Finite Group Databasebeta

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Dic₁₃₃₈₃ #72289142

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial

Aliases

Splittings

Subgroups

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1500 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,122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,1135)(381,1134)(382,1133)(383,1132)(384,1131)(385,1130)(386,1129)(387,1128)(388,1127)(389,1126)(390,1125)(391,1124)(392,1123)(393,1122)(394,1121)(395,1120)(396,1119)(397,1118)(398,1117)(399,1116)(400,1115)(401,1114)(402,1113)(403,1112)(404,1111)(405,1110)(406,1109)(407,1108)(408,1107)(409,1106)(410,1105)(411,1104)(412,1103)(413,1102)(414,1101)(415,1100)(416,1099)(417,1098)(418,1097)(419,1096)(420,1095)(421,1094)(422,1093)(423,1092)(424,1091)(425,1090)(426,1089)(427,1088)(428,1087)(429,1086)(430,1085)(431,1084)(432,1083)(433,1082)(434,1081)(435,1080)(436,1079)(437,1078)(438,1077)(439,1076)(440,1075)(441,1074)(442,1073)(443,1072)(444,1071)(445,1070)(446,1069)(447,1068)(448,1067)(449,1066)(450,1065)(451,1064)(452,1063)(453,1062)(454,1061)(455,1060)(456,1059)(457,1058)(458,1057)(459,1056)(460,1055)(461,1054)(462,1053)(463,1052)(464,1051)(465,1050)(466,1049)(467,1048)(468,1047)(469,1046)(470,1045)(471,1044)(472,1043)(473,1042)(474,1041)(475,1040)(476,1039)(477,1038)(478,1037)(479,1036)(480,1035)(481,1034)(482,1033)(483,1032)(484,1031)(485,1030)(486,1029)(487,1028)(488,1027)(489,1026)(490,1025)(491,1024)(492,1023)(493,1022)(494,1021)(495,1020)(496,1019)(497,1018)(498,1017)(499,1016)(500,1015)(501,1014)(502,1013)(503,1012)(504,1011)(505,1010)(506,1009)(507,1008)(508,1007)(509,1006)(510,1005)(511,1004)(512,1003)(513,1002)(514,1001)(515,1000)(516,999)(517,998)(518,997)(519,996)(520,995)(521,994)(522,993)(523,992)(524,991)(525,990)(526,989)(527,988)(528,987)(529,986)(530,985)(531,984)(532,983)(533,982)(534,981)(535,980)(536,979)(537,978)(538,977)(539,976)(540,975)(541,974)(542,973)(543,972)(544,971)(545,970)(546,969)(547,968)(548,967)(549,966)(550,965)(551,964)(552,963)(553,962)(554,961)(555,960)(556,959)(557,958)(558,957)(559,956)(560,955)(561,954)(562,953)(563,952)(564,951)(565,950)(566,949)(567,948)(568,947)(569,946)(570,945)(571,944)(572,943)(573,942)(574,941)(575,940)(576,939)(577,938)(578,937)(579,936)(580,935)(581,934)(582,933)(583,932)(584,931)(585,930)(586,929)(587,928)(588,927)(589,926)(590,925)(591,924)(592,923)(593,922)(594,921)(595,920)(596,919)(597,918)(598,917)(599,916)(600,915)(601,914)(602,913)(603,912)(604,911)(605,910)(606,909)(607,908)(608,907)(609,906)(610,905)(611,904)(612,903)(613,902)(614,901)(615,900)(616,899)(617,898)(618,897)(619,896)(620,895)(621,894)(622,893)(623,892)(624,891)(625,890)(626,889)(627,888)(628,887)(629,886)(630,885)(631,884)(632,883)(633,882)(634,881)(635,880)(636,879)(637,878)(638,877)(639,876)(640,875)(641,874)(642,873)(643,872)(644,871)(645,870)(646,869)(647,868)(648,867)(649,866)(650,865)(651,864)(652,863)(653,862)(654,861)(655,860)(656,859)(657,858)(658,857)(659,856)(660,855)(661,854)(662,853)(663,852)(664,851)(665,850)(666,849)(667,848)(668,847)(669,846)(670,845)(671,844)(672,843)(673,842)(674,841)(675,840)(676,839)(677,838)(678,837)(679,836)(680,835)(681,834)(682,833)(683,832)(684,831)(685,830)(686,829)(687,828)(688,827)(689,826)(690,825)(691,824)(692,823)(693,822)(694,821)(695,820)(696,819)(697,818)(698,817)(699,816)(700,815)(701,814)(702,813)(703,812)(704,811)(705,810)(706,809)(707,808)(708,807)(709,806)(710,805)(711,804)(712,803)(713,802)(714,801)(715,800)(716,799)(717,798)(718,797)(719,796)(720,795)(721,794)(722,793)(723,792)(724,791)(725,790)(726,789)(727,788)(728,787)(729,786)(730,785)(731,784)(732,783)(733,782)(734,781)(735,780)(736,779)(737,778)(738,777)(739,776)(740,775)(741,774)(742,773)(743,772)(744,771)(745,770)(746,769)(747,768)(748,767)(749,766)(750,765)(751,764)(752,763)(753,762)(754,761)(755,760)(756,759)(757,758)

Polycyclic presentation:

2^2,3,1487,3|a^2=b,c^3=e,c^a=c^2e^2,d^a=d^1486,e^a=e^2

Conjugacy classes & character table

13386 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 3, 1487, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2);;
gap> SetConjugate(coll, 4, 1, F.4^1486);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);