Finite Group Databasebeta

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C₃⋊(D₁₈₆₇⋊C₃) #46777307

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

Aliases

Splittings

Subgroups

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1870 points:

(2,3)(5,9,10,6,7,11)(8,17,18,12,13,27)(14,30,31,20,21,44)(15,29,25,23,19,16)(22,49,50,33,34,67)(24,47,48,36,32,41)(26,28,46,38,39,42)(35,72,73,52,53,96)(37,70,71,55,51,64)(40,45,69,57,58,65)(43,85,83,61,62,60)(54,101,102,75,76,131)(56,99,100,78,74,93)(59,68,98,80,81,94)(63,114,115,86,87,89)(66,120,112,90,91,84)(77,136,137,104,105,172)(79,134,135,107,103,128)(82,97,133,109,110,129)(88,149,150,116,117,124)(92,156,154,121,122,119)(95,161,147,125,126,113)(106,177,178,139,140,219)(108,175,176,142,138,169)(111,132,174,144,145,170)(118,190,191,151,152,165)(123,197,198,157,158,160)(127,203,195,162,163,155)(130,208,188,166,167,148)(141,224,225,180,181,272)(143,222,223,183,179,216)(146,173,221,185,186,217)(153,237,238,192,193,212)(159,244,245,199,200,207)(164,251,249,204,205,202)(168,256,242,209,210,196)(171,261,235,213,214,189)(182,277,278,227,228,331)(184,275,276,230,226,269)(187,220,274,232,233,270)(194,290,291,239,240,265)(201,297,298,246,247,260)(206,304,305,252,253,255)(211,310,302,257,258,250)(215,315,295,262,263,243)(218,320,288,266,267,236)(229,336,337,280,281,396)(231,334,335,283,279,328)(234,273,333,285,286,329)(241,349,350,292,293,324)(248,356,357,299,300,319)(254,363,364,306,307,314)(259,370,368,311,312,309)(264,375,361,316,317,303)(268,380,354,321,322,296)(271,385,347,325,326,289)(282,401,402,339,340,467)(284,399,400,342,338,393)(287,332,398,344,345,394)(294,414,415,351,352,389)(301,421,422,358,359,384)(308,428,429,365,366,379)(313,435,436,371,372,374)(318,441,433,376,377,369)(323,446,426,381,382,362)(327,451,419,386,387,355)(330,456,412,390,391,348)(341,472,473,404,405,544)(343,470,471,407,403,464)(346,397,469,409,410,465)(353,485,486,416,417,460)(360,492,493,423,424,455)(367,499,500,430,431,450)(373,506,507,437,438,445)(378,513,511,442,443,440)(383,518,504,447,448,434)(388,523,497,452,453,427)(392,528,490,457,458,420)(395,533,483,461,462,413)(406,549,550,475,476,627)(408,547,548,478,474,541)(411,468,546,480,481,542)(418,562,563,487,488,537)(425,569,570,494,495,532)(432,576,577,501,502,527)(439,583,584,508,509,522)(444,590,591,514,515,517)(449,596,588,519,520,512)(454,601,581,524,525,505)(459,606,574,529,530,498)(463,611,567,534,535,491)(466,616,560,538,539,484)(477,632,633,552,553,716)(479,630,631,555,551,624)(482,545,629,557,558,625)(489,645,646,564,565,620)(496,652,653,571,572,615)(503,659,660,578,579,610)(510,666,667,585,586,605)(516,673,674,592,593,600)(521,680,678,597,598,595)(526,685,671,602,603,589)(531,690,664,607,608,582)(536,695,657,612,613,575)(540,700,650,617,618,568)(543,705,643,621,622,561)(554,721,722,635,636,811)(556,719,720,638,634,713)(559,628,718,640,641,714)(566,734,735,647,648,709)(573,741,742,654,655,704)(580,748,749,661,662,699)(587,755,756,668,669,694)(594,762,763,675,676,689)(599,769,770,681,682,684)(604,775,767,686,687,679)(609,780,760,691,692,672)(614,785,753,696,697,665)(619,790,746,701,702,658)(623,795,739,706,707,651)(626,800,732,710,711,644)(637,816,817,724,725,912)(639,814,815,727,723,808)(642,717,813,729,730,809)(649,829,830,736,737,804)(656,836,837,743,744,799)(663,843,844,750,751,794)(670,850,851,757,758,789)(677,857,858,764,765,784)(683,864,865,771,772,779)(688,871,869,776,777,774)(693,876,862,781,782,768)(698,881,855,786,787,761)(703,886,848,791,792,754)(708,891,841,796,797,747)(712,896,834,801,802,740)(715,901,827,805,806,733)(726,917,918,819,820,1019)(728,915,916,822,818,909)(731,812,914,824,825,910)(738,930,931,831,832,905)(745,937,938,838,839,900)(752,944,945,845,846,895)(759,951,952,852,853,890)(766,958,959,859,860,885)(773,965,966,866,867,880)(778,972,973,872,873,875)(783,978,970,877,878,870)(788,983,963,882,883,863)(793,988,956,887,888,856)(798,993,949,892,893,849)(803,998,942,897,898,842)(807,1003,935,902,903,835)(810,1008,928,906,907,828)(821,1024,1025,920,921,1132)(823,1022,1023,923,919,1016)(826,913,1021,925,926,1017)(833,1037,1038,932,933,1012)(840,1044,1045,939,940,1007)(847,1051,1052,946,947,1002)(854,1058,1059,953,954,997)(861,1065,1066,960,961,992)(868,1072,1073,967,968,987)(874,1079,1080,974,975,982)(879,1086,1084,979,980,977)(884,1091,1077,984,985,971)(889,1096,1070,989,990,964)(894,1101,1063,994,995,957)(899,1106,1056,999,1000,950)(904,1111,1049,1004,1005,943)(908,1116,1042,1009,1010,936)(911,1121,1035,1013,1014,929)(922,1137,1138,1027,1028,1251)(924,1135,1136,1030,1026,1129)(927,1020,1134,1032,1033,1130)(934,1150,1151,1039,1040,1125)(941,1157,1158,1046,1047,1120)(948,1164,1165,1053,1054,1115)(955,1171,1172,1060,1061,1110)(962,1178,1179,1067,1068,1105)(969,1185,1186,1074,1075,1100)(976,1192,1193,1081,1082,1095)(981,1199,1200,1087,1088,1090)(986,1205,1197,1092,1093,1085)(991,1210,1190,1097,1098,1078)(996,1215,1183,1102,1103,1071)(1001,1220,1176,1107,1108,1064)(1006,1225,1169,1112,1113,1057)(1011,1230,1162,1117,1118,1050)(1015,1235,1155,1122,1123,1043)(1018,1240,1148,1126,1127,1036)(1029,1256,1257,1140,1141,1376)(1031,1254,1255,1143,1139,1248)(1034,1133,1253,1145,1146,1249)(1041,1269,1270,1152,1153,1244)(1048,1276,1277,1159,1160,1239)(1055,1283,1284,1166,1167,1234)(1062,1290,1291,1173,1174,1229)(1069,1297,1298,1180,1181,1224)(1076,1304,1305,1187,1188,1219)(1083,1311,1312,1194,1195,1214)(1089,1318,1319,1201,1202,1209)(1094,1325,1323,1206,1207,1204)(1099,1330,1316,1211,1212,1198)(1104,1335,1309,1216,1217,1191)(1109,1340,1302,1221,1222,1184)(1114,1345,1295,1226,1227,1177)(1119,1350,1288,1231,1232,1170)(1124,1355,1281,1236,1237,1163)(1128,1360,1274,1241,1242,1156)(1131,1365,1267,1245,1246,1149)(1142,1381,1382,1259,1260,1507)(1144,1379,1380,1262,1258,1373)(1147,1252,1378,1264,1265,1374)(1154,1394,1395,1271,1272,1369)(1161,1401,1402,1278,1279,1364)(1168,1408,1409,1285,1286,1359)(1175,1415,1416,1292,1293,1354)(1182,1422,1423,1299,1300,1349)(1189,1429,1430,1306,1307,1344)(1196,1436,1437,1313,1314,1339)(1203,1443,1444,1320,1321,1334)(1208,1450,1451,1326,1327,1329)(1213,1456,1448,1331,1332,1324)(1218,1461,1441,1336,1337,1317)(1223,1466,1434,1341,1342,1310)(1228,1471,1427,1346,1347,1303)(1233,1476,1420,1351,1352,1296)(1238,1481,1413,1356,1357,1289)(1243,1486,1406,1361,1362,1282)(1247,1491,1399,1366,1367,1275)(1250,1496,1392,1370,1371,1268)(1261,1512,1513,1384,1385,1633)(1263,1510,1511,1387,1383,1504)(1266,1377,1509,1389,1390,1505)(1273,1524,1525,1396,1397,1500)(1280,1531,1532,1403,1404,1495)(1287,1538,1539,1410,1411,1490)(1294,1545,1546,1417,1418,1485)(1301,1552,1553,1424,1425,1480)(1308,1559,1560,1431,1432,1475)(1315,1566,1567,1438,1439,1470)(1322,1571,1572,1445,1446,1465)(1328,1578,1579,1452,1453,1460)(1333,1585,1583,1457,1458,1455)(1338,1590,1576,1462,1463,1449)(1343,1595,1516,1467,1468,1442)(1348,1600,1564,1472,1473,1435)(1353,1605,1557,1477,1478,1428)(1358,1609,1550,1482,1483,1421)(1363,1613,1543,1487,1488,1414)(1368,1618,1536,1492,1493,1407)(1372,1623,1529,1497,1498,1400)(1375,1627,1522,1501,1502,1393)(1386,1601,1602,1515,1474,1708)(1388,1635,1636,1517,1514,1631)(1391,1508,1634,1519,1520,1632)(1398,1641,1642,1526,1527,1629)(1405,1648,1649,1533,1534,1626)(1412,1655,1656,1540,1541,1622)(1419,1660,1661,1547,1548,1617)(1426,1665,1666,1554,1555,1612)(1433,1670,1671,1561,1562,1608)(1440,1675,1676,1568,1569,1604)(1447,1680,1681,1573,1574,1599)(1454,1685,1686,1580,1581,1594)(1459,1692,1693,1586,1587,1589)(1464,1698,1690,1591,1592,1584)(1469,1703,1638,1596,1597,1577)(1479,1712,1521,1606,1607,1565)(1484,1714,1640,1610,1611,1558)(1489,1716,1647,1614,1615,1551)(1494,1719,1654,1619,1620,1544)(1499,1722,1653,1624,1625,1537)(1503,1720,1646,1621,1628,1530)(1506,1717,1639,1616,1630,1523)(1518,1709,1710,1637,1603,1726)(1528,1727,1728,1643,1644,1725)(1535,1729,1730,1650,1651,1724)(1542,1733,1734,1657,1658,1723)(1549,1738,1739,1662,1663,1721)(1556,1743,1744,1667,1668,1718)(1563,1748,1749,1672,1673,1715)(1570,1753,1754,1677,1678,1713)(1575,1758,1759,1682,1683,1711)(1582,1763,1764,1687,1688,1707)(1588,1768,1769,1694,1695,1702)(1593,1773,1732,1699,1700,1697)(1598,1778,1645,1704,1705,1691)(1652,1779,1780,1731,1706,1789)(1659,1790,1791,1735,1736,1788)(1664,1792,1793,1740,1741,1787)(1669,1796,1797,1745,1746,1786)(1674,1801,1802,1750,1751,1785)(1679,1806,1807,1755,1756,1784)(1684,1811,1812,1760,1761,1783)(1689,1816,1817,1765,1766,1782)(1696,1821,1795,1770,1771,1781)(1701,1825,1737,1774,1775,1777)(1742,1826,1827,1794,1776,1834)(1747,1835,1836,1798,1799,1833)(1752,1837,1838,1803,1804,1832)(1757,1841,1842,1808,1809,1831)(1762,1846,1847,1813,1814,1830)(1767,1851,1840,1818,1819,1829)(1772,1855,1800,1822,1823,1828)(1805,1856,1857,1839,1824,1861)(1810,1862,1863,1843,1844,1860)(1815,1864,1865,1848,1849,1859)(1820,1867,1845,1852,1853,1858)(1850,1868,1869,1866,1854,1870);(1,2,3)(4,5,8,14,22,35,54,77,106,141,182,229,282,341,406,477,554,637,726,821,922,1029,1142,1261,1386,1516,1441,1316,1197,1084,973,865,763,667,577,493,415,335,225,274,320,315,310,304,297,290,275,224,273,330,392,459,531,609,693,783,879,981,1089,1203,1322,1447,1575,1684,1762,1815,1850,1865,1842,1802,1744,1661,1539,1402,1270,1136,918,1021,1121,1116,1111,1106,1101,1096,1091,1086,1079,1072,1065,1058,1051,1044,1037,1022,917,1020,1131,1247,1368,1494,1621,1501,1265,1028,1139,1153,1160,1167,1174,1181,1188,1195,1202,1207,1212,1217,1222,1227,1232,1237,1242,1246,1249,1251,1373,1500,1626,1723,1787,1833,1861,1867,1864,1862,1856,1822,1770,1694,1586,1457,1331,1211,1097,989,887,791,701,617,538,410,281,338,352,359,366,372,377,382,387,391,394,396,464,537,615,699,789,885,987,1095,1209,1329,1455,1584,1691,1643,1526,1396,1271,1152,1039,932,831,736,647,564,487,416,351,292,239,192,151,116,86,61,39,13,19,27,41,60,84,113,148,189,236,289,348,413,484,561,644,733,828,929,1036,1149,1268,1393,1523,1640,1557,1427,1302,1183,1070,963,862,767,678,591,507,429,357,291,223,137,174,208,203,197,190,175,136,173,218,268,323,383,449,521,599,683,773,868,969,1076,1189,1308,1433,1563,1674,1752,1805,1840,1812,1754,1671,1553,1416,1284,1158,1038,916,722,813,901,896,891,886,881,876,871,864,857,850,843,836,829,814,721,812,911,1015,1124,1238,1358,1484,1520,1260,1383,1397,1404,1411,1418,1425,1432,1439,1446,1453,1458,1463,1468,1473,1478,1483,1488,1493,1498,1502,1505,1507,1631,17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Polycyclic presentation:

2,3^2,1867|c^a=c^2,d^a=d^1866,d^b=d^1032

Conjugacy classes & character table

942 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33606, 9);