Finite Group DatabasebetaFinite Group Databasebeta| order | 36020 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| Fitting, socle | C2×C1801 ⋊ C2×C5 |
| derived, Sylow | C1801 ⋊ C22×C5 |
| Hall | C1801⋊C5 ⋊ C22 |
| Hall | C2×D1801 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D1801⋊C5 |
| commutator, cosocle, hypocenter | C1801 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C1801 | C2×C5 |
| LCS | G ▷ C1801 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1803 |
| transitive degree | μt(G) = 3602 |
A minimal faithful permutation representation — 2 generators on 1803 points:
(4,8,9,27,32,5,6,14,16,56)(7,21,22,89,94,11,12,43,45,174)(10,37,38,39,67,18,19,77,79,81)(13,50,51,208,213,24,25,105,107,411)(15,61,62,256,237,29,30,128,130,118)(17,71,72,68,69,34,35,73,33,80)(20,83,84,267,185,40,41,161,163,366)(23,99,100,384,141,47,48,195,197,284)(26,112,113,445,450,53,54,224,226,772)(28,122,123,57,239,58,59,244,119,120)(31,135,136,514,519,64,65,271,273,343)(36,147,148,186,187,74,75,295,95,368)(42,168,169,595,422,86,87,330,332,730)(44,179,180,635,473,91,92,353,355,238)(46,189,190,142,143,96,97,373,70,286)(49,202,203,404,367,102,103,398,400,659)(52,218,219,749,285,109,110,432,434,542)(55,231,232,804,809,115,116,461,463,1282)(60,250,251,175,619,125,126,486,344,345)(63,265,266,82,316,132,133,369,158,159)(66,278,279,910,914,138,139,530,532,709)(76,301,302,423,424,150,151,556,214,732)(78,309,101,393,394,154,155,199,200,240)(85,326,247,248,153,165,166,124,389,307)(88,337,338,1010,783,171,172,609,610,1249)(90,347,348,121,476,176,177,624,241,242)(93,360,361,1064,1024,182,183,647,649,618)(98,379,380,370,371,192,193,676,188,663)(104,405,406,1139,731,205,206,700,701,1187)(106,416,417,1168,826,210,211,719,721,474)(108,426,427,287,288,215,216,737,144,545)(111,439,440,1226,660,221,222,763,764,1086)(114,455,456,1264,543,228,229,793,795,931)(117,468,469,1270,935,234,235,796,818,548)(127,492,493,412,1153,253,254,853,710,711)(129,297,376,377,314,258,259,191,261,157)(131,375,292,293,324,262,263,146,245,164)(134,508,509,507,1026,268,269,880,877,878)(137,524,525,681,575,275,276,901,902,965)(140,536,537,1390,1003,281,282,919,921,1237)(145,298,299,390,391,290,149,257,198,246)(152,236,560,784,785,304,305,471,451,1251)(156,568,220,758,759,311,312,436,437,620)(160,576,357,358,733,317,318,181,600,643)(162,582,526,1050,828,321,322,904,905,475)(167,591,566,204,662,327,328,310,402,403)(170,604,605,627,308,334,335,349,754,564)(173,615,616,1229,1290,340,341,765,1020,1100)(178,629,630,346,1028,350,351,1038,621,622)(184,654,655,1516,1520,363,364,1077,1079,1152)(194,561,562,734,735,382,306,930,425,1190)(196,685,686,692,830,386,387,1112,1114,477)(201,672,410,555,664,395,396,300,506,401)(207,706,707,1142,1250,408,409,702,906,1199)(209,713,714,243,832,413,414,1158,478,479)(212,726,727,1582,1431,419,420,1179,1181,1025)(217,743,744,665,666,429,430,1201,372,1091)(223,769,770,1013,1174,442,443,611,722,1256)(225,777,778,748,535,447,448,1244,1205,827)(227,787,788,460,546,452,453,1254,289,803)(230,800,801,810,1087,458,459,464,1277,1291)(233,812,813,1664,932,465,466,1295,1297,1399)(249,842,580,510,1031,483,484,320,882,883)(252,849,682,683,590,489,490,385,1107,984)(255,858,859,773,1630,495,496,1340,1238,1239)(260,864,865,892,893,499,500,1055,520,661)(264,871,872,937,1032,503,504,550,625,881)(270,885,886,991,1248,511,512,1361,782,1442)(272,558,740,741,573,516,517,428,640,315)(274,895,896,837,838,521,522,668,481,974)(277,907,908,699,1476,527,528,1379,1136,1137)(280,806,807,1206,966,534,462,1284,1285,1421)(283,926,894,1090,639,539,540,1371,1372,1056)(291,342,559,755,756,547,303,515,435,626)(294,487,333,356,739,444,549,601,602,641)(296,939,940,1051,840,551,552,636,903,482)(313,959,457,1272,1273,570,571,797,798,1154)(319,968,723,724,1252,577,578,418,1012,1176)(323,977,808,1507,1428,584,585,1287,1288,1027)(325,554,678,673,674,587,588,381,378,392)(329,987,975,909,1189,592,593,583,1141,1381)(331,995,996,1587,529,597,598,1449,1451,829)(336,1005,1006,1115,1089,606,607,687,1228,1529)(339,1015,1016,1160,565,612,613,715,1269,953)(352,1044,1045,712,1577,632,633,1493,1155,1156)(354,1053,1054,868,869,637,638,897,502,544)(359,1059,1060,879,1314,644,645,1504,825,1357)(362,1072,1073,1106,967,651,652,1514,1076,1376)(365,1084,675,1537,603,657,658,1101,1102,1002)(374,1097,861,862,847,669,670,498,1113,488)(383,969,970,779,1253,679,579,1423,786,1246)(388,1117,1118,1220,1425,688,689,1543,1545,1029)(397,1105,990,1404,1191,693,680,941,1001,1380)(399,1131,768,1500,513,696,697,1232,1233,831)(407,1144,1145,1359,567,703,704,884,1216,955)(415,1162,1163,623,1382,716,717,1528,913,1030)(421,1058,938,1039,1362,642,729,1009,1403,1487)(431,925,1204,1192,1193,746,747,920,736,1462)(433,1210,1211,1126,497,751,752,1283,1511,833)(438,1221,1222,1355,1092,760,761,1098,876,1533)(441,1231,1063,1438,569,766,767,1108,986,957)(446,1241,1242,480,1315,774,775,983,834,835)(449,1247,646,1506,978,780,781,1061,1062,1430)(454,1258,1259,1093,1094,790,791,1639,667,1534)(467,1301,1302,1694,1527,815,816,1373,1532,1172)(470,1307,1308,889,890,819,820,1088,518,1369)(472,1312,725,1225,1083,823,824,1177,1178,1321)(485,1322,992,993,1579,843,844,596,1443,1444)(491,1330,1129,705,1478,850,851,695,1146,1147)(494,1336,1207,1208,985,855,856,750,1607,1436)(501,1317,1318,805,1383,866,836,1509,915,1188)(505,1352,1353,1406,1483,873,874,946,1159,1505)(523,1085,1333,1479,1480,898,899,1104,1033,1143)(531,949,776,1257,964,911,912,789,1173,574)(533,1384,1385,1306,1319,916,917,1195,839,1429)(538,1396,1397,1621,1422,923,924,1219,1569,1409)(541,742,1377,589,1348,928,929,1200,982,936)(553,971,1405,1271,1482,942,943,1065,1286,1034)(557,1354,822,1588,1320,875,947,1169,1311,841)(563,1411,757,1358,945,950,951,1217,1218,944)(572,1419,814,1667,1668,961,962,1298,1299,1631)(581,1363,1364,628,1484,972,887,1486,1035,1036)(586,1432,1398,1730,1568,979,980,1492,1704,1576)(594,1440,1441,1452,1598,988,989,997,1378,1186)(599,1454,1455,1690,976,998,999,1360,1715,1427)(608,1459,1460,1236,1591,1007,1008,1132,1175,1629)(614,1468,1469,1613,1530,1017,1018,1212,1656,1737)(617,1471,1472,1624,954,1021,1022,1243,1666,1413)(631,1489,1109,1110,1004,1041,1042,684,1049,1457)(634,1498,1119,1240,1745,1047,1048,1546,1547,1627)(648,1508,1296,1502,1400,1066,1067,1386,1057,933)(650,1512,1416,1326,1327,1069,1070,1096,846,1450)(653,1515,728,1138,1596,1074,1075,1183,1184,1566)(656,1524,1525,1606,1375,1081,1082,1580,1453,900)(671,1370,1366,1367,1488,891,1099,888,1612,1040)(677,1325,1412,1392,1329,867,952,1052,1230,848)(690,1549,1550,1653,1761,1120,1121,1749,1750,1578)(691,1426,948,927,1203,1123,1124,1408,745,1374)(694,1557,1558,1313,1597,1127,1128,1407,1463,1185)(698,1562,1280,1581,973,1133,1134,1161,1658,1424)(708,1572,1289,1712,956,1149,1150,1445,1652,1414)(718,1263,1418,1157,1633,1165,1166,1351,1281,1245)(720,821,1589,1349,1350,1170,1171,1309,870,934)(738,1603,1345,1346,1334,1196,1197,863,1544,854)(753,1615,1616,1623,1709,1213,1214,1665,1770,1477)(762,1526,845,1679,1461,1223,1224,1323,1324,1011)(771,1628,1182,1571,958,1234,1235,1148,1595,1415)(792,1643,1611,1599,1600,1261,1262,1676,1194,1719)(794,1648,1649,1510,963,1266,1267,1531,1068,1316)(799,1654,1560,1688,1535,1274,1275,1604,1356,1741)(802,1657,1448,1710,960,1278,1279,1608,1439,1417)(811,1659,1660,1344,1536,1292,1293,1764,1095,1686)(817,1675,1393,1394,1635,1304,1305,922,1701,1466)(852,1681,1567,1000,1513,1331,1332,1140,1071,1456)(857,1632,1625,1584,1585,1337,1338,1227,1167,1763)(860,1685,1645,1646,1437,1342,1343,1265,1784,1708)(918,1697,1542,1724,1725,1387,1388,1605,1481,1759)(981,1698,1696,1743,1713,1434,1435,1748,1783,1777)(994,1389,1433,1037,1700,1446,1447,1699,1391,1485)(1014,1310,1641,1339,1538,1464,1465,1677,1103,1682)(1019,1687,1683,1636,1637,1347,1470,1341,1255,1779)(1023,1721,1722,1786,1738,1473,1474,1650,1739,1594)(1043,1729,1559,1151,1727,1490,1491,1130,1573,1574)(1046,1731,1609,1610,1458,1495,1496,1209,1767,1716)(1078,1501,1720,1335,1202,1517,1518,1702,1590,1401)(1080,1736,1561,1678,1680,1521,1522,1602,1328,1714)(1111,1554,1718,1164,1655,1519,1539,1467,1276,1583)(1116,1586,1744,1365,1728,1540,1541,1570,1523,1691)(1122,1754,1617,1792,1801,1552,1553,1671,1771,1775)(1125,1707,1410,1268,1642,1555,1556,1705,1260,1651)(1135,1756,1674,1769,1758,1563,1564,1614,1789,1762)(1180,1368,1765,1494,1198,1592,1593,1692,1503,1402)(1215,1663,1753,1788,1778,1618,1619,1695,1740,1634)(1294,1790,1791,1734,1766,1661,1662,1802,1601,1755)(1300,1793,1794,1475,1742,1669,1670,1780,1689,1723)(1303,1622,1795,1706,1420,1672,1673,1773,1711,1644)(1395,1732,1626,1620,1798,1497,1703,1774,1726,1772)(1499,1751,1752,1785,1717,1733,1551,1647,1800,1768)(1548,1638,1799,1575,1796,1746,1747,1776,1693,1760)(1565,1797,1787,1781,1782,1735,1757,1684,1640,1803);(1,2)(3,4,7,13,26,55,117,236,472,825,877,163,265,506,876,1356,1689,1473,1021,615,276,526,906,1378,1515,578,479,833,495,253,125,58,9,17,36,76,152,306,563,952,1323,843,483,247,122,245,481,839,1304,815,465,231,462,808,1289,1443,1059,989,844,1110,1047,632,350,176,22,46,98,194,383,680,504,670,1098,1414,955,566,309,286,544,933,1401,430,745,926,924,1398,1490,1041,629,1039,1332,1541,1584,1165,716,413,51,108,217,431,748,1137,878,345,620,1027,1427,975,582,974,1426,1297,1666,761,522,897,1374,1696,1746,1546,1117,1544,1682,1747,1636,453,789,774,113,227,454,792,1263,1571,1359,627,749,1206,1606,1441,995,1450,1713,1668,1630,1233,905,355,640,1057,1503,1493,1562,1741,1534,1740,1750,798,532,915,232,464,811,1294,1663,1760,1691,1042,1211,463,735,1191,1379,1628,1776,1794,1779,1629,1381,643,732,1188,1488,1038,1486,1727,1772,1802,1800,1279,807,1286,469,793,1265,1647,1536,1742,1564,1496,1616,813,1200,1051,901,1231,1627,1775,1803,1680,999,1060,1505,1482,756,624,1031,1478,1585,1764,1778,1633,448,779,784,445,773,1240,1632,1687,1121,1551,1753,1302,1371,892,514,758,1220,1623,1556,1319,585,978,1431,1024,519,94,185,367,660,1087,1527,1170,211,418,725,778,764,1228,1330,1459,633,1046,1497,1393,919,929,740,426,738,1198,1350,288,312,569,958,1416,896,190,375,672,1100,1282,1056,637,92,181,359,646,1063,1107,842,987,1062,1447,1387,916,530,235,470,821,1310,1678,1326,549,126,252,491,852,1333,380,678,379,677,866,499,258,30,63,134,270,513,830,476,576,967,1375,898,521,271,515,888,1368,1693,1573,1149,706,269,510,884,1360,1524,562,928,539,281,138,64,29,6,10,20,42,88,173,342,301,557,873,503,262,128,257,498,863,1347,1017,612,337,528,909,1139,1449,1072,970,835,1316,961,570,311,154,19,39,82,160,319,579,971,1317,834,478,241,119,16,33,70,144,289,458,228,112,225,449,782,332,600,1001,1324,1539,1552,1120,688,386,48,101,201,397,694,1128,874,1099,1413,953,564,307,38,80,157,315,574,747,923,536,920,1391,1035,605,347,625,1033,1481,1726,1618,1213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conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36020, 9);