Finite Group Databasebeta

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Dic₁₂₆₈₁ #72288047

• 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 1422 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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,1018)(420,1017)(421,1016)(422,1015)(423,1014)(424,1013)(425,1012)(426,1011)(427,1010)(428,1009)(429,1008)(430,1007)(431,1006)(432,1005)(433,1004)(434,1003)(435,1002)(436,1001)(437,1000)(438,999)(439,998)(440,997)(441,996)(442,995)(443,994)(444,993)(445,992)(446,991)(447,990)(448,989)(449,988)(450,987)(451,986)(452,985)(453,984)(454,983)(455,982)(456,981)(457,980)(458,979)(459,978)(460,977)(461,976)(462,975)(463,974)(464,973)(465,972)(466,971)(467,970)(468,969)(469,968)(470,967)(471,966)(472,965)(473,964)(474,963)(475,962)(476,961)(477,960)(478,959)(479,958)(480,957)(481,956)(482,955)(483,954)(484,953)(485,952)(486,951)(487,950)(488,949)(489,948)(490,947)(491,946)(492,945)(493,944)(494,943)(495,942)(496,941)(497,940)(498,939)(499,938)(500,937)(501,936)(502,935)(503,934)(504,933)(505,932)(506,931)(507,930)(508,929)(509,928)(510,927)(511,926)(512,925)(513,924)(514,923)(515,922)(516,921)(517,920)(518,919)(519,918)(520,917)(521,916)(522,915)(523,914)(524,913)(525,912)(526,911)(527,910)(528,909)(529,908)(530,907)(531,906)(532,905)(533,904)(534,903)(535,902)(536,901)(537,900)(538,899)(539,898)(540,897)(541,896)(542,895)(543,894)(544,893)(545,892)(546,891)(547,890)(548,889)(549,888)(550,887)(551,886)(552,885)(553,884)(554,883)(555,882)(556,881)(557,880)(558,879)(559,878)(560,877)(561,876)(562,875)(563,874)(564,873)(565,872)(566,871)(567,870)(568,869)(569,868)(570,867)(571,866)(572,865)(573,864)(574,863)(575,862)(576,861)(577,860)(578,859)(579,858)(580,857)(581,856)(582,855)(583,854)(584,853)(585,852)(586,851)(587,850)(588,849)(589,848)(590,847)(591,846)(592,845)(593,844)(594,843)(595,842)(596,841)(597,840)(598,839)(599,838)(600,837)(601,836)(602,835)(603,834)(604,833)(605,832)(606,831)(607,830)(608,829)(609,828)(610,827)(611,826)(612,825)(613,824)(614,823)(615,822)(616,821)(617,820)(618,819)(619,818)(620,817)(621,816)(622,815)(623,814)(624,813)(625,812)(626,811)(627,810)(628,809)(629,808)(630,807)(631,806)(632,805)(633,804)(634,803)(635,802)(636,801)(637,800)(638,799)(639,798)(640,797)(641,796)(642,795)(643,794)(644,793)(645,792)(646,791)(647,790)(648,789)(649,788)(650,787)(651,786)(652,785)(653,784)(654,783)(655,782)(656,781)(657,780)(658,779)(659,778)(660,777)(661,776)(662,775)(663,774)(664,773)(665,772)(666,771)(667,770)(668,769)(669,768)(670,767)(671,766)(672,765)(673,764)(674,763)(675,762)(676,761)(677,760)(678,759)(679,758)(680,757)(681,756)(682,755)(683,754)(684,753)(685,752)(686,751)(687,750)(688,749)(689,748)(690,747)(691,746)(692,745)(693,744)(694,743)(695,742)(696,741)(697,740)(698,739)(699,738)(700,737)(701,736)(702,735)(703,734)(704,733)(705,732)(706,731)(707,730)(708,729)(709,728)(710,727)(711,726)(712,725)(713,724)(714,723)(715,722)(716,721)(717,720)(718,719)

Polycyclic presentation:

2^2,3,1409,3|a^2=b,c^3=e,c^a=c^2e^2,d^a=d^1408,e^a=e^2

Conjugacy classes & character table

12684 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, 1409, 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.4^1408);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);