5 ], [ 13/4, 3/4, -121/20, 57/5, -17/20 ], [ 241/4, 7/4, -1477/20, 319/5, -189/20 ] ] , [ [ 19/2, 0, -3, -19/2, -1/2 ], [ -74/5, 129/20, 7/4, 159/4, 9/10 ], [ 53/10, 4/5, -4, 9/2, -9/10 ], [ 37/10, -41/20, -7/4, -29/4, -3/5 ], [ 137/5, -457/20, 37/4, -559/4, 3/10 ] ] ]); fi; if n in [5..8] then return MatExamples( n-4 ); fi; if n=9 then return POL_AlmostCrystallographicGroup( 3, 17, [1,0,-3,7] ); fi; if n=10 then return POL_AlmostCrystallographicGroup( 3, 16, [3,4,-1,0] ); fi; if n=11 then return POL_AlmostCrystallographicGroup( 3, 15, [-1,1,0,0] ); fi; if n=12 then return POL_AlmostCrystallographicGroup( 3, 14, [-2,-1,2,-1] ); fi; if n=13 then return POL_AlmostCrystallographicGroup( 4, 2, [0,-3,-2,1,-1,-2,4] ); fi; if n=14 then return POL_AlmostCrystallographicGroup( 4, 13, [-2,0,-2,-2,-1,-1] ); fi; if n=15 then return POL_AlmostCrystallographicGroup( 4, 37, [ 1,2,0,-3,-4] ); fi; if n=16 then return POL_AlmostCrystallographicGroup( 4, 86,[0,-2,-1,-2,-1] ); fi; if n=17 then return POL_AlmostCrystallographicGroup( 4, 10,[-2,-2,-1] ); fi; if n=18 then return POL_AlmostCrystallographicGroup( 4, 11,[-4,-2,-1] ); fi; if n=19 then return POL_AlmostCrystallographicGroup( 4, 12,[0,1,3,-1] ); fi; if n=20 then return POL_KroneckerProduct( MatExamples(4), -1 ); fi; if n in [21..24] then return POL_KroneckerProduct( PolExamples( n-12 ), 1 ); fi; if n = 25 then #Marco Costantini example r := [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ] ]; s := [ [ 1, 0, 0, 0, 0, 0, 0, 1 ], [ 0, E(7)^6, 0, 0, 0, 0, 0, 0 ], [ 0, 0, E(7), 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ] ]; return Group( [r,s^7] ); fi; if n = 26 then # subgroup of a group which is conjugate to a direct product # of PolExamples(1) and PolExamples(6); gens := [ [ [ -403, 2500, 32, -633, -1638, 800, 2073, 406 ], [ 3, 37, 20, -9, 5, -11, 29, 7 ], [ -65, -16, 5, 3, -73, 129, -43, 59 ] , [ -37, -2936, 80, 735, 1366, 54, -2676, 28 ], [ 2, -788, 0, 197, 364, -4, -707, -12 ], [ -207, 20, 16, -9, -263, 411, -71, 191 ], [ -13, -1276, 0, 319, 567, 26, -1152, -6 ], [ 8, 972, 0, -243, -434, -16, 877, 7 ] ], [ [ -615, 40, -64, -12, -850, 1248, -241, 515 ], [ -9, 37, -4, -9, -31, 19, 29, 7 ], [ -39, 347, -7, -87, -205, 80, 293, 41 ], [ 437, 3676, -16, -918, -1054, -870, 3495, -290 ], [ 106, 748, 0, -187, -189, -212, 719, -72 ], [ -149, 1184, -32, -297, -727, 307, 995, 151 ], [ 187, 1376, 0, -344, -359, -374, 1319, -126 ], [ -106, -816, 0, 204, 217, 212, -779, 70 ] ], [ [ -352, -632, -64, 150, -246, 722, -699, 251 ], [ 17, -107, -8, 27, 63, -32, -87, -21 ], [ -59, -16, -7, 3, -73, 120, -37, 51 ], [ -58, 84, -32, -20, -93, 124, 64, 87 ], [ -2, 220, 0, -55, -90, 4, 199, 18 ], [ -188, -16, -32, 0, -256, 385, -82, 160 ], [ -36, 260, 0, -65, -144, 72, 221, 57 ], [ 3, -228, 0, 57, 94, -6, -206, -20 ] ], [ [ 893, 7836, 0, -1959, -2319, -1784, 7446, -585 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 76, 456, 1, -114, -102, -152, 444, -54 ], [ -210, -4128, 0, 1033, 1567, 420, -3805, 87 ], [ -158, -1776, 0, 444, 587, 316, -1668, 96 ], [ 228, 1368, 0, -342, -306, -455, 1332, -162 ], [ -142, -2088, 0, 522, 746, 284, -1940, 75 ], [ 190, 2100, 0, -525, -690, -380, 1974, -116 ] ], [ [ -1656, -60, 0, 15, -1925, 3314, -920, 1473 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ -46, 0, 1, 0, -50, 92, -26, 42 ], [ 1626, 236, 0, -58, 1847, -3252, 1043, -1434 ], [ 468, 20, 0, -5, 551, -936, 259, -414 ], [ -138, 0, 0, 0, -150, 277, -78, 126 ], [ 700, 76, 0, -19, 805, -1400, 428, -618 ], [ -519, 0, 0, 0, -621, 1038, -267, 460 ] ], [ [ -427, -4608, 0, 1152, 1464, 856, -4359, 207 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ -104, -360, 1, 90, 36, 208, -372, 90 ], [ -525, 2300, 0, -574, -1560, 1050, 1845, 660 ], [ -48, 876, 0, -219, -421, 96, 771, 96 ], [ -312, -1080, 0, 270, 108, 625, -1116, 270 ], [ -171, 1104, 0, -276, -657, 342, 923, 234 ], [ 58, -936, 0, 234, 457, -116, -821, -110 ] ], [ [ -479, 2668, -64, -669, -1882, 976, 2189, 434 ], [ -9, 37, -4, -9, -31, 19, 29, 7 ], [ -71, -13, -7, 3, -91, 144, -43, 59 ], [ -85, -2936, -16, 735, 1222, 174, -2676, 28 ], [ 2, -788, 0, 197, 364, -4, -707, -12 ], [ -245, 104, -32, -27, -385, 499, -13, 205 ], [ -13, -1276, 0, 319, 567, 26, -1152, -6 ], [ 8, 972, 0, -243, -434, -16, 877, 7 ] ], [ [ -539, -128, 32, 24, -606, 1072, -357, 487 ], [ 3, 37, 20, -9, 5, -11, 29, 7 ], [ -33, 344, 5, -87, -187, 65, 293, 41 ], [ 485, 3676, 80, -918, -910, -990, 3495, -290 ], [ 106, 748, 0, -187, -189, -212, 719, -72 ], [ -111, 1100, 16, -279, -605, 219, 937, 137 ], [ 187, 1376, 0, -344, -359, -374, 1319, -126 ], [ -106, -816, 0, 204, 217, 212, -779, 70 ] ], [ [ -352, -632, -64, 150, -246, 722, -699, 251 ], [ 17, -107, -8, 27, 63, -32, -87, -21 ], [ -59, -16, -7, 3, -73, 120, -37, 51 ], [ -58, 84, -32, -20, -93, 124, 64, 87 ], [ -2, 220, 0, -55, -90, 4, 199, 18 ], [ -188, -16, -32, 0, -256, 385, -82, 160 ], [ -36, 260, 0, -65, -144, 72, 221, 57 ], [ 3, -228, 0, 57, 94, -6, -206, -20 ] ] ]; return Group( gens ); fi; if n = 27 then # subgroup of a group which is conjugate to a direct product # of PolExamples(3) and PolExamples(5); gens := [ [ [ -396/5, 49, -4/5, 0, 147, -292, -2611/5, 1799/10, 6, -2/5, 2, -373/2, 882, 0, 0, 16/5, -2, -2, 389/2, -59 ], [ 539/10, -97/2, 78/5, 2, -283/2, 283, 3619/10, -2541/20, 15, 44/5, 2, 623/4, -849, 2, -5, -132/5, 0, 20, -623/4, 91/2 ], [ -9, 1, -10, 1, 8, -17, -51, 17, -2, -4, 2, -16, 51, 1, 1, 12, -1, -5, 15, -5 ], [ -117/10, 5/2, -49/5, 1, 19/2, -18, -737/10, 503/20, 1, -22/5, 2, -81/4, 55, 1, 0, 66/5, -1, -1, 85/4, -13/2 ], [ 32, -2, 18, -2, -10, 16, 192, -64, -11, 8, -4, 49, -52, -2, 3, -26, 2, -10, -51, 17 ], [ -30, 2, 6, 2, 10, -25, -162, 54, 11, 4, 4, -55, 73, 2, -2, -10, -2, 6, 48, -16 ], [ 169/10, -19/2, 18/5, 0, -57/2, 55, 1159/10, -791/20, -4, 9/5, -1, 141/4, -169, 0, 0, -32/5, 2, 1, -161/4, 25/2 ], [ 547/10, -57/2, 94/5, 0, -171/2, 166, 3657/10, -2493/20, -9, 47/5, -2, 463/4, -507, 0, 0, -151/5, 5, 2, -507/4, 79/2 ], [ 653/5, -41, 117/5, -1, -125, 248, 3958/5, -1326/5, -2, 56/5, -3, 257, -746, -2, 0, -173/5, 2, 2, -259, 85 ], [ -87/5, 11, -28/5, 1, 23, -35, -652/5, 219/5, 1, -24/5, -1, -32, 111, -2, -2, 72/5, 2, 10, 43, -14 ], [ 10, 0, 0, 0, 0, 1, 57, -19, 0, 0, 0, 16, -2, -1, 0, 0, 0, 0, -15, 5 ] , [ -2, 0, -8, 0, 0, 1, -12, 4, -3, -4, -1, -4, -4, 0, 0, 11, 1, 1, 3, -1 ], [ -15, 1, -1, 1, 5, -11, -84, 28, 5, 0, 2, -26, 33, 1, -1, 1, -1, 3, 24, -8 ], [ -66, 5, -5, 6, 24, -48, -396, 132, 15, 0, 7, -107, 144, 5, -5, 0, 0, 20, 107, -36 ], [ 2139/5, -124, 236/5, -3, -381, 751, 13014/5, -4358/5, -20, 108/5, -13, 814, -2267, -11, 1, -359/5, 10, 6, -832, 273 ], [ -3, 1, -8, 1, 3, -3, -24, 8, -1, -4, 1, -3, 11, 0, 0, 12, 0, 0, 6, -2 ], [ 1241/5, -81, 234/5, -1, -248, 491, 7556/5, -2532/5, -3, 112/5, -3, 492, -1477, -3, 1, -341/5, 2, -2, -497, 163 ], [ 6, 0, -8, 0, 0, -1, 39, -13, -3, -4, -1, 6, 1, -1, 0, 11, 1, 1, -9, 3 ], [ -639/10, 97/2, -158/5, -2, 283/2, -281, -4249/10, 2961/20, -8, -84/5, -3, -683/4, 843, -2, 2, 247/5, 0, -7, 683/4, -101/2 ], [ -243/10, 97/2, -191/5, -5, 271/2, -268, -2003/10, 1487/20, -12, -108/5, -5, -433/4, 807, -5, 5, 324/5, -3, -20, 445/4, -59/2 ] ], [ [ -1561/10, -2/5, -134/5, 0, -26/5, 72/5, -852, 2699/10, 2, -67/5, 2, -2431/10, -196/5, 0, 0, 211/5, -2, -2, 2451/10, -489/5 ], [ 3529/20, -21/5, 53/5, 0, -63/5, 151/5, 981, -6261/20, 0, 53/10, 2, 5589/20, -428/5, -3, 0, -109/10, 0, -2, -5589/20, 1071/10 ], [ -22, 0, 0, 0, 0, -2, -129, 43, 0, 0, 1, -32, 5, 2, 0, -1, 0, -2, 33, -11 ], [ 113/20, -7/5, 1/5, 0, -21/5, 47/5, 33, -217/20, 0, 1/10, 1, 233/20, -136/5, 0, 0, 7/10, 0, -1, -213/20, 37/10 ], [ -10, 0, 0, 0, 0, -1, -66, 22, 0, 0, -2, -15, 2, 1, 0, -3, 0, 2, 15, -5 ], [ 0, 0, 0, 0, -3, 6, 6, -2, 0, 0, 2, 0, -18, 0, 0, 2, 0, -2, 0, 0 ], [ 519/20, 19/5, 48/5, 1, 77/5, -164/5, 130, -791/20, 2, 53/10, -1, 719/20, 482/5, 1, -1, -179/10, 1, 6, -739/20, 171/10 ], [ 1517/20, 42/5, 104/5, 3, 176/5, -372/5, 378, -2293/20, 7, 119/10, -2, 2177/20, 1096/5, 3, -3, -407/10, 2, 17, -2217/20, 503/10 ], [ 1004/5, -64/5, 72/5, 0, -192/5, 374/5, 1101, -1751/5, 2, 36/5, -2, 1639/5, -1132/5, 0, -1, -118/5, 1, 6, -1644/5, 632/5 ], [ -31/5, 1/5, -128/5, 0, 3/5, 29/5, -18, 9/5, -3, -64/5, 1, -76/5, -77/5, -4, 0, 202/5, 0, 4, 66/5, -48/5 ], [ 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0 ], [ 2, 2, 8, 0, 8, -18, 12, -4, -1, 4, -1, 0, 52, 0, 0, -13, 1, 1, -1, 1 ], [ 0, 0, 0, 0, -1, 2, 3, -1, 0, 0, 1, 0, -6, 0, 0, 1, 0, -1, 0, 0 ], [ -3, 0, 0, 1, -1, 7, -3, 1, 0, 0, 6, -5, -16, 0, 0, 5, 0, -5, 5, -2 ], [ 3062/5, -157/5, 376/5, 0, -436/5, 737/5, 3384, -5388/5, 2, 188/5, -10, 4877/5, -2346/5, 0, -3, -639/5, 7, 23, -4972/5, 1921/5 ], [ 0, -1, -8, 0, -3, 7, 0, 0, -1, -4, 1, 1, -21, 0, 0, 12, 0, 0, -1, 0 ] , [ 2008/5, -133/5, 104/5, 0, -409/5, 803/5, 2202, -3502/5, 2, 52/5, -2, 3288/5, -2424/5, 0, -1, -171/5, 1, 7, -3293/5, 1264/5 ], [ 2, 2, 8, 0, 8, -21, 18, -6, -1, 4, -1, -3, 58, 0, 0, -14, 1, 1, -1, 1 ], [ -2629/20, 26/5, -8/5, -1, 78/5, -176/5, -708, 4441/20, -4, -13/10, -3, -4269/20, 508/5, -1, 1, 19/10, 1, -2, 4249/20, -841/10 ], [ -1923/20, -8/5, -36/5, -3, -54/5, 83/5, -501, 3067/20, -9, -51/10, -5, -3003/20, -274/5, -3, 3, 133/10, 0, -10, 3003/20, -637/10 ] ], [ [ -59, 10, -12, 0, 66, -154, -312, 104, 2, 0, 0, -120, 446, 6, -2, -8, 2, 4, 100, -30 ], [ -114, 9, -3, 0, 31, -65, -648, 216, 5, 0, 7, -186, 195, 10, 5, 7, -10, -30, 179, -57 ], [ 27, -3, -6, 1, -14, 33, 150, -50, 2, -4, 1, 46, -97, -3, -3, 11, 3, 15, -43, 13 ], [ -9, 0, -1, 0, 3, -7, -48, 16, 3, 0, 1, -15, 20, 1, -1, 0, 0, 3, 13, -4 ], [ 26, 0, 15, 0, 4, -15, 162, -54, -3, 8, -3, 34, 39, -2, -1, -27, 4, 6, -39, 13 ], [ -70, 4, -17, 0, 12, -25, -396, 132, 5, -8, 5, -115, 75, 6, 3, 29, -8, -18, 109, -35 ], [ 42, -4, 13, 0, -26, 59, 232, -77, -1, 4, -2, 74, -172, -4, 0, -10, 1, 3, -67, 21 ], [ 96, -7, 33, 0, -45, 100, 537, -178, -2, 12, -6, 162, -293, -9, -1, -34, 4, 11, -151, 48 ], [ 31, 0, 0, 1, 1, -3, 177, -59, -2, 0, -3, 47, 8, -3, -1, -2, 3, 7, -46, 15 ], [ -45, 3, 14, -5, 22, -57, -246, 82, -10, 9, -5, -74, 164, 6, 6, -28, -3, -30, 68, -20 ], [ 22, -2, 2, 0, -11, 25, 120, -40, -2, 0, -1, 39, -73, -2, 0, 0, 1, 2, -35, 11 ], [ 42, -5, 5, 0, -29, 68, 228, -76, -2, 0, -1, 78, -197, -4, 0, 2, 1, 3, -68, 21 ], [ -24, 1, -8, 0, 2, -3, -138, 46, 2, -4, 2, -38, 10, 2, 1, 14, -3, -6, 37, -12 ], [ -110, 5, -1, 0, 13, -28, -612, 204, 6, 0, 10, -182, 82, 10, 4, 9, -9, -26, 170, -55 ], [ 276, -25, 24, 2, -139, 318, 1506, -502, -16, 0, -19, 489, -926, -26, -3, 1, 14, 38, -438, 137 ], [ 3, -1, 0, -1, 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5, -10, -18, 6, 3, 0, 6, -6, 30, 1, -1, 0, 0, 4, 6, -2 ], [ 2902/5, -177/5, 96/5, 0, -516/5, 1042/5, 3213, -5103/5, 4, 48/5, -10, 4687/5, -3136/5, 0, -4, -139/5, 4, 15, -4752/5, 1841/5 ], [ 2, 0, 8, 0, 0, -1, 12, -4, -1, 4, 0, 3, 2, 0, 0, -12, 1, 0, -3, 1 ], [ 1978/5, -123/5, 64/5, 0, -364/5, 713/5, 2181, -3467/5, 0, 32/5, -3, 3218/5, -2154/5, 0, 0, -91/5, 0, 0, -3238/5, 1249/5 ], [ -2, 0, 0, 0, 0, 2, -9, 3, 1, 0, -1, -4, -5, 0, -1, 0, 1, 4, 3, -1 ], [ -2949/20, 16/5, -93/5, -1, 48/5, -81/5, -801, 5061/20, -1, -93/10, -2, -4689/20, 253/5, 0, 0, 279/10, 0, 0, 4689/20, -921/10 ], [ -2603/20, -8/5, -51/5, -3, -24/5, 48/5, -699, 4387/20, 0, -51/10, -5, -4023/20, -144/5, 0, 0, 153/10, 0, 0, 4023/20, -807/10 ] ], [ [ -19, -4, -50, 0, -8, 16, -114, 38, -2, -24, 0, -30, -52, 2, 0, 68, -2, 4, 26, -10 ], [ -69, 11, -5, 5, 48, -102, -405, 135, 12, 0, 0, -122, 303, 5, -2, -5, -3, 8, 116, -37 ], [ 14, -3, 1, -2, -12, 27, 81, -27, -1, 0, 0, 29, -79, -1, -1, 2, 2, 3, -25, 8 ], [ -7, 0, -1, 0, 3, -6, -42, 14, 3, 0, 0, -10, 18, 1, -1, 0, 0, 4, 10, -3 ], [ 37, -4, 3, -1, -21, 44, 219, -73, -8, 0, 0, 60, -133, -3, 2, 1, 1, -6, -60, 19 ], [ -24, 2, -2, 0, 15, -36, -126, 42, 6, 0, 0, -45, 104, 2, -2, -3, 0, 6, 38, -12 ], [ 22, 2, 25, 0, 4, -8, 130, -43, -1, 12, 0, 32, 25, -2, 0, -34, 2, -1, -31, 11 ], [ 56, 4, 50, 0, 8, -16, 330, -109, -4, 24, 0, 81, 49, -5, 0, -68, 5, -1, -80, 28 ], [ 25, -3, -6, -1, -15, 29, 150, -50, -8, -4, 0, 40, -89, -2, 2, 12, 1, -7, -41, 13 ], [ -58, 6, -3, 4, 30, -66, -342, 114, 11, 1, 0, -103, 194, 5, -1, -7, -4, 6, 95, -31 ], [ 1, -1, 1, -1, -6, 12, 9, -3, -3, 0, 0, 3, -36, 0, 1, 1, 0, -5, -3, 1 ], [ 10, 1, 17, 0, 0, 2, 54, -18, 0, 8, 0, 18, -3, -1, 0, -22, 1, -1, -14, 5 ], [ -12, 1, -1, 0, 7, -16, -66, 22, 3, 0, 0, -21, 47, 1, -1, -1, 0, 3, 19, -6 ], [ -65, 10, -6, 5, 48, -97, -390, 130, 16, 0, -1, -111, 290, 5, -6, -6, 1, 25, 110, -35 ], [ 132, -11, 51, -8, -65, 127, 792, -264, -26, 20, -1, 217, -380, -11, 6, -52, 8, -28, -213, 70 ], [ -10, 0, -1, 0, 2, -4, -60, 20, 3, 0, 0, -15, 12, 1, -1, 0, 0, 4, 15, -5 ], [ 38, -5, -13, -2, -23, 42, 234, -78, -13, -8, -1, 58, -132, -3, 3, 23, 2, -10, -63, 20 ], [ 11, 0, 17, -1, -3, 6, 66, -22, 0, 8, 0, 19, -16, -1, 0, -22, 1, -2, -17, 6 ], [ 28, -3, 19, -2, -18, 38, 162, -54, -4, 8, 0, 50, -110, -2, 2, -20, 0, -10, -45, 15 ], [ 39, -10, 5, -5, -45, 90, 234, -78, -9, 0, 0, 74, -267, -2, 5, 5, -3, -23, -71, 23 ] ] ]; return Group( gens ); fi; if n = 30 then # subgroup of a group which is conjugate to a direct product # of PolExamples(1) and PolExamples(6); gens := [ [ [ -402, -128, 32, 24, -452, 798, -293, 349 ], [ 3, 37, 20, -9, 5, -11, 29, 7 ], [ -67, 56, 5, -15, -105, 133, 21, 65 ], [ -114, 660, 80, -164, -325, 208, 528, 199 ], [ -2, 220, 0, -55, -90, 4, 199, 18 ], [ -213, 236, 16, -63, -359, 423, 121, 209 ], [ -36, 260, 0, -65, -144, 72, 221, 57 ], [ 3, -228, 0, 57, 94, -6, -206, -20 ] ], [ [ -446, 1144, -64, -288, -1114, 910, 833, 407 ], [ -9, 37, -4, -9, -31, 19, 29, 7 ], [ -125, 155, -7, -39, -223, 252, 77, 113 ], [ -510, 428, -16, -106, -779, 1024, 104, 457 ], [ -108, 28, 0, -7, -134, 216, -35, 96 ], [ -407, 608, -32, -153, -781, 823, 347, 367 ], [ -190, 92, 0, -23, -254, 380, -23, 171 ], [ 159, 12, 0, -3, 176, -318, 98, -140 ] ], [ [ -621, 2704, -64, -684, -1991, 1260, 2143, 599 ], [ 17, -107, -8, 27, 63, -32, -87, -21 ], [ -1, 248, -7, -63, -117, 4, 225, 3 ], [ 856, -1252, -32, 314, 1560, -1704, -694, -819 ], [ 194, -408, 0, 102, 411, -388, -268, -186 ], [ -14, 776, -32, -198, -388, 37, 704, 16 ], [ 322, -444, 0, 111, 581, -644, -237, -303 ], [ -222, 480, 0, -120, -476, 444, 319, 214 ] ], [ [ 751, 1404, 0, -351, 316, -1500, 1588, -669 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 36, 48, 1, -12, 28, -72, 58, -30 ], [ -609, -1316, 0, 330, -135, 1218, -1457, 564 ], [ -184, -380, 0, 95, -53, 368, -423, 168 ], [ 108, 144, 0, -36, 84, -215, 174, -90 ], [ -267, -568, 0, 142, -66, 534, -629, 246 ], [ 200, 420, 0, -105, 55, -400, 466, -182 ] ], [ [ -127, 2424, 0, -606, -1351, 256, 2117, 75 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 4, 0, 1, 0, 2, -8, 2, -6 ], [ 208, -2844, 0, 712, 1634, -416, -2456, -159 ], [ 50, -808, 0, 202, 455, -100, -702, -36 ], [ 12, 0, 0, 0, 6, -23, 6, -18 ], [ 84, -1220, 0, 305, 695, -168, -1055, -63 ], [ -54, 936, 0, -234, -520, 108, 815, 40 ] ], [ [ -402, -128, 32, 24, -452, 798, -293, 349 ], [ 3, 37, 20, -9, 5, -11, 29, 7 ], [ -67, 56, 5, -15, -105, 133, 21, 65 ], [ -114, 660, 80, -164, -325, 208, 528, 199 ], [ -2, 220, 0, -55, -90, 4, 199, 18 ], [ -213, 236, 16, -63, -359, 423, 121, 209 ], [ -36, 260, 0, -65, -144, 72, 221, 57 ], [ 3, -228, 0, 57, 94, -6, -206, -20 ] ], [ [ -611, 600, 0, -150, -979, 1224, 257, 615 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ -276, -1536, 1, 384, 338, 552, -1510, 210 ], [ -1604, -13100, 0, 3276, 3766, 3208, -12532, 1065 ], [ -302, -3016, 0, 754, 947, 604, -2854, 180 ], [ -828, -4608, 0, 1152, 1014, 1657, -4530, 630 ], [ -608, -5252, 0, 1313, 1551, 1216, -5007, 393 ], [ 422, 3960, 0, -990, -1212, -844, 3759, -260 ] ], [ [ 7, 2628, -992, -657, -2051, 236, 2575, -430 ], [ -21, 37, -28, -9, -67, 49, 29, 7 ], [ -131, 324, -123, -81, -435, 293, 261, 57 ], [ -1520, -540, -112, 136, -1868, 3068, -1127, 1264 ], [ -316, -168, 0, 42, -347, 632, -288, 270 ], [ -483, 1116, -496, -279, -1591, 1091, 899, 199 ], [ -566, -276, 0, 69, -630, 1132, -494, 486 ], [ 342, 60, 0, -15, 441, -684, 201, -287 ] ], [ [ -562, 440, -64, -120, -996, 1142, 171, 461 ], [ 22, -143, -12, 36, 82, -41, -116, -28 ], [ -87, 127, -7, -33, -173, 176, 79, 79 ], [ -38, -60, -48, 16, -17, 88, -52, 59 ], [ -2, 220, 0, -55, -90, 4, 199, 18 ], [ -293, 520, -32, -135, -631, 595, 353, 265 ], [ -36, 260, 0, -65, -144, 72, 221, 57 ], [ 3, -228, 0, 57, 94, -6, -206, -20 ] ] ]; return Group( gens ); fi; if n= 31 then gens := GeneratorsOfGroup( MatExamples(4) ); return Group( gens{[1,2]} ); fi; return fail; end; ############################################################################# ## #F POL_RandomRationalTriangularGroup( dim, numberOfGens ) ## POL_RandomRationalTriangularGroup := function( dim, numberOfGens ) local gens, i, g, j, k, range, kk,ll; range := 7; ll := [-range..range]; kk := [-range..range]; ll := Filtered( ll, (x->x<>0) ); kk := Filtered( kk, (x->x<>0) ); gens:=[]; for i in [1..numberOfGens] do g:=IdentityMat(dim,Rationals); for j in [1..dim] do for k in [j..dim] do if j<=k then g[j][k] := RandomList( ll )/RandomList( kk ); else g[j][k] := 0; fi; od; od; Add(gens,g); od; return gens; end; ############################################################################# ## #F POL_RandomRationalUnipotentGroup( dim, numberOfGens ) ## POL_RandomRationalUnipotentGroup := function( dim, numberOfGens ) local G, x; G := POL_RandomRationalTriangularGroup( dim, numberOfGens ); x := RandomInvertibleMat( dim, Rationals ); return G^x; end; ############################################################################# ## #F POL_RandomSubgroup( G, numberOfgens ) ## ## POL_RandomSubgroup := function( G, numberOfNewGens ) local gens,newGens, i; gens := GeneratorsOfGroup( G ); newGens := []; for i in [1..numberOfNewGens] do Add( newGens, POL_RandomGroupElement( gens ) ); od; return Group( newGens ); end; ############################################################################# ## #F POL_DirectProduct( G1, G2 ) ## ## POL_DirectProduct:= function( G1, G2 ) local n1, n2, gens, k, i, j, matrix, x, G, gens1, gens2; gens1 := GeneratorsOfGroup( G1 ); gens2 := GeneratorsOfGroup( G2 ); n1 := Length( gens1[1] ); n2 := Length( gens2[2] ); gens := []; for k in [1..Length( gens1 )] do matrix := IdentityMat( n1+n2 ); for i in [1..n1] do for j in [1..n1] do matrix[i][j] := gens1[k][i][j]; od; od; Add( gens, matrix ); od; for k in [1..Length( gens2 )] do matrix := IdentityMat( n1+n2 ); for i in [1..n2] do for j in [1..n2] do matrix[i+n1][j+n1] := gens2[k][i][j]; od; od; Add( gens, matrix ); od; return Group( gens ); end; ############################################################################# ## #F POL_SemidirectProductVectorSpace( G ) ## POL_SemidirectProductVectorSpace := function( G ) local gens, n, newGens, g, h, i, v; gens := GeneratorsOfGroup( G ); n := Length( gens[1] ); newGens := []; for g in gens do h := POL_CopyVectorList( g ); for i in [1..n] do Add( h[i], 0 ); od; v := List( [1..n+1], x->0 ); v[n+1] := 1; Add( h, v ); Add( newGens, h ); od; for i in [1..n] do h := IdentityMat( n+1 ); h[i][n+1] := 1; Add( newGens, h ); od; return Group( newGens ); end; POL_IrredPol := function( n ) local x,p; x := Indeterminate(Rationals,"x":old); if n = 1 then p := x^2 + 6*x + 3; return p; fi; if n = 2 then p := x^3 + 14*x^2 - 49*x + 7; return p; fi; if n = 3 then p := x^2+1; return p; fi; if n = 4 then p := x^3+2*x+2; return p; fi; if n = 5 then p := x^3+2*x+2; return p; fi; if n = 6 then p := -4*x^2-2*x+1; return p; fi; end; POL_AbelianIrreducibleGens := function( n ) local k,l,ll,range,pol,mat,n_gens,gens,i,exps,l_comb,g; pol := POL_IrredPol( n ); mat := CompanionMat( pol ); range := 3; l := [-range..range]; l := Filtered( l, (x -> x<>0) ); # k := [-range..range]; k := [1]; k := Filtered( k, (x -> x<>0) ); exps := [0,1,2]; n_gens := 3; gens := []; for i in [1..n_gens] do l_comb := 2; ll := List( [1..l_comb], x-> RandomList(l)/RandomList(k) ); g := List( [1..l_comb] , x-> ll[x]*mat^RandomList(exps) ); Add( gens, Sum( g ) ); od; return gens; end; POL_RandomUnitGroupGens := function( n ) local F, U, u_gens, n_gens, gens; F := ExampleMatField( n ); U := UnitGroup( F ); u_gens := GeneratorsOfGroup( U ); n_gens := 3; gens := List( [1..n_gens], x->POL_RandomGroupElement( u_gens )); return gens; end; POL_TriangularizableGens := function( n_blocks, n_gens ) local n_fields,a_gens,i,ll,dim,j,k,start,rr,gens,g,start1,start2, rr1,rr2, u_entries, mat; n_fields := 6; a_gens := []; for i in [1..n_blocks] do Add( a_gens, POL_RandomUnitGroupGens( RandomList( [1..n_fields] )) ); od; #get dimension of composed matrix ll := List( [1..n_blocks], x-> Length( a_gens[x][1] )); dim := Sum( ll ); #get final generators gens := []; for i in [1..n_gens] do g := NullMat( dim,dim ); #write blocks for j in [1..n_blocks] do for k in [j..n_blocks] do if j = k then #place block on diagonal # set starting point start := Sum( ll{[1..(j-1)]} ); # get range rr := [1..ll[j]]+start; g{rr}{rr} := RandomList( a_gens[j] ); elif j <= k then # write upper part #set starting points start1 := Sum( ll{[1..(j-1)]} ); start2 := Sum( ll{[1..(k-1)]} ); #get range rr1 := [1..ll[j]]+start1; rr2 := [1..ll[k]]+start2; u_entries := [0,0,0,1/2,0]; mat := RandomMat( ll[j], ll[k], u_entries ); g{rr1}{rr2} := mat; fi; od; od; Add( gens, g ); od; return gens; end; ############################################################################# ## #F POL_PolExamples2( n ) .............Examples for polycyc rat. matrix groups ## POL_PolExamples2 := function( n ) local i,M,P, nat, G, gens, d, l, l1, l2, l3,h1,h2,H,g,h; # check if aclib is needed l1 := Concatenation( [9..16],[21..28],[37..40]); l2 := l1 + 40; l3 := [82..100]; l := Concatenation( l1, l2 ,l3); l1 := l+100; l := Concatenation( l, l1 ); l1 := l+300; l := Concatenation( l, l1 ); l1 := l+400; l := Concatenation( l, l1 ); if POL_AlmostCrystallographicGroup = false then if n in l then Print( "package 'aclib' is needed for this example.\n" ); return fail; fi; fi; # some unipotent groups if n=1 then return Group( [ [ [ 1, -4, 1, 2 ], [ 0, 1, 5, 1 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ] , [ [ 1, -1, -2, 4 ], [ 0, 1, -1, 1 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ] , [ [ 1, -4, -2, 1 ], [ 0, 1, -2, -3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ] ] ]); fi; if n=2 then return Group( [ [ [ 1, 5, -3, 1, -3 ], [ 0, 1, 2, 0, 1 ], [ 0, 0, 1, 2, -2 ], [ 0, 0, 0, 1, -2 ], [ 0, 0, 0, 0, 1 ] ] , [ [ 1, 3, -3, -1, 1 ], [ 0, 1, 0, -2, 2 ], [ 0, 0, 1, 4, 1 ], [ 0, 0, 0, 1, -1 ], [ 0, 0, 0, 0, 1 ] ] ]); fi; if n=3 then return Group( [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ] , [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ]); fi; if n=4 then return Group( [ [ [ 5, 2, -8, 17, -1 ], [ -69/4, -15/4, 449/20, -163/5, 53/20 ], [ -2, 4, 9/5, 63/5, 3/5 ], [ 13/4, 3/4, -121/20, 57/5, -17/20 ], [ 241/4, 7/4, -1477/20, 319/5, -189/20 ] ] , [ [ 19/2, 0, -3, -19/2, -1/2 ], [ -74/5, 129/20, 7/4, 159/4, 9/10 ], [ 53/10, 4/5, -4, 9/2, -9/10 ], [ 37/10, -41/20, -7/4, -29/4, -3/5 ], [ 137/5, -457/20, 37/4, -559/4, 3/10 ] ] ]); fi; if n in [5..8] then return MatExamples( n-4 ); fi; if n=9 then return POL_AlmostCrystallographicGroup( 3, 17, [1,0,-3,7] ); fi; if n=10 then return POL_AlmostCrystallographicGroup( 3, 16, [3,4,-1,0] ); fi; if n=11 then return POL_AlmostCrystallographicGroup( 3, 15, [-1,1,0,0] ); fi; if n=12 then return POL_AlmostCrystallographicGroup( 3, 14, [-2,-1,2,-1] ); fi; if n=13 then return POL_AlmostCrystallographicGroup( 4, 2, [0,-3,-2,1,-1,-2,4] ); fi; if n=14 then return POL_AlmostCrystallographicGroup( 4, 13, [-2,0,-2,-2,-1,-1] ); fi; if n=15 then return POL_AlmostCrystallographicGroup( 4, 37, [ 1,2,0,-3,-4] ); fi; if n=16 then return POL_AlmostCrystallographicGroup( 4, 86,[0,-2,-1,-2,-1] ); fi; if n in [17..20] then return POL_KroneckerProduct( MatExamples( n-16), -1 ); fi; if n in [21..28] then return POL_KroneckerProduct( PolExamples( n-12 ), 1 ); fi; if n=29 then return POL_DirectProduct( PolExamples( 1 ), PolExamples( 2 ) ); fi; if n=30 then return POL_DirectProduct( PolExamples( 1 ), PolExamples( 4 ) ); fi; if n=31 then return POL_DirectProduct( PolExamples( 2 ), PolExamples( 8 ) ); fi; if n=32 then return POL_DirectProduct( PolExamples( 6 ), PolExamples( 7 ) ); fi; if n=33 then return POL_DirectProduct( PolExamples( 1 ), PolExamples( 2 ) ); fi; if n=34 then return POL_DirectProduct( PolExamples( 1 ), PolExamples( 4 ) ); fi; if n=35 then return POL_DirectProduct( PolExamples( 2 ), PolExamples( 8 ) ); fi; if n=36 then return POL_DirectProduct( PolExamples( 6 ), PolExamples( 7 ) ); fi; if n=37 then return POL_DirectProduct( PolExamples( 1 ), PolExamples( 9 ) ); fi; if n=38 then return POL_DirectProduct( PolExamples( 10 ), PolExamples( 11 ) ); fi; if n=39 then return POL_DirectProduct( PolExamples( 17 ), PolExamples( 21 ) ); fi; if n=40 then return POL_DirectProduct( PolExamples( 2 ), PolExamples( 27 ) ); fi; if n in [41..81] then return POL_SemidirectProductVectorSpace( PolExamples( n-40 ) ); fi; if n in [82..100] then return POL_AlmostCrystallographicGroup( 4, n-5, false ); fi; if n in [101..200] then P := SymmetricGroup( 3 ); M := POL_PolExamples2( n-100 ); return WreathProductOfMatrixGroup( M, P ); fi; if n in [201..300] then P := SmallGroup( 8, 3 ); nat := RegularActionHomomorphism( P ); P := Image( nat ); M := POL_PolExamples2( n-200 ); return WreathProductOfMatrixGroup( M, P ); fi; if n in [301..600] then G := POL_PolExamples2( n-300 ); gens := GeneratorsOfGroup( G ); d := Length( gens[1] ); M := RandomInvertibleMat( Integers, d ); return G^M; fi; if n in [601..1200] then return POL_RandomSubgroup( POL_PolExamples2( n-600), 2 ); fi; # examples which can be nonpolycyclic or nonsolvable if n = 1201 then h1 := [[2,0],[0,1]]; h2 := [[1,1],[0,1]]; H := Group( [h1,h2] ); return H; fi; if n = 1202 then gens := [ [ [ -1/4, 5/4, -4/3 ], [ 1, 0, 1 ], [ 0, -5/3, -1 ] ], [ [ 2, 3/5, 4 ], [ 2/3, 0, 1 ], [ 5/3, 0, 1/2 ] ] ]; return Group( gens ); fi; if n = 1203 then gens := [ [ [ 2, 4/3, 1/2, 1/2 ], [ -2, -1/2, -1/4, 1 ], [ -1/2, 0, 0, 3/2 ], [ 0, 1/3, 0, 3/2 ] ], [ [ 2, 2, 1, 2 ], [ -2/5, 3/2, 0, -1/2 ], [ -3/2, 0, 1/3, 4 ], [ -1/4, 0, -3, 0 ] ] ]; return Group( gens ); fi; if n = 1204 then gens := [ [ [ 3, 1, 3, -1 ], [ 2, -1, 1, 1 ], [ -1, 0, 1, 3 ], [ 1, 1, -1, 0 ] ], [ [ 0, -3, -1, 2 ], [ -3, -3, 0, -1 ], [ -1, -2, -1, 0 ], [ 1, -1, 2, -5 ] ] ]; return Group( gens ); fi; if n= 1205 then gens := [ [ [ -7/3, 77/3, 0, 0, 1, 0 ], [ -77/3, 994/3, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, -28/3 ], [ 0, 0, 2/3, 0, 0, 22/3 ], [ 0, 0, 0, 2/3, 0, -10/3 ], [ 0, 0, 0, 0, 2/3, -2 ] ], [ [ 1/3, 8/3, 0, 0, 0, 0 ], [ -8/3, 35, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, -28/3 ], [ 0, 0, 2/3, 0, 0, 22/3 ], [ 0, 0, 0, 2/3, 0, -10/3 ], [ 0, 0, 0, 0, 2/3, -2 ]] ]; return Group( gens ); fi; if n = 1206 then gens := [ [ [ -6, -1/2, 3/4, -5/3 ], [ 0, -3, 3, -3/2 ], [ 0, 0, 1/3, -4/7 ], [ 0, 0, 0, 1 ] ], [ [ 7/4, 2, 1/2, -5/3 ], [ 0, 5/6, -6, 1 ], [ 0, 0, 3, 2/7 ], [ 0, 0, 0, 2/3 ] ] ]; return Group( gens ); fi; if n = 1207 then gens := [ [ [ 1, 5/7, 3/4, -5, 2/5 ], [ 0, -1, 3/2, 2, 4/7 ], [ 0, 0, 2/3, 1, -7/5 ], [ 0, 0, 0, 1/5, 4/7 ], [ 0, 0, 0, 0, 2/5 ] ], [ [ 2, -2/7, -4/7, 1, 3/7 ], [ 0, -1/5, -1/3, 1, 1/2 ], [ 0, 0, -4/3, -3, 2 ], [ 0, 0, 0, 7, -1/6 ], [ 0, 0, 0, 0, -2 ] ] ]; return Group( gens ); fi; if n = 1208 then gens := [ [ [ 3/2, -3, -4/5, 1/3, -3, -5/6, 4, 5/3 ], [ 0, -1, 5/7, -1/2, 3/7, -1/3, -4/5, 1/2 ], [ 0, 0, -1/4, 3/5, 5/2, -1/6, -1, 1 ], [ 0, 0, 0, 1/5, 7/4, -2, -7/4, -1 ], [ 0, 0, 0, 0, 5/3, -5/2, -1, -1/6 ], [ 0, 0, 0, 0, 0, -1, 5/2, -1 ], [ 0, 0, 0, 0, 0, 0, 3, 1 ], [ 0, 0, 0, 0, 0, 0, 0, -2/7 ] ], [ [ 3/2, 1, -3/2, 1, -1, -2/5, 1/5, -1 ], [ 0, 1, 4/7, 1, -3/5, -5/7, 5/2, 3 ], [ 0, 0, -1/3, 1, 3/2, -4/3, 3/2, 2/3 ], [ 0, 0, 0, 3, -3, -7/5, 5/2, 1/2 ], [ 0, 0, 0, 0, 3/5, -2, 4/3, 1 ], [ 0, 0, 0, 0, 0, -2, 3, 3 ], [ 0, 0, 0, 0, 0, 0, -1/2, -1/2 ], [ 0, 0, 0, 0, 0, 0, 0, 5/3 ] ] ]; return Group( gens ); fi; if n = 1209 then gens := [ [ [ 5/6, -2, 1, 4, -1, 2 ], [ 2/3, -19/6, -1, 4, -3, 0 ], [ 0, 0, 5/6, -2, -1, -2 ], [ 0, 0, 2/3, -19/6, 0, -3 ], [ 0, 0, 0, 0, 5/6, -2 ], [ 0, 0, 0, 0, 2/3, -19/6 ] ], [ [ -1/2, -12/5, 3, -1, 1, -2 ], [ 4/5, -53/10, -2, -4, -2, 1 ], [ 0, 0, -1/2, -12/5, -2, -4 ], [ 0, 0, 4/5, -53/10, 2, 3 ], [ 0, 0, 0, 0, -1/2, -12/5 ], [ 0, 0, 0, 0, 4/5, -53/10 ] ], [ [ 4733/600, -2037/50, -2, -1, 4, -2 ], [ 679/50, -8831/120, 0, 0, -3, -2 ], [ 0, 0, 4733/600, -2037/50, 0, 2 ], [ 0, 0, 679/50, -8831/120, 4, 1 ], [ 0, 0, 0, 0, 4733/600, -2037/50 ], [ 0, 0, 0, 0, 679/50, -8831/120 ] ], [ [ 1, 0, -3, -2, -2, -4 ], [ 0, 1, 3, 2, 1, 3 ], [ 0, 0, 1, 0, -2, 1 ], [ 0, 0, 0, 1, 2, 0 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ] ] ]; return Group( gens ); fi; if n = 1210 then # takes 6 minutes to test polycyclicity gens := [ [ [ 63/20, -5/2, 121/2, -1475, -5, 1, 1, 1 ], [ 0, -237/20, 721/2, -17579/2, 1, -3, 2, -2 ], [ 2/5, -15/2, 3393/20, -8129/2, 1, -1, 1, -1 ], [ 1/2, -121/10, 295, -144107/20, -3, -1, 0, 0 ], [ 0, 0, 0, 0, 63/20, -5/2, 121/2, -1475 ], [ 0, 0, 0, 0, 0, -237/20, 721/2, -17579/2 ], [ 0, 0, 0, 0, 2/5, -15/2, 3393/20, -8129/2 ], [ 0, 0, 0, 0, 1/2, -121/10, 295, -144107/20 ] ], [ [ -281351/400, 34831/2, -8511437/20, 20798633/2, 3, 0, -2, 0 ], [ -85011/20, 41515849/400, -12680078/5, 1239406543/20, -1, 0, -1, 1 ], [ -196313/100, 959919/20, -469170371/400, 286619339/10, -1, -4, 1, 1 ], [ -34831/10, 8511437/100, -20798633/10, 20329462629/400, 1, -2, 4, -2 ], [ 0, 0, 0, 0, -281351/400, 34831/2, -8511437/20, 20798633/2 ], [ 0, 0, 0, 0, -85011/20, 41515849/400, -12680078/5, 1239406543/20 ], [ 0, 0, 0, 0, -196313/100, 959919/20, -469170371/400, 286619339/10 ], [ 0, 0, 0, 0, -34831/10, 8511437/100, -20798633/10, 20329462629/400 ] ], [ [ -5, -3, 75, -3665/2, 1, 0, -1, -2 ], [ 1/2, -23, 447, -10920, 0, -3, 1, 2 ], [ 0, -17/2, 202, -10101/2, -3, -1, 0, 1 ], [ 3/5, -15, 733/2, -17921/2, 0, 1, 2, 3 ], [ 0, 0, 0, 0, -5, -3, 75, -3665/2 ], [ 0, 0, 0, 0, 1/2, -23, 447, -10920 ], [ 0, 0, 0, 0, 0, -17/2, 202, -10101/2 ], [ 0, 0, 0, 0, 3/5, -15, 733/2, -17921/2 ] ], [ [ -50, -30, 750, -18325, 3, -2, 0, 2 ], [ 5, -230, 4470, -109200, 0, 1, 5, 3 ], [ 0, -85, 2020, -50505, 3, 0, 1, 4 ], [ 6, -150, 3665, -89605, 2, -3, 1, -1 ] , [ 0, 0, 0, 0, -50, -30, 750, -18325 ], [ 0, 0, 0, 0, 5, -230, 4470, -109200 ], [ 0, 0, 0, 0, 0, -85, 2020, -50505 ], [ 0, 0, 0, 0, 6, -150, 3665, -89605 ] ] ]; return Group( gens ); fi; ############################################################### # some easy examples if n = -1 then g := 2*IdentityMat( 2, 1 ); gens := [g]; return Group( gens ); fi; if n = -2 then g := 2*IdentityMat( 3, 1 ); gens := [g]; return Group( gens ); fi; if n = -3 then g := 1345*IdentityMat( 5, 1 ); gens := [g]; return Group( gens ); fi; if n = -4 then g := 23* IdentityMat( 5, 1 ); h := 12* IdentityMat( 5, 1 ); gens := [g,h]; return Group( gens ); fi; if n = -5 then g := IdentityMat( 3,1 ); g[3][3] := 34; h := IdentityMat( 3,1 ); h[1][1] := 23; gens := [g,h]; return Group( gens ); fi; if n = -6 then g := IdentityMat(4,1 ); gens := [g]; return Group( gens ); fi; if n = -7 then g := [[2]]; h := [[3]]; gens := [g,h]; return Group( gens ); fi; if n = -8 then g := [[1]]; gens := [g]; return Group( gens ); fi; ################################################################## # some abelian irrdeucible examples if n = - 9 then gens := [ [ [ -21/5, 0, 56, -168 ], [ 0, -21/5, -44, 188 ], [ -4, 0, 79/5, -104 ], [ 0, -4, 12, -101/5 ] ], [ [ -1/2, 0, 434/5, -1302/5 ], [ 0, -1/2, -341/5, 1457/5 ], [ -31/5, 0, 61/2, -806/5 ], [ 0, -31/5, 93/5, -253/10 ] ], [ [ -5/6, 0, 14/5, -77/5 ], [ 1/2, -5/6, -11/5, 149/10 ], [ -1/5, 1/2, 1/6, -77/10 ], [ 0, -1/5, 11/10, -47/15 ] ] ]; return Group( gens ); fi; if n = - 10 then gens := [ [ [ -37/20, 0, 0, -70/3 ], [ 5/3, -37/20, 0, 55/3 ], [ 0, 5/3, -37/20, -25/3 ], [ 0, 0, 5/3, -137/20 ] ], [ [ 3/4, 0, 0, -56 ], [ 4, 3/4, 0, 44 ], [ 0, 4, 3/4, -20 ], [ 0, 0, 4, -45/4 ] ], [ [ 7/2, 0, 28, -84 ], [ 0, 7/2, -22, 94 ], [ -2, 0, 27/2, -52 ], [ 0, -2, 6, -9/2 ] ] ]; return Group( gens ); fi; if n = -11 then gens := [ [ [ -1, 185/12 ], [ -185/12, 2393/12 ] ], [ [ -1/2, -24 ], [ 24, -625/2 ] ] , [ [ 1/3, -1/2 ], [ 1/2, -37/6 ] ] ]; return( Group(gens ) ); fi; if n = -12 then gens := [ [ [ 0, 0, 0, 0, 0, 119/6 ], [ -17/12, 0, 0, 0, 0, -187/12 ], [ 0, -17/12, 0, 0, 0, 0 ], [ 0, 0, -17/12, 0, 0, -17 ], [ 0, 0, 0, -17/12, 0, 0 ], [ 0, 0, 0, 0, -17/12, 85/12 ] ], [ [ 0, 0, 0, 0, 196/15, -238/3 ], [ 1, 0, 0, 0, -154/15, 377/5 ], [ -14/15, 1, 0, 0, 0, -154/15 ], [ 0, -14/15, 1, 0, -56/5, 68 ], [ 0, 0, -14/15, 1, 0, -56/5 ], [ 0, 0, 0, -14/15, 17/3, -85/3 ] ], [ [ 2, 0, 0, 0, -84/5, 98 ], [ -1, 2, 0, 0, 66/5, -469/5 ], [ 6/5, -1, 2, 0, 0, 66/5 ], [ 0, 6/5, -1, 2, 72/5, -84 ], [ 0, 0, 6/5, -1, 2, 72/5 ], [ 0, 0, 0, 6/5, -7, 37 ] ] ]; return Group( gens ); fi; ################################################################## # some triangularizable examples if n = -13 then gens := [ [ [ 9/10, 39/10, 0, 0 ], [ -39/10, 258/5, 0, 0 ], [ 0, 0, 4, -39 ], [ 0, 0, 39, -503 ] ], [ [ 9/10, 39/10, 0, 0 ], [ -39/10, 258/5, 0, 0 ], [ 0, 0, 4, -39 ], [ 0, 0, 39, -503 ] ] ]; return Group( gens ); fi; if n = -14 then gens := [ [ [ 7, 6, 0, 0 ], [ -6, 85, 0, 0 ], [ 0, 0, 0, 5 ], [ 0, 0, -5, 65 ] ], [ [ -7, 78, 0, 0 ], [ -78, 1007, 0, 0 ], [ 0, 0, -6, 81 ], [ 0, 0, -81, 1047 ] ] ]; return Group( gens ); fi; if n = -15 then gens := [ [ [ -9, 36, 0, 0, 0 ], [ -12, 63, 0, 0, 0 ], [ 0, 0, -1, 6, 0 ], [ 0, 0, 0, 5, 6 ], [ 0, 0, -3, 0, 5 ] ], [ [ -6, 24, 0, 0, 0 ], [ -8, 42, 0, 0, 0 ], [ 0, 0, -1, 6, 0 ], [ 0, 0, 0, 5, 6 ], [ 0, 0, -3, 0, 5 ] ] ]; return Group( gens ); fi; # takes some more time, example form referee if n = -16 then gens := [ [ [ 1, -3, 8, -2, 0, 2 ], [ 5, -14, 37, -3, -2, -2 ], [ -3, 8, -22, 0, -5, 2 ], [ 0, 0, 0, 0, 1, -1 ], [ 0, 0, 0, 0, 5, -4 ], [ 0, 0, 0, 1, -1, 6 ] ], [ [ 1, 0, 0, -1, 1, -2 ], [ 0, 1, 0, -3, 0, 5 ], [ 0, 0, 1, -1, 2, 1 ], [ 0, 0, 0, 2, -1, -1 ], [ 0, 0, 0, -2, -3, -6 ], [ 0, 0, 0, -1, -1, -2 ] ] ]; return Group( gens ); fi; # finite examples if n = -17 then return ImfMatrixGroup( 4,1,1 ); fi; if n = -18 then return ImfMatrixGroup(6,2,1); fi; ################################################################ #difficult examples ??? if n = -100 then gens := [ [ [ -100, 51, 70, -231, 1, 0, 0, 1 ], [ -169, 82, 112, -385, 0, 0, 0, 0 ], [ -59, 27, 37, -132, 0, 0, 0, 0 ], [ -80, 38, 52, -181, 0, 1, 0, 0 ], [ 0, 0, 0, 0, -19, 96, 70, -231 ], [ 0, 0, 0, 0, -34, 163, 112, -385 ], [ 0, 0, 0, 0, 31, -153, -107, 363 ], [ 0, 0, 0, 0, 10, -52, -38, 125 ] ], [ [ -1, 0, -1, 0, 0, 0, 0, 0 ], [ -2, 3, 5, -11, 0, 0, 1, 0 ], [ -1, 0, 0, 0, 0, 0, 1, 0 ], [ -1, 1, 2, -4, 1, 0, 1, 0 ], [ 0, 0, 0, 0, -19, 96, 70, -231 ], [ 0, 0, 0, 0, -34, 163, 112, -385 ], [ 0, 0, 0, 0, 31, -153, -107, 363 ], [ 0, 0, 0, 0, 10, -52, -38, 125 ] ] ]; return Group( gens ); fi; if n = -101 then gens:= [[[1,-3,8,-2,0,2], [5,-14,37,-3,-2,-2], [-3,8,-22,0,-5,2], [0,0,0,0,1,-1], [0,0,0,0,5,-4], [0,0,0,1,-1,6]], [[1,0,0,-1,1,-2], [0,1,0,-3,0,5], [0,0,1,-1,2,1], [0,0,0,2,-1,-1], [0,0,0,-2,-3,-6], [0,0,0,-1,-1,-2]]]; return Group( gens ); fi; if n = -102 then gens :=[[[-984,113,11,19,0,-2,-2,5,1,1,-1], [1160,146,223,201,0,1,0,-2,-3,2,3], [124,30,36,33,2,0,2,-2,1,-5,0], [113,11,19,17,3,0,4,-1,1,1,-1], [0,0,0,0, 619,1178,2242, 2,-2,-1,-2], [0,0,0,0, 3420, 6509,12388, 2,-2,-2,-4], [0,0,0,0, 1178, 2242, 4267, 2,0,1,-3], [0,0,0,0, 0,0,0, 5274, 4861, 4561, 4272], [0,0,0,0, 0,0,0, 57443, 53884,50471,47281], [0,0,0,0, 0,0,0, 9422,8833,8274,7751], [0,0,0,0, 0,0,0, 4861,4561,4272,4002]], [[4272,4002,3749,3512, -1,-2,2,-1,1,1,2], [47281,44292,41492,38869, 4,-1,-1,3,-2,3,1], [7751,7261,6802,6372, 2,-2,5,4,-1,-1,-3], [4002,3749,3512,3290, -1,1,0,-2,0,1,-3], [0,0,0,0, 0,0,1, -1,-1,-2,2], [0,0,0,0, 1,0,5, 3,-1,2,2], [0,0,0,0, 0,1,-1, -1,2,-1,0], [0,0,0,0, 0,0,0, 4272,4002,3749,3512 ], [0,0,0,0, 0,0,0, 47281,44292,41492,38869], [0,0,0,0, 0,0,0, 7751,7261,6802,6372], [0,0,0,0, 0,0,0, 4002,3749,3512,3290]] ]; return Group( gens ); fi; if n = -103 then gens := [ [ [ -12, -3, -18, 22, 0, 1/2, 0, 0, 0, 0, 0, 0 ], [ -1, 16, 46, -88, 1/2, 0, 1/2, 0, 0, 0, 0, 0 ], [ -2, -10, -23, 33, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 4, 14, -29, 0, 0, 0, 1/2, 1/2, 1/2, 0, 0 ], [ 0, 0, 0, 0, -1, -1, -1, 0, 0, 1/2, 0, 0 ], [ 0, 0, 0, 0, -5, 3, 5, -11, 1/2, 0, 0, 0 ], [ 0, 0, 0, 0, -3, 4, 5, -11, 0, 0, 1/2, 0 ], [ 0, 0, 0, 0, -3, 3, 4, -9, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, -15, 48, 36 -121 ], [ 0, 0, 0, 0, 0, 0, 0, 0, -26, 83, 62, -209 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 11, -71, -49, 165 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, -22, -14, 47 ] ], [ [ -12, -3, -18, 22, 0, 0, 0, 0, 0, 0, 0, 0 ], --> -------------------- --> maximum size reached --> -------------------- [ Dauer der Verarbeitung: 0.14 Sekunden (vorverarbeitet) ] |
2026-04-02