Finite Group Databasebeta

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Dic₁₅₀₀₁ #72292389

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial

Aliases

Splittings

Subgroups

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 2154 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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2,1615)(553,1614)(554,1613)(555,1612)(556,1611)(557,1610)(558,1609)(559,1608)(560,1607)(561,1606)(562,1605)(563,1604)(564,1603)(565,1602)(566,1601)(567,1600)(568,1599)(569,1598)(570,1597)(571,1596)(572,1595)(573,1594)(574,1593)(575,1592)(576,1591)(577,1590)(578,1589)(579,1588)(580,1587)(581,1586)(582,1585)(583,1584)(584,1583)(585,1582)(586,1581)(587,1580)(588,1579)(589,1578)(590,1577)(591,1576)(592,1575)(593,1574)(594,1573)(595,1572)(596,1571)(597,1570)(598,1569)(599,1568)(600,1567)(601,1566)(602,1565)(603,1564)(604,1563)(605,1562)(606,1561)(607,1560)(608,1559)(609,1558)(610,1557)(611,1556)(612,1555)(613,1554)(614,1553)(615,1552)(616,1551)(617,1550)(618,1549)(619,1548)(620,1547)(621,1546)(622,1545)(623,1544)(624,1543)(625,1542)(626,1541)(627,1540)(628,1539)(629,1538)(630,1537)(631,1536)(632,1535)(633,1534)(634,1533)(635,1532)(636,1531)(637,1530)(638,1529)(639,1528)(640,1527)(641,1526)(642,1525)(643,1524)(644,1523)(645,1522)(646,1521)(647,1520)(648,1519)(649,1518)(650,1517)(651,1516)(652,1515)(653,1514)(654,1513)(655,1512)(656,1511)(657,1510)(658,1509)(659,1508)(660,1507)(661,1506)(662,1505)(663,1504)(664,1503)(665,1502)(666,1501)(667,1500)(668,1499)(669,1498)(670,1497)(671,1496)(672,1495)(673,1494)(674,1493)(675,1492)(676,1491)(677,1490)(678,1489)(679,1488)(680,1487)(681,1486)(682,1485)(683,1484)(684,1483)(685,1482)(686,1481)(687,1480)(688,1479)(689,1478)(690,1477)(691,1476)(692,1475)(693,1474)(694,1473)(695,1472)(696,1471)(697,1470)(698,1469)(699,1468)(700,1467)(701,1466)(702,1465)(703,1464)(704,1463)(705,1462)(706,1461)(707,1460)(708,1459)(709,1458)(710,1457)(711,1456)(712,1455)(713,1454)(714,1453)(715,1452)(716,1451)(717,1450)(718,1449)(719,1448)(720,1447)(721,1446)(722,1445)(723,1444)(724,1443)(725,1442)(726,1441)(727,1440)(728,1439)(729,1438)(730,1437)(731,1436)(732,1435)(733,1434)(734,1433)(735,1432)(736,1431)(737,1430)(738,1429)(739,1428)(740,1427)(741,1426)(742,1425)(743,1424)(744,1423)(745,1422)(746,1421)(747,1420)(748,1419)(749,1418)(750,1417)(751,1416)(752,1415)(753,1414)(754,1413)(755,1412)(756,1411)(757,1410)(758,1409)(759,1408)(760,1407)(761,1406)(762,1405)(763,1404)(764,1403)(765,1402)(766,1401)(767,1400)(768,1399)(769,1398)(770,1397)(771,1396)(772,1395)(773,1394)(774,1393)(775,1392)(776,1391)(777,1390)(778,1389)(779,1388)(780,1387)(781,1386)(782,1385)(783,1384)(784,1383)(785,1382)(786,1381)(787,1380)(788,1379)(789,1378)(790,1377)(791,1376)(792,1375)(793,1374)(794,1373)(795,1372)(796,1371)(797,1370)(798,1369)(799,1368)(800,1367)(801,1366)(802,1365)(803,1364)(804,1363)(805,1362)(806,1361)(807,1360)(808,1359)(809,1358)(810,1357)(811,1356)(812,1355)(813,1354)(814,1353)(815,1352)(816,1351)(817,1350)(818,1349)(819,1348)(820,1347)(821,1346)(822,1345)(823,1344)(824,1343)(825,1342)(826,1341)(827,1340)(828,1339)(829,1338)(830,1337)(831,1336)(832,1335)(833,1334)(834,1333)(835,1332)(836,1331)(837,1330)(838,1329)(839,1328)(840,1327)(841,1326)(842,1325)(843,1324)(844,1323)(845,1322)(846,1321)(847,1320)(848,1319)(849,1318)(850,1317)(851,1316)(852,1315)(853,1314)(854,1313)(855,1312)(856,1311)(857,1310)(858,1309)(859,1308)(860,1307)(861,1306)(862,1305)(863,1304)(864,1303)(865,1302)(866,1301)(867,1300)(868,1299)(869,1298)(870,1297)(871,1296)(872,1295)(873,1294)(874,1293)(875,1292)(876,1291)(877,1290)(878,1289)(879,1288)(880,1287)(881,1286)(882,1285)(883,1284)(884,1283)(885,1282)(886,1281)(887,1280)(888,1279)(889,1278)(890,1277)(891,1276)(892,1275)(893,1274)(894,1273)(895,1272)(896,1271)(897,1270)(898,1269)(899,1268)(900,1267)(901,1266)(902,1265)(903,1264)(904,1263)(905,1262)(906,1261)(907,1260)(908,1259)(909,1258)(910,1257)(911,1256)(912,1255)(913,1254)(914,1253)(915,1252)(916,1251)(917,1250)(918,1249)(919,1248)(920,1247)(921,1246)(922,1245)(923,1244)(924,1243)(925,1242)(926,1241)(927,1240)(928,1239)(929,1238)(930,1237)(931,1236)(932,1235)(933,1234)(934,1233)(935,1232)(936,1231)(937,1230)(938,1229)(939,1228)(940,1227)(941,1226)(942,1225)(943,1224)(944,1223)(945,1222)(946,1221)(947,1220)(948,1219)(949,1218)(950,1217)(951,1216)(952,1215)(953,1214)(954,1213)(955,1212)(956,1211)(957,1210)(958,1209)(959,1208)(960,1207)(961,1206)(962,1205)(963,1204)(964,1203)(965,1202)(966,1201)(967,1200)(968,1199)(969,1198)(970,1197)(971,1196)(972,1195)(973,1194)(974,1193)(975,1192)(976,1191)(977,1190)(978,1189)(979,1188)(980,1187)(981,1186)(982,1185)(983,1184)(984,1183)(985,1182)(986,1181)(987,1180)(988,1179)(989,1178)(990,1177)(991,1176)(992,1175)(993,1174)(994,1173)(995,1172)(996,1171)(997,1170)(998,1169)(999,1168)(1000,1167)(1001,1166)(1002,1165)(1003,1164)(1004,1163)(1005,1162)(1006,1161)(1007,1160)(1008,1159)(1009,1158)(1010,1157)(1011,1156)(1012,1155)(1013,1154)(1014,1153)(1015,1152)(1016,1151)(1017,1150)(1018,1149)(1019,1148)(1020,1147)(1021,1146)(1022,1145)(1023,1144)(1024,1143)(1025,1142)(1026,1141)(1027,1140)(1028,1139)(1029,1138)(1030,1137)(1031,1136)(1032,1135)(1033,1134)(1034,1133)(1035,1132)(1036,1131)(1037,1130)(1038,1129)(1039,1128)(1040,1127)(1041,1126)(1042,1125)(1043,1124)(1044,1123)(1045,1122)(1046,1121)(1047,1120)(1048,1119)(1049,1118)(1050,1117)(1051,1116)(1052,1115)(1053,1114)(1054,1113)(1055,1112)(1056,1111)(1057,1110)(1058,1109)(1059,1108)(1060,1107)(1061,1106)(1062,1105)(1063,1104)(1064,1103)(1065,1102)(1066,1101)(1067,1100)(1068,1099)(1069,1098)(1070,1097)(1071,1096)(1072,1095)(1073,1094)(1074,1093)(1075,1092)(1076,1091)(1077,1090)(1078,1089)(1079,1088)(1080,1087)(1081,1086)(1082,1085)(1083,1084)

Polycyclic presentation:

2^2,7,2143|a^2=b,c^a=c^6,d^a=d^2142

Conjugacy classes & character table

15004 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");;
gap> coll := SingleCollector(F, [2, 2, 7, 2143]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^2142);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);