Finite Group Databasebeta

?

← Prev (#72292331) Next (#72292333) →

Dic₁₄₉₆₇ #72292332

• 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 1676 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,1676)(16,1675)(17,1674)(18,1673)(19,1672)(20,1671)(21,1670)(22,1669)(23,1668)(24,1667)(25,1666)(26,1665)(27,1664)(28,1663)(29,1662)(30,1661)(31,1660)(32,1659)(33,1658)(34,1657)(35,1656)(36,1655)(37,1654)(38,1653)(39,1652)(40,1651)(41,1650)(42,1649)(43,1648)(44,1647)(45,1646)(46,1645)(47,1644)(48,1643)(49,1642)(50,1641)(51,1640)(52,1639)(53,1638)(54,1637)(55,1636)(56,1635)(57,1634)(58,1633)(59,1632)(60,1631)(61,1630)(62,1629)(63,1628)(64,1627)(65,1626)(66,1625)(67,1624)(68,1623)(69,1622)(70,1621)(71,1620)(72,1619)(73,1618)(74,1617)(75,1616)(76,1615)(77,1614)(78,1613)(79,1612)(80,1611)(81,1610)(82,1609)(83,1608)(84,1607)(85,1606)(86,1605)(87,1604)(88,1603)(89,1602)(90,1601)(91,1600)(92,1599)(93,1598)(94,1597)(95,1596)(96,1595)(97,1594)(98,1593)(99,1592)(100,1591)(101,1590)(102,1589)(103,1588)(104,1587)(105,1586)(106,1585)(107,1584)(108,1583)(109,1582)(110,1581)(111,1580)(112,1579)(113,1578)(114,1577)(115,1576)(116,1575)(117,1574)(118,1573)(119,1572)(120,1571)(121,1570)(122,1569)(123,1568)(124,1567)(125,1566)(126,1565)(127,1564)(128,1563)(129,1562)(130,1561)(131,1560)(132,1559)(133,1558)(134,1557)(135,1556)(136,1555)(137,1554)(138,1553)(139,1552)(140,1551)(141,1550)(142,1549)(143,1548)(144,1547)(145,1546)(146,1545)(147,1544)(148,1543)(149,1542)(150,1541)(151,1540)(152,1539)(153,1538)(154,1537)(155,1536)(156,1535)(157,1534)(158,1533)(159,1532)(160,1531)(161,1530)(162,1529)(163,1528)(164,1527)(165,1526)(166,1525)(167,1524)(168,1523)(169,1522)(170,1521)(171,1520)(172,1519)(173,1518)(174,1517)(175,1516)(176,1515)(177,1514)(178,1513)(179,1512)(180,1511)(181,1510)(182,1509)(183,1508)(184,1507)(185,1506)(186,1505)(187,1504)(188,1503)(189,1502)(190,1501)(191,1500)(192,1499)(193,1498)(194,1497)(195,1496)(196,1495)(197,1494)(198,1493)(199,1492)(200,1491)(201,1490)(202,1489)(203,1488)(204,1487)(205,1486)(206,1485)(207,1484)(208,1483)(209,1482)(210,1481)(211,1480)(212,1479)(213,1478)(214,1477)(215,1476)(216,1475)(217,1474)(218,1473)(219,1472)(220,1471)(221,1470)(222,1469)(223,1468)(224,1467)(225,1466)(226,1465)(227,1464)(228,1463)(229,1462)(230,1461)(231,1460)(232,1459)(233,1458)(234,1457)(235,1456)(236,1455)(237,1454)(238,1453)(239,1452)(240,1451)(241,1450)(242,1449)(243,1448)(244,1447)(245,1446)(246,1445)(247,1444)(248,1443)(249,1442)(250,1441)(251,1440)(252,1439)(253,1438)(254,1437)(255,1436)(256,1435)(257,1434)(258,1433)(259,1432)(260,1431)(261,1430)(262,1429)(263,1428)(264,1427)(265,1426)(266,1425)(267,1424)(268,1423)(269,1422)(270,1421)(271,1420)(272,1419)(273,1418)(274,1417)(275,1416)(276,1415)(277,1414)(278,1413)(279,1412)(280,1411)(281,1410)(282,1409)(283,1408)(284,1407)(285,1406)(286,1405)(287,1404)(288,1403)(289,1402)(290,1401)(291,1400)(292,1399)(293,1398)(294,1397)(295,1396)(296,1395)(297,1394)(298,1393)(299,1392)(300,1391)(301,1390)(302,1389)(303,1388)(304,1387)(305,1386)(306,1385)(307,1384)(308,1383)(309,1382)(310,1381)(311,1380)(312,1379)(313,1378)(314,1377)(315,1376)(316,1375)(317,1374)(318,1373)(319,1372)(320,1371)(321,1370)(322,1369)(323,1368)(324,1367)(325,1366)(326,1365)(327,1364)(328,1363)(329,1362)(330,1361)(331,1360)(332,1359)(333,1358)(334,1357)(335,1356)(336,1355)(337,1354)(338,1353)(339,1352)(340,1351)(341,1350)(342,1349)(343,1348)(344,1347)(345,1346)(346,1345)(347,1344)(348,1343)(349,1342)(350,1341)(351,1340)(352,1339)(353,1338)(354,1337)(355,1336)(356,1335)(357,1334)(358,1333)(359,1332)(360,1331)(361,1330)(362,1329)(363,1328)(364,1327)(365,1326)(366,1325)(367,1324)(368,1323)(369,1322)(370,1321)(371,1320)(372,1319)(373,1318)(374,1317)(375,1316)(376,1315)(377,1314)(378,1313)(379,1312)(380,1311)(381,1310)(382,1309)(383,1308)(384,1307)(385,1306)(386,1305)(387,1304)(388,1303)(389,1302)(390,1301)(391,1300)(392,1299)(393,1298)(394,1297)(395,1296)(396,1295)(397,1294)(398,1293)(399,1292)(400,1291)(401,1290)(402,1289)(403,1288)(404,1287)(405,1286)(406,1285)(407,1284)(408,1283)(409,1282)(410,1281)(411,1280)(412,1279)(413,1278)(414,1277)(415,1276)(416,1275)(417,1274)(418,1273)(419,1272)(420,1271)(421,1270)(422,1269)(423,1268)(424,1267)(425,1266)(426,1265)(427,1264)(428,1263)(429,1262)(430,1261)(431,1260)(432,1259)(433,1258)(434,1257)(435,1256)(436,1255)(437,1254)(438,1253)(439,1252)(440,1251)(441,1250)(442,1249)(443,1248)(444,1247)(445,1246)(446,1245)(447,1244)(448,1243)(449,1242)(450,1241)(451,1240)(452,1239)(453,1238)(454,1237)(455,1236)(456,1235)(457,1234)(458,1233)(459,1232)(460,1231)(461,1230)(462,1229)(463,1228)(464,1227)(465,1226)(466,1225)(467,1224)(468,1223)(469,1222)(470,1221)(471,1220)(472,1219)(473,1218)(474,1217)(475,1216)(476,1215)(477,1214)(478,1213)(479,1212)(480,1211)(481,1210)(482,1209)(483,1208)(484,1207)(485,1206)(486,1205)(487,1204)(488,1203)(489,1202)(490,1201)(491,1200)(492,1199)(493,1198)(494,1197)(495,1196)(496,1195)(497,1194)(498,1193)(499,1192)(500,1191)(501,1190)(502,1189)(503,1188)(504,1187)(505,1186)(506,1185)(507,1184)(508,1183)(509,1182)(510,1181)(511,1180)(512,1179)(513,1178)(514,1177)(515,1176)(516,1175)(517,1174)(518,1173)(519,1172)(520,1171)(521,1170)(522,1169)(523,1168)(524,1167)(525,1166)(526,1165)(527,1164)(528,1163)(529,1162)(530,1161)(531,1160)(532,1159)(533,1158)(534,1157)(535,1156)(536,1155)(537,1154)(538,1153)(539,1152)(540,1151)(541,1150)(542,1149)(543,1148)(544,1147)(545,1146)(546,1145)(547,1144)(548,1143)(549,1142)(550,1141)(551,1140)(552,1139)(553,1138)(554,1137)(555,1136)(556,1135)(557,1134)(558,1133)(559,1132)(560,1131)(561,1130)(562,1129)(563,1128)(564,1127)(565,1126)(566,1125)(567,1124)(568,1123)(569,1122)(570,1121)(571,1120)(572,1119)(573,1118)(574,1117)(575,1116)(576,1115)(577,1114)(578,1113)(579,1112)(580,1111)(581,1110)(582,1109)(583,1108)(584,1107)(585,1106)(586,1105)(587,1104)(588,1103)(589,1102)(590,1101)(591,1100)(592,1099)(593,1098)(594,1097)(595,1096)(596,1095)(597,1094)(598,1093)(599,1092)(600,1091)(601,1090)(602,1089)(603,1088)(604,1087)(605,1086)(606,1085)(607,1084)(608,1083)(609,1082)(610,1081)(611,1080)(612,1079)(613,1078)(614,1077)(615,1076)(616,1075)(617,1074)(618,1073)(619,1072)(620,1071)(621,1070)(622,1069)(623,1068)(624,1067)(625,1066)(626,1065)(627,1064)(628,1063)(629,1062)(630,1061)(631,1060)(632,1059)(633,1058)(634,1057)(635,1056)(636,1055)(637,1054)(638,1053)(639,1052)(640,1051)(641,1050)(642,1049)(643,1048)(644,1047)(645,1046)(646,1045)(647,1044)(648,1043)(649,1042)(650,1041)(651,1040)(652,1039)(653,1038)(654,1037)(655,1036)(656,1035)(657,1034)(658,1033)(659,1032)(660,1031)(661,1030)(662,1029)(663,1028)(664,1027)(665,1026)(666,1025)(667,1024)(668,1023)(669,1022)(670,1021)(671,1020)(672,1019)(673,1018)(674,1017)(675,1016)(676,1015)(677,1014)(678,1013)(679,1012)(680,1011)(681,1010)(682,1009)(683,1008)(684,1007)(685,1006)(686,1005)(687,1004)(688,1003)(689,1002)(690,1001)(691,1000)(692,999)(693,998)(694,997)(695,996)(696,995)(697,994)(698,993)(699,992)(700,991)(701,990)(702,989)(703,988)(704,987)(705,986)(706,985)(707,984)(708,983)(709,982)(710,981)(711,980)(712,979)(713,978)(714,977)(715,976)(716,975)(717,974)(718,973)(719,972)(720,971)(721,970)(722,969)(723,968)(724,967)(725,966)(726,965)(727,964)(728,963)(729,962)(730,961)(731,960)(732,959)(733,958)(734,957)(735,956)(736,955)(737,954)(738,953)(739,952)(740,951)(741,950)(742,949)(743,948)(744,947)(745,946)(746,945)(747,944)(748,943)(749,942)(750,941)(751,940)(752,939)(753,938)(754,937)(755,936)(756,935)(757,934)(758,933)(759,932)(760,931)(761,930)(762,929)(763,928)(764,927)(765,926)(766,925)(767,924)(768,923)(769,922)(770,921)(771,920)(772,919)(773,918)(774,917)(775,916)(776,915)(777,914)(778,913)(779,912)(780,911)(781,910)(782,909)(783,908)(784,907)(785,906)(786,905)(787,904)(788,903)(789,902)(790,901)(791,900)(792,899)(793,898)(794,897)(795,896)(796,895)(797,894)(798,893)(799,892)(800,891)(801,890)(802,889)(803,888)(804,887)(805,886)(806,885)(807,884)(808,883)(809,882)(810,881)(811,880)(812,879)(813,878)(814,877)(815,876)(816,875)(817,874)(818,873)(819,872)(820,871)(821,870)(822,869)(823,868)(824,867)(825,866)(826,865)(827,864)(828,863)(829,862)(830,861)(831,860)(832,859)(833,858)(834,857)(835,856)(836,855)(837,854)(838,853)(839,852)(840,851)(841,850)(842,849)(843,848)(844,847)(845,846)

Polycyclic presentation:

2^2,3,1663,3|a^2=b,c^3=e,c^a=c^2e^2,d^a=d^1662,e^a=e^2

Conjugacy classes & character table

14970 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, 1663, 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.4^1662);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);