Finite Group DatabasebetaFinite Group Databasebeta| order | 31014 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1723⋊C3) |
| unambiguous semidirect product | (C1723⋊C3)⋊D3 |
| unambiguous semidirect product | D5169⋊C3 |
| Fitting, socle, derived | C3×C1723 ⋊ C2×C3 |
| Hall | C3×C1723⋊C3 ⋊ C2 |
| Sylow | C1723 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1723 | C2×C3 |
| LCS | G ▷ C3×C1723 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1726 |
| transitive degree | μt(G) = 5169 |
A minimal faithful permutation representation — 2 generators on 1726 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)(922,1137,1138,1027,1028,1249)(924,1135,1136,1030,1026,1129)(927,1020,1134,1032,1033,1130)(934,1148,1149,1039,1040,1125)(941,1155,1156,1046,1047,1120)(948,1162,1163,1053,1054,1115)(955,1169,1170,1060,1061,1110)(962,1176,1177,1067,1068,1105)(969,1183,1184,1074,1075,1100)(976,1190,1191,1081,1082,1095)(981,1197,1198,1087,1088,1090)(986,1203,1195,1092,1093,1085)(991,1208,1188,1097,1098,1078)(996,1213,1181,1102,1103,1071)(1001,1218,1174,1107,1108,1064)(1006,1223,1167,1112,1113,1057)(1011,1228,1160,1117,1118,1050)(1015,1233,1153,1122,1123,1043)(1018,1238,1146,1126,1127,1036)(1029,1254,1255,1140,1141,1365)(1031,1252,1253,1142,1139,1246)(1034,1133,1251,1143,1144,1247)(1041,1261,1262,1150,1151,1242)(1048,1266,1267,1157,1158,1237)(1055,1273,1274,1164,1165,1232)(1062,1280,1281,1171,1172,1227)(1069,1287,1288,1178,1179,1222)(1076,1294,1295,1185,1186,1217)(1083,1301,1302,1192,1193,1212)(1089,1308,1309,1199,1200,1207)(1094,1315,1313,1204,1205,1202)(1099,1320,1306,1209,1210,1196)(1104,1325,1299,1214,1215,1189)(1109,1330,1292,1219,1220,1182)(1114,1335,1285,1224,1225,1175)(1119,1340,1278,1229,1230,1168)(1124,1345,1271,1234,1235,1161)(1128,1350,1264,1239,1240,1154)(1131,1355,1259,1243,1244,1147)(1145,1250,1366,1256,1257,1363)(1152,1367,1370,1258,1263,1359)(1159,1373,1374,1268,1269,1354)(1166,1377,1378,1275,1276,1349)(1173,1384,1385,1282,1283,1344)(1180,1391,1392,1289,1290,1339)(1187,1398,1399,1296,1297,1334)(1194,1405,1406,1303,1304,1329)(1201,1412,1413,1310,1311,1324)(1206,1419,1420,1316,1317,1319)(1211,1425,1417,1321,1322,1314)(1216,1430,1410,1326,1327,1307)(1221,1435,1403,1331,1332,1300)(1226,1440,1396,1336,1337,1293)(1231,1445,1389,1341,1342,1286)(1236,1450,1382,1346,1347,1279)(1241,1455,1375,1351,1352,1272)(1245,1460,1371,1356,1357,1265)(1248,1465,1368,1360,1361,1260)(1270,1469,1470,1369,1364,1464)(1277,1473,1474,1379,1380,1459)(1284,1477,1478,1386,1387,1454)(1291,1481,1482,1393,1394,1449)(1298,1488,1489,1400,1401,1444)(1305,1495,1496,1407,1408,1439)(1312,1502,1503,1414,1415,1434)(1318,1509,1510,1421,1422,1429)(1323,1516,1514,1426,1427,1424)(1328,1521,1507,1431,1432,1418)(1333,1526,1500,1436,1437,1411)(1338,1531,1493,1441,1442,1404)(1343,1536,1486,1446,1447,1397)(1348,1541,1479,1451,1452,1390)(1353,1546,1475,1456,1457,1383)(1358,1551,1471,1461,1462,1376)(1362,1555,1381,1466,1467,1372)(1388,1556,1557,1472,1468,1550)(1395,1560,1561,1483,1484,1545)(1402,1564,1565,1490,1491,1540)(1409,1568,1569,1497,1498,1535)(1416,1575,1576,1504,1505,1530)(1423,1582,1583,1511,1512,1525)(1428,1589,1590,1517,1518,1520)(1433,1595,1587,1522,1523,1515)(1438,1600,1580,1527,1528,1508)(1443,1605,1573,1532,1533,1501)(1448,1610,1566,1537,1538,1494)(1453,1615,1562,1542,1543,1487)(1458,1620,1558,1547,1548,1480)(1463,1624,1485,1552,1553,1476)(1492,1625,1626,1559,1554,1619)(1499,1629,1630,1570,1571,1614)(1506,1633,1634,1577,1578,1609)(1513,1637,1638,1584,1585,1604)(1519,1644,1645,1591,1592,1599)(1524,1651,1649,1596,1597,1594)(1529,1656,1642,1601,1602,1588)(1534,1661,1635,1606,1607,1581)(1539,1666,1631,1611,1612,1574)(1544,1671,1627,1616,1617,1567)(1549,1675,1572,1621,1622,1563)(1579,1676,1677,1628,1623,1670)(1586,1680,1681,1639,1640,1665)(1593,1684,1685,1646,1647,1660)(1598,1688,1689,1652,1653,1655)(1603,1694,1686,1657,1658,1650)(1608,1699,1682,1662,1663,1643)(1613,1704,1678,1667,1668,1636)(1618,1708,1641,1672,1673,1632)(1648,1709,1710,1679,1674,1703)(1654,1713,1714,1690,1691,1698)(1659,1717,1715,1695,1696,1693)(1664,1719,1711,1700,1701,1687)(1669,1723,1692,1705,1706,1683)(1697,1724,1725,1712,1707,1718)(1702,1726,1722,1720,1721,1716);(1,2,3)(4,5,8,14,22,35,54,77,106,141,182,229,282,341,406,477,554,637,726,821,922,1029,1142,1030,923,822,727,638,555,478,407,342,283,230,183,142,107,78,55,36,23,7,11,16,26,40,59,82,111,146,187,234,287,346,411,482,559,642,731,826,927,1034,1145,1258,1150,1039,932,831,736,647,564,487,416,351,292,239,192,151,116,86,61,39,13,19,27,41,60,84,113,148,189,236,289,348,413,484,561,644,733,828,929,1036,1147,1260,1369,1268,1157,1046,939,838,743,654,571,494,423,358,299,246,199,157,121,90,58,21,32,42,44,64,89,119,155,196,243,296,355,420,491,568,651,740,835,936,1043,1154,1265,1372,1379,1275,1164,1053,946,845,750,661,578,501,430,365,306,252,204,162,125,81,34,51,62,65,67,93,124,160,202,250,303,362,427,498,575,658,747,842,943,1050,1161,1272,1376,1472,1386,1282,1171,1060,953,852,757,668,585,508,437,371,311,257,209,166,110,53,74,87,91,94,96,128,165,207,255,309,369,434,505,582,665,754,849,950,1057,1168,1279,1383,1476,1483,1393,1289,1178,1067,960,859,764,675,592,514,442,376,316,262,213,145,76,103,117,122,126,129,131,169,212,260,314,374,440,512,589,672,761,856,957,1064,1175,1286,1390,1480,1559,1490,1400,1296,1185,1074,967,866,771,681,597,519,447,381,321,266,186,105,138,152,158,163,167,170,172,216,265,319,379,445,517,595,679,768,863,964,1071,1182,1293,1397,1487,1563,1570,1497,1407,1303,1192,1081,974,872,776,686,602,524,452,386,325,233,140,179,193,200,205,210,214,217,219,269,324,384,450,522,600,684,774,870,971,1078,1189,1300,1404,1494,1567,1628,1577,1504,1414,1310,1199,1087,979,877,781,691,607,529,457,390,286,181,226,240,247,253,258,263,267,270,272,328,389,455,527,605,689,779,875,977,1085,1196,1307,1411,1501,1574,1632,1639,1584,1511,1421,1316,1204,1092,984,882,786,696,612,534,461,345,228,279,293,300,307,312,317,322,326,329,331,393,460,532,610,694,784,880,982,1090,1202,1314,1418,1508,1581,1636,1679,1646,1591,1517,1426,1321,1209,1097,989,887,791,701,617,538,410,281,338,352,359,366,372,377,382,387,391,394,396,464,537,615,699,789,885,987,1095,1207,1319,1424,1515,1588,1643,1683,1690,1652,1596,1522,1431,1326,1214,1102,994,892,796,706,621,481,340,403,417,424,431,438,443,448,453,458,462,465,467,541,620,704,794,890,992,1100,1212,1324,1429,1520,1594,1650,1687,1712,1695,1657,1601,1527,1436,1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,147,112,83,48,18,29,17,28,43,63,88,118,153,194,241,294,353,418,489,566,649,738,833,934,1041,1152,1256,1143,1032,925,824,729,640,557,480,409,344,285,232,185,144,109,80,57,38,25,10,9,15,24,37,56,79,108,143,184,231,284,343,408,479,556,639,728,823,924,1031,1140,1027,920,819,724,635,552,475,404,339,280,227,180,139,104,75,52,33,20,12,6)Polycyclic presentation:
2,3^2,1723|c^a=c^2,d^a=d^1722,d^b=d^41870 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31014, 9);