Finite Group Databasebeta

?

← Prev (#72272914) Next (#72272916) →

Dic₈₅₇₄ #72272915

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Splittings

Subgroups

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1440 points:

(1,2,3,4)(5,8,7,6)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440);(1,5,3,7)(2,6,4,8)(10,11)(13,1440)(14,1439)(15,1438)(16,1437)(17,1436)(18,1435)(19,1434)(20,1433)(21,1432)(22,1431)(23,1430)(24,1429)(25,1428)(26,1427)(27,1426)(28,1425)(29,1424)(30,1423)(31,1422)(32,1421)(33,1420)(34,1419)(35,1418)(36,1417)(37,1416)(38,1415)(39,1414)(40,1413)(41,1412)(42,1411)(43,1410)(44,1409)(45,1408)(46,1407)(47,1406)(48,1405)(49,1404)(50,1403)(51,1402)(52,1401)(53,1400)(54,1399)(55,1398)(56,1397)(57,1396)(58,1395)(59,1394)(60,1393)(61,1392)(62,1391)(63,1390)(64,1389)(65,1388)(66,1387)(67,1386)(68,1385)(69,1384)(70,1383)(71,1382)(72,1381)(73,1380)(74,1379)(75,1378)(76,1377)(77,1376)(78,1375)(79,1374)(80,1373)(81,1372)(82,1371)(83,1370)(84,1369)(85,1368)(86,1367)(87,1366)(88,1365)(89,1364)(90,1363)(91,1362)(92,1361)(93,1360)(94,1359)(95,1358)(96,1357)(97,1356)(98,1355)(99,1354)(100,1353)(101,1352)(102,1351)(103,1350)(104,1349)(105,1348)(106,1347)(107,1346)(108,1345)(109,1344)(110,1343)(111,1342)(112,1341)(113,1340)(114,1339)(115,1338)(116,1337)(117,1336)(118,1335)(119,1334)(120,1333)(121,1332)(122,1331)(123,1330)(124,1329)(125,1328)(126,1327)(127,1326)(128,1325)(129,1324)(130,1323)(131,1322)(132,1321)(133,1320)(134,1319)(135,1318)(136,1317)(137,1316)(138,1315)(139,1314)(140,1313)(141,1312)(142,1311)(143,1310)(144,1309)(145,1308)(146,1307)(147,1306)(148,1305)(149,1304)(150,1303)(151,1302)(152,1301)(153,1300)(154,1299)(155,1298)(156,1297)(157,1296)(158,1295)(159,1294)(160,1293)(161,1292)(162,1291)(163,1290)(164,1289)(165,1288)(166,1287)(167,1286)(168,1285)(169,1284)(170,1283)(171,1282)(172,1281)(173,1280)(174,1279)(175,1278)(176,1277)(177,1276)(178,1275)(179,1274)(180,1273)(181,1272)(182,1271)(183,1270)(184,1269)(185,1268)(186,1267)(187,1266)(188,1265)(189,1264)(190,1263)(191,1262)(192,1261)(193,1260)(194,1259)(195,1258)(196,1257)(197,1256)(198,1255)(199,1254)(200,1253)(201,1252)(202,1251)(203,1250)(204,1249)(205,1248)(206,1247)(207,1246)(208,1245)(209,1244)(210,1243)(211,1242)(212,1241)(213,1240)(214,1239)(215,1238)(216,1237)(217,1236)(218,1235)(219,1234)(220,1233)(221,1232)(222,1231)(223,1230)(224,1229)(225,1228)(226,1227)(227,1226)(228,1225)(229,1224)(230,1223)(231,1222)(232,1221)(233,1220)(234,1219)(235,1218)(236,1217)(237,1216)(238,1215)(239,1214)(240,1213)(241,1212)(242,1211)(243,1210)(244,1209)(245,1208)(246,1207)(247,1206)(248,1205)(249,1204)(250,1203)(251,1202)(252,1201)(253,1200)(254,1199)(255,1198)(256,1197)(257,1196)(258,1195)(259,1194)(260,1193)(261,1192)(262,1191)(263,1190)(264,1189)(265,1188)(266,1187)(267,1186)(268,1185)(269,1184)(270,1183)(271,1182)(272,1181)(273,1180)(274,1179)(275,1178)(276,1177)(277,1176)(278,1175)(279,1174)(280,1173)(281,1172)(282,1171)(283,1170)(284,1169)(285,1168)(286,1167)(287,1166)(288,1165)(289,1164)(290,1163)(291,1162)(292,1161)(293,1160)(294,1159)(295,1158)(296,1157)(297,1156)(298,1155)(299,1154)(300,1153)(301,1152)(302,1151)(303,1150)(304,1149)(305,1148)(306,1147)(307,1146)(308,1145)(309,1144)(310,1143)(311,1142)(312,1141)(313,1140)(314,1139)(315,1138)(316,1137)(317,1136)(318,1135)(319,1134)(320,1133)(321,1132)(322,1131)(323,1130)(324,1129)(325,1128)(326,1127)(327,1126)(328,1125)(329,1124)(330,1123)(331,1122)(332,1121)(333,1120)(334,1119)(335,1118)(336,1117)(337,1116)(338,1115)(339,1114)(340,1113)(341,1112)(342,1111)(343,1110)(344,1109)(345,1108)(346,1107)(347,1106)(348,1105)(349,1104)(350,1103)(351,1102)(352,1101)(353,1100)(354,1099)(355,1098)(356,1097)(357,1096)(358,1095)(359,1094)(360,1093)(361,1092)(362,1091)(363,1090)(364,1089)(365,1088)(366,1087)(367,1086)(368,1085)(369,1084)(370,1083)(371,1082)(372,1081)(373,1080)(374,1079)(375,1078)(376,1077)(377,1076)(378,1075)(379,1074)(380,1073)(381,1072)(382,1071)(383,1070)(384,1069)(385,1068)(386,1067)(387,1066)(388,1065)(389,1064)(390,1063)(391,1062)(392,1061)(393,1060)(394,1059)(395,1058)(396,1057)(397,1056)(398,1055)(399,1054)(400,1053)(401,1052)(402,1051)(403,1050)(404,1049)(405,1048)(406,1047)(407,1046)(408,1045)(409,1044)(410,1043)(411,1042)(412,1041)(413,1040)(414,1039)(415,1038)(416,1037)(417,1036)(418,1035)(419,1034)(420,1033)(421,1032)(422,1031)(423,1030)(424,1029)(425,1028)(426,1027)(427,1026)(428,1025)(429,1024)(430,1023)(431,1022)(432,1021)(433,1020)(434,1019)(435,1018)(436,1017)(437,1016)(438,1015)(439,1014)(440,1013)(441,1012)(442,1011)(443,1010)(444,1009)(445,1008)(446,1007)(447,1006)(448,1005)(449,1004)(450,1003)(451,1002)(452,1001)(453,1000)(454,999)(455,998)(456,997)(457,996)(458,995)(459,994)(460,993)(461,992)(462,991)(463,990)(464,989)(465,988)(466,987)(467,986)(468,985)(469,984)(470,983)(471,982)(472,981)(473,980)(474,979)(475,978)(476,977)(477,976)(478,975)(479,974)(480,973)(481,972)(482,971)(483,970)(484,969)(485,968)(486,967)(487,966)(488,965)(489,964)(490,963)(491,962)(492,961)(493,960)(494,959)(495,958)(496,957)(497,956)(498,955)(499,954)(500,953)(501,952)(502,951)(503,950)(504,949)(505,948)(506,947)(507,946)(508,945)(509,944)(510,943)(511,942)(512,941)(513,940)(514,939)(515,938)(516,937)(517,936)(518,935)(519,934)(520,933)(521,932)(522,931)(523,930)(524,929)(525,928)(526,927)(527,926)(528,925)(529,924)(530,923)(531,922)(532,921)(533,920)(534,919)(535,918)(536,917)(537,916)(538,915)(539,914)(540,913)(541,912)(542,911)(543,910)(544,909)(545,908)(546,907)(547,906)(548,905)(549,904)(550,903)(551,902)(552,901)(553,900)(554,899)(555,898)(556,897)(557,896)(558,895)(559,894)(560,893)(561,892)(562,891)(563,890)(564,889)(565,888)(566,887)(567,886)(568,885)(569,884)(570,883)(571,882)(572,881)(573,880)(574,879)(575,878)(576,877)(577,876)(578,875)(579,874)(580,873)(581,872)(582,871)(583,870)(584,869)(585,868)(586,867)(587,866)(588,865)(589,864)(590,863)(591,862)(592,861)(593,860)(594,859)(595,858)(596,857)(597,856)(598,855)(599,854)(600,853)(601,852)(602,851)(603,850)(604,849)(605,848)(606,847)(607,846)(608,845)(609,844)(610,843)(611,842)(612,841)(613,840)(614,839)(615,838)(616,837)(617,836)(618,835)(619,834)(620,833)(621,832)(622,831)(623,830)(624,829)(625,828)(626,827)(627,826)(628,825)(629,824)(630,823)(631,822)(632,821)(633,820)(634,819)(635,818)(636,817)(637,816)(638,815)(639,814)(640,813)(641,812)(642,811)(643,810)(644,809)(645,808)(646,807)(647,806)(648,805)(649,804)(650,803)(651,802)(652,801)(653,800)(654,799)(655,798)(656,797)(657,796)(658,795)(659,794)(660,793)(661,792)(662,791)(663,790)(664,789)(665,788)(666,787)(667,786)(668,785)(669,784)(670,783)(671,782)(672,781)(673,780)(674,779)(675,778)(676,777)(677,776)(678,775)(679,774)(680,773)(681,772)(682,771)(683,770)(684,769)(685,768)(686,767)(687,766)(688,765)(689,764)(690,763)(691,762)(692,761)(693,760)(694,759)(695,758)(696,757)(697,756)(698,755)(699,754)(700,753)(701,752)(702,751)(703,750)(704,749)(705,748)(706,747)(707,746)(708,745)(709,744)(710,743)(711,742)(712,741)(713,740)(714,739)(715,738)(716,737)(717,736)(718,735)(719,734)(720,733)(721,732)(722,731)(723,730)(724,729)(725,728)(726,727)

Polycyclic presentation:

2^3,3,1429|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^1428

Conjugacy classes & character table

8577 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, 2, 3, 1429]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^1428);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);