Finite Group DatabasebetaFinite Group Databasebeta| order | 31338 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1741⋊C3) |
| unambiguous semidirect product | (C1741⋊C3)⋊D3 |
| unambiguous semidirect product | D5223⋊C3 |
| Fitting, socle, derived | C3×C1741 ⋊ C2×C3 |
| Hall | C3×C1741⋊C3 ⋊ C2 |
| Sylow | C1741 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1741 | C2×C3 |
| LCS | G ▷ C3×C1741 | |
| UCS | C1 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1744 |
| transitive degree | μt(G) = 5223 |
A minimal faithful permutation representation — 2 generators on 1744 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)(1029,1256,1257,1140,1141,1374)(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)(1069,1295,1296,1180,1181,1224)(1076,1302,1303,1187,1188,1219)(1083,1309,1310,1194,1195,1214)(1089,1316,1317,1201,1202,1209)(1094,1323,1321,1206,1207,1204)(1099,1328,1314,1211,1212,1198)(1104,1333,1307,1216,1217,1191)(1109,1338,1300,1221,1222,1184)(1114,1343,1293,1226,1227,1177)(1119,1348,1288,1231,1232,1170)(1124,1353,1281,1236,1237,1163)(1128,1358,1274,1241,1242,1156)(1131,1363,1267,1245,1246,1149)(1142,1379,1380,1259,1260,1488)(1144,1377,1378,1262,1258,1371)(1147,1252,1376,1264,1265,1372)(1154,1388,1389,1271,1272,1367)(1161,1393,1394,1278,1279,1362)(1168,1398,1399,1285,1286,1357)(1175,1382,1403,1261,1292,1352)(1182,1406,1407,1297,1298,1347)(1189,1411,1412,1304,1305,1342)(1196,1418,1419,1311,1312,1337)(1203,1425,1426,1318,1319,1332)(1208,1432,1433,1324,1325,1327)(1213,1438,1430,1329,1330,1322)(1218,1443,1423,1334,1335,1315)(1223,1448,1416,1339,1340,1308)(1228,1453,1409,1344,1345,1301)(1233,1458,1404,1349,1350,1294)(1238,1463,1401,1354,1355,1289)(1243,1467,1396,1359,1360,1282)(1247,1472,1391,1364,1365,1275)(1250,1477,1386,1368,1369,1268)(1263,1400,1471,1287,1381,1485)(1266,1375,1489,1383,1384,1486)(1273,1395,1476,1280,1390,1481)(1299,1490,1501,1385,1408,1462)(1306,1504,1505,1413,1414,1457)(1313,1508,1509,1420,1421,1452)(1320,1515,1516,1427,1428,1447)(1326,1522,1523,1434,1435,1442)(1331,1529,1527,1439,1440,1437)(1336,1534,1520,1444,1445,1431)(1341,1539,1513,1449,1450,1424)(1346,1544,1506,1454,1455,1417)(1351,1549,1502,1459,1460,1410)(1356,1554,1499,1464,1465,1405)(1361,1558,1497,1468,1469,1402)(1366,1562,1495,1473,1474,1397)(1370,1566,1493,1478,1479,1392)(1373,1570,1491,1482,1483,1387)(1415,1574,1575,1492,1487,1553)(1422,1586,1587,1510,1511,1548)(1429,1590,1591,1517,1518,1543)(1436,1594,1595,1524,1525,1538)(1441,1601,1602,1530,1531,1533)(1446,1607,1599,1535,1536,1528)(1451,1612,1592,1540,1541,1521)(1456,1617,1588,1545,1546,1514)(1461,1622,1584,1550,1551,1507)(1466,1627,1582,1555,1556,1503)(1470,1631,1580,1559,1560,1500)(1475,1635,1578,1563,1564,1498)(1480,1639,1576,1567,1568,1496)(1484,1626,1512,1571,1572,1494)(1519,1643,1644,1577,1573,1621)(1526,1655,1656,1596,1597,1616)(1532,1659,1660,1603,1604,1611)(1537,1663,1661,1608,1609,1606)(1542,1668,1657,1613,1614,1600)(1547,1673,1653,1618,1619,1593)(1552,1678,1651,1623,1624,1589)(1557,1682,1649,1628,1629,1585)(1561,1686,1647,1632,1633,1583)(1565,1690,1645,1636,1637,1581)(1569,1677,1598,1640,1641,1579)(1605,1694,1695,1646,1642,1672)(1610,1706,1707,1664,1665,1667)(1615,1709,1704,1669,1670,1662)(1620,1711,1702,1674,1675,1658)(1625,1715,1700,1679,1680,1654)(1630,1719,1698,1683,1684,1652)(1634,1723,1696,1687,1688,1650)(1638,1710,1666,1691,1692,1648)(1671,1727,1728,1697,1693,1708)(1676,1736,1733,1712,1713,1705)(1681,1737,1731,1716,1717,1703)(1685,1738,1729,1720,1721,1701)(1689,1735,1714,1724,1725,1699)(1718,1742,1743,1730,1726,1734)(1722,1744,1741,1739,1740,1732);(1,2,3)(4,5,8,14,22,35,54,77,106,141,182,229,282,341,406,477,554,637,726,821,922,1029,1142,1261,1173,1060,953,852,757,668,585,508,437,371,311,257,209,166,110,53,74,87,91,94,96,128,165,207,255,309,369,434,505,582,665,754,849,950,1057,1170,1289,1402,1498,1579,1596,1524,1434,1324,1206,1092,984,882,786,696,612,534,461,345,228,279,293,300,307,312,317,322,326,329,331,393,460,532,610,694,784,880,982,1090,1204,1322,1431,1521,1593,1654,1701,1730,1716,1679,1623,1550,1459,1349,1231,1117,1009,906,730,553,634,648,655,662,669,676,682,687,692,697,702,707,711,714,716,808,905,1007,1115,1229,1347,1457,1548,1621,1677,1690,1686,1682,1678,1673,1668,1663,1659,1655,1643,1571,1482,1265,1028,1139,1153,1160,1167,1174,1181,1188,1195,1202,1207,1212,1217,1222,1227,1232,1237,1242,1246,1249,1251,1371,1481,1389,1255,1025,1134,1240,1235,1230,1225,1220,1215,1210,1205,1199,1192,1185,1178,1171,1164,1157,1150,1135,1024,1133,1250,1370,1480,1569,1642,1665,1670,1675,1680,1684,1688,1692,1695,1656,1591,1509,1412,1296,1172,1052,938,830,720,550,629,705,700,695,690,685,680,673,666,659,652,645,630,549,628,715,807,904,1006,1114,1228,1346,1456,1547,1620,1676,1714,1698,1651,1588,1513,1423,1314,1197,1084,973,865,763,667,577,493,415,335,225,274,320,315,310,304,297,290,275,224,273,330,392,459,531,609,693,783,879,981,1089,1203,1320,1429,1519,1576,1495,1396,1281,1162,1049,942,841,746,657,574,497,426,361,302,249,198,150,100,50,69,85,70,49,68,95,127,164,206,254,308,367,432,503,580,663,752,847,948,1055,1168,1287,1262,1143,1030,923,822,727,638,555,478,407,342,283,230,183,142,107,78,55,36,23,7,11,16,26,40,59,82,111,146,187,234,287,346,411,482,559,642,731,826,927,1034,1147,1266,1385,1297,1180,1067,960,859,764,675,592,514,442,376,316,262,213,145,76,103,117,122,126,129,131,169,212,260,314,374,440,512,589,672,761,856,957,1064,1177,1294,1405,1500,1581,1646,1603,1530,1439,1329,1211,1097,989,887,791,701,617,538,410,281,338,352,359,366,372,377,382,387,391,394,396,464,537,615,699,789,885,987,1095,1209,1327,1437,1528,1600,1658,1703,1732,1739,1720,1683,1628,1555,1464,1354,1236,1122,1013,825,636,723,737,744,751,758,765,772,777,782,7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,762,755,748,741,734,719,632,717,810,908,1011,1119,1233,1351,1461,1552,1625,1681,1718,1729,1700,1653,1592,1520,1430,1321,1200,1080,966,858,756,660,570,486,400,278,333,385,380,375,370,363,356,349,334,277,332,395,463,536,614,698,788,884,986,1094,1208,1326,1436,1526,1598,1578,1497,1401,1288,1169,1056,949,848,753,664,581,504,433,368,305,245,191,135,73,98,120,114,99,72,97,130,168,211,259,313,373,439,510,587,670,759,854,955,1062,1175,1259,1140,1027,920,819,724,635,552,475,404,339,280,227,180,139,104,75,52,33,20,12,6)Polycyclic presentation:
2,3^2,1741|c^a=c^2,d^a=d^1740,d^b=d^356879 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31338, 9);