Finite Group DatabasebetaFinite Group Databasebeta| order | 20110 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D2011⋊C5 |
| unambiguous semidirect product | (C2011⋊C5)⋊C2 |
| Fitting, socle, derived, Sylow | C2011 ⋊ C2×C5 |
| Hall | C2011⋊C5 ⋊ C2 |
| Hall | D2011 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C2011 | C2×C5 |
| LCS | G ▷ C2011 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 2011 |
| transitive degree | μt(G) = 2011 |
A minimal faithful permutation representation — 2 generators on 2011 points:
(2,6,7,25,30,3,4,12,14,54)(5,19,20,87,92,9,10,41,43,172)(8,35,36,37,65,16,17,75,77,79)(11,48,49,206,211,22,23,103,105,410)(13,59,60,254,235,27,28,126,128,116)(15,69,70,66,67,32,33,71,31,78)(18,81,82,265,183,38,39,159,161,364)(21,97,98,383,139,45,46,193,195,282)(24,110,111,444,449,51,52,222,224,797)(26,120,121,55,237,56,57,242,117,118)(29,133,134,513,518,62,63,269,271,341)(34,145,146,184,185,72,73,293,93,366)(40,166,167,603,421,84,85,328,330,751)(42,177,178,645,472,89,90,351,353,236)(44,187,188,140,141,94,95,371,68,284)(47,200,201,403,365,100,101,397,399,669)(50,216,217,770,283,107,108,431,433,542)(53,229,230,828,833,113,114,460,462,1335)(58,248,249,173,629,123,124,485,342,343)(61,263,264,80,314,130,131,367,156,157)(64,276,277,937,942,136,137,529,531,728)(74,299,300,422,423,148,149,562,212,753)(76,307,99,392,393,152,153,197,198,238)(83,324,245,246,151,163,164,122,388,305)(86,335,336,1028,807,169,170,615,617,1305)(88,345,346,119,475,174,175,634,239,240)(91,358,359,1087,1050,180,181,657,659,628)(96,377,378,368,369,190,191,686,186,673)(102,404,405,1161,752,203,204,715,717,1219)(104,415,416,1202,849,208,209,738,740,473)(106,425,426,285,286,213,214,758,142,545)(109,438,439,1265,670,219,220,784,786,605)(112,454,455,1017,543,226,227,816,602,961)(115,466,467,1356,1250,232,233,841,842,1731)(125,491,492,411,1187,251,252,873,729,730)(127,295,374,375,312,256,257,189,259,155)(129,373,290,291,322,260,261,144,243,162)(132,507,508,506,1052,266,267,900,897,898)(135,523,524,695,583,273,274,924,926,1001)(138,536,537,1476,1416,279,280,952,954,1286)(143,296,297,389,390,288,147,255,196,244)(150,567,568,808,809,302,303,990,450,1307)(154,576,218,779,780,309,310,435,436,630)(158,584,355,356,754,315,316,179,606,653)(160,590,525,1072,851,319,320,928,929,474)(165,599,574,202,672,325,326,308,401,402)(168,610,611,637,306,332,333,347,775,572)(171,622,623,1544,1341,338,339,1041,1043,1680)(176,639,640,344,1054,348,349,1063,631,632)(182,664,665,1282,1172,361,362,793,1097,1186)(192,689,690,755,756,380,381,1119,424,1223)(194,699,700,706,853,385,386,1134,1136,476)(199,682,559,560,674,394,395,298,505,400)(205,722,723,817,1306,407,408,1173,1175,1319)(207,732,733,241,855,412,413,1191,477,478)(210,745,746,1693,1557,418,419,1213,1215,1051)(215,764,765,675,676,428,429,1236,370,1101)(221,791,792,1757,1220,441,442,1278,1280,1383)(223,802,803,1768,571,446,447,1296,1298,850)(225,811,812,546,547,451,452,1311,287,964)(228,822,823,1366,604,457,458,1326,852,329)(231,838,839,864,865,464,465,1351,484,1379)(234,846,847,1203,1612,469,470,1361,1362,1682)(247,863,588,509,1058,482,483,318,903,750)(250,870,696,697,598,488,489,384,1130,1013)(253,744,877,798,1413,494,495,1211,1287,1288)(258,883,884,915,916,498,499,1077,519,671)(262,891,892,973,1059,502,503,554,635,902)(268,905,906,1019,1673,510,511,1417,1419,1527)(270,563,761,762,581,515,516,427,650,313)(272,918,919,860,861,520,521,678,480,1008)(275,931,932,714,1009,526,527,1447,1159,594)(278,947,948,1248,1002,533,534,1468,1470,1520)(281,589,957,1421,638,539,540,907,1483,1062)(289,564,565,776,777,549,301,514,434,636)(292,486,331,354,760,551,552,607,608,651)(294,975,976,1073,471,555,556,646,927,481)(304,830,831,1342,1343,570,461,1338,834,1678)(311,663,456,1323,1160,578,579,819,820,1188)(317,1004,742,743,1308,585,586,417,1033,1210)(321,535,949,1474,1559,592,593,950,1472,1053)(323,558,688,683,684,595,596,379,376,391)(327,1016,538,933,1222,600,601,591,956,1450)(334,1024,1025,1138,1099,612,613,701,1269,1032)(337,1037,1038,1194,573,619,620,734,1018,995)(340,1046,1047,1809,1732,625,626,1552,1454,1337)(350,843,1068,731,1152,642,643,1358,1189,967)(352,1075,1076,888,889,647,648,920,501,544)(357,840,1085,899,998,654,655,1354,1412,580)(360,824,1095,1129,1003,661,662,1329,960,1521)(363,575,719,720,904,667,668,406,1169,1170)(372,1107,880,881,868,679,680,497,1135,487)(382,1124,1125,1309,1310,692,693,1639,810,1777)(387,1140,1141,1259,1560,702,703,1647,1649,1055)(396,1149,1150,1494,1224,707,708,977,1020,1449)(398,1155,790,1578,1368,711,712,1275,1276,854)(409,1180,1181,1539,1554,725,726,1665,1048,1198)(414,1196,1197,633,878,735,736,796,1056,496)(420,490,1218,1153,721,748,749,871,710,1171)(430,1242,1243,1225,1226,767,768,1720,757,1703)(432,1251,1252,1148,1370,772,773,1733,1174,856)(437,1260,1261,1410,1102,781,782,1108,896,1619)(440,1273,1274,1522,577,788,789,1131,1015,666)(443,1284,1071,1455,1775,794,795,1575,1576,1870)(445,1290,1291,479,1372,799,800,1387,857,858)(448,1303,1304,1646,1489,805,806,1432,1139,1558)(453,1245,1246,1103,1104,814,769,1726,677,1620)(459,660,1331,1608,1382,825,826,1093,1106,867)(463,1345,1346,965,966,835,836,1805,548,656)(468,641,1360,1132,1133,844,845,1066,698,1643)(493,1217,1249,541,1014,747,876,771,958,959)(500,1398,1399,1459,1460,885,886,1603,943,1221)(504,1406,1407,1506,1675,893,894,985,1192,1595)(512,1302,1422,1532,1844,908,909,1772,1850,1915)(517,1428,1429,1605,1606,912,913,1686,1092,1098)(522,1437,1438,1563,1564,921,922,1115,1060,1168)(528,955,1451,1657,1830,934,935,1481,1779,1378)(530,991,1314,1315,999,939,940,813,1207,582)(532,1462,1463,1376,1377,944,945,1228,862,951)(550,992,993,1321,1027,968,569,938,818,1193)(553,874,618,741,1313,970,971,1034,1035,1208)(557,1496,1497,1599,1566,978,979,1088,1471,1061)(561,901,718,930,1232,982,983,1166,1167,1445)(566,925,787,1137,1111,987,988,1270,1271,1165)(587,1523,1300,1301,1677,1005,1006,804,1547,1195)(597,1397,1121,1233,1234,1010,1011,1585,763,1444)(609,1529,1238,1112,1113,1021,1022,1439,685,1629)(614,1499,1500,1283,1701,1026,980,1156,1759,1762)(616,621,1534,1540,1736,1030,1031,1039,1253,1367)(624,1533,1551,1380,996,1029,1045,1292,866,1517)(627,1525,1555,1316,1953,1012,1049,1640,1782,1783)(644,1513,1574,1289,895,1069,1070,1906,1696,1408)(649,1583,1584,1433,1434,1078,1079,1241,917,1100)(652,1591,1491,1492,1568,1082,1083,974,1064,1420)(658,1601,1602,1589,1484,1089,1090,1464,1081,962)(681,1627,1425,1426,1570,1109,1110,911,1734,1065)(687,1634,1579,1580,1384,1116,1117,1074,1272,869)(691,1582,778,1415,981,1122,1123,1256,1257,1501)(694,1199,1200,1804,1674,1127,737,1553,1344,1190)(704,1363,1535,1788,709,1143,1144,1823,1778,1151)(705,1524,986,1239,1240,1145,1146,1508,766,1436)(713,1498,1330,1884,1645,1157,1158,1795,1796,1561)(716,994,1469,1328,1735,1163,1164,1515,1793,1369)(724,1662,1663,1813,997,1177,1178,1352,1526,1518)(727,1668,859,1829,1336,1183,1184,1373,1374,829)(739,1684,1685,1403,1404,1204,1205,1609,890,963)(759,1708,1394,1395,1389,1229,1230,882,1648,875)(774,1737,1738,1743,1392,1254,1255,1905,1989,879)(783,1745,1746,1911,1704,1262,1263,1635,1528,1761)(785,1750,1751,1935,1827,1266,1267,1957,1848,1371)(801,1765,1766,1057,1423,1293,1294,1614,1562,910)(815,1747,1748,1705,1706,1317,1264,1991,1227,1982)(821,1784,1756,1842,1621,1324,1318,1709,1411,1896)(827,1799,1800,2004,1993,1332,1333,1877,1858,1960)(832,1477,1353,1386,969,1162,1340,1814,872,1488)(837,1810,1792,1622,1623,1348,1349,1997,1105,1895)(848,1142,1825,1971,1644,1364,1365,1651,1753,1567)(887,1656,1838,1886,1774,1400,1401,1764,1480,1776)(914,1652,1855,1692,1664,1430,1431,1972,1615,1700)(923,1859,1716,1967,1538,1440,1441,1637,1676,1920)(936,1803,1424,1320,1973,1452,1453,1811,1785,1655)(941,1357,1871,1295,1375,1456,1457,1819,1697,1616)(946,1875,1876,1926,1927,1465,1466,1717,1565,1908)(953,1339,1807,1808,1007,1478,1479,1347,1770,1000)(972,1212,1044,1299,1806,1128,1490,1548,1549,1771)(984,1418,1176,1327,1781,1503,1504,1659,1660,1791)(989,1448,1281,1277,1713,1510,1511,1760,1754,1755)(1023,1914,1722,1714,1715,1530,1531,1725,1235,1752)(1036,1919,1727,1630,1631,1536,1537,1952,1114,1961)(1040,1216,1921,1955,1831,1541,1542,1698,1625,1388)(1042,1179,1872,1572,1402,1154,1546,1458,1067,1826)(1080,1939,1632,1873,1550,1586,1587,1962,1461,1702)(1084,1944,1900,1901,1334,1592,1593,1507,1679,1801)(1086,1802,1185,1916,1409,1596,1597,1670,1671,1841)(1091,1945,1893,1907,1948,1604,1594,1730,1607,1617)(1094,1950,1951,1929,1930,1610,1611,1633,1569,1661)(1096,1763,1598,1885,1891,1613,1285,1887,1689,1485)(1118,1758,1946,1931,1828,1279,1636,1600,1794,1571)(1120,1707,1968,1350,1393,1268,1638,1683,1812,1385)(1126,1947,1322,1922,1502,1641,1642,1787,1543,1897)(1147,1475,1514,1728,1729,1653,1654,1712,1247,1949)(1182,1923,1669,1767,1519,1545,1667,1917,1381,1909)(1201,1942,1815,1694,1695,1681,1590,1486,1214,1979)(1206,1861,1853,1856,1857,1687,1442,2003,1435,1618)(1209,1954,1892,1820,1821,1690,1624,1493,1359,1851)(1231,1749,1852,1742,1932,1710,1711,1427,1988,1573)(1237,1878,1936,1937,1832,1467,1718,1581,2009,1390)(1244,1719,1258,1658,1509,1723,1724,1740,1741,1903)(1297,1941,1786,1839,1840,1769,1588,1980,1405,1487)(1312,1867,1835,1836,1834,1688,1780,1396,1824,1391)(1325,1995,1996,2008,1983,1789,1790,1984,1912,2011)(1355,1869,1446,1943,1894,1816,1817,1866,1843,1495)(1414,1516,1473,1798,1956,1845,1846,1881,1882,1626)(1443,1992,1650,1744,1958,1863,1864,1969,1890,1628)(1482,1577,1879,1994,1904,1888,1889,1933,1928,1512)(1505,1773,1666,1918,1925,1898,1899,1975,1556,1739)(1672,1691,1862,1874,1865,1910,1854,1963,1902,1924)(1699,1940,1978,1880,1847,1837,1981,1964,1822,1976)(1721,2007,1965,1818,1934,1883,1986,1977,2002,1833)(1797,2000,1970,1990,2006,1998,1999,1959,1868,1966)(1849,1913,2005,1985,2010,1974,2001,1860,1938,1987);(1,2,5,11,24,53,115,234,471,390,242,324,598,1014,586,478,856,1371,1828,1570,1063,1483,1872,1456,939,137,278,535,951,1475,1804,1342,828,1197,794,441,219,107,48,104,210,420,750,346,636,1061,1567,1643,1379,961,542,282,79,54,116,236,473,850,570,302,148,72,32,12,26,58,125,253,496,879,1393,1384,868,485,610,844,466,303,569,994,1516,1908,1932,1358,1301,1544,1766,1613,362,666,576,545,963,1486,1814,1351,733,1193,1026,612,332,166,329,605,669,364,172,341,628,1051,1558,1488,968,549,288,33,71,144,292,553,972,1127,692,380,190,94,41,88,176,350,644,1071,1577,1934,1832,1389,873,1037,1539,1046,1401,1500,1155,1619,1101,1618,1906,800,1292,1291,1386,749,488,248,486,867,1383,1219,751,410,728,1186,1171,719,404,716,1165,395,261,371,678,1106,1625,1855,1979,1897,1501,596,257,293,554,974,1493,1893,1728,1245,767,428,213,103,207,414,737,1201,792,1281,1761,1703,1834,1211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35,1368,853,475,584,1003,1522,326,246,383,695,1129,958,539,279,136,62,27,4,8,18,40,86,171,340,627,1010,595,322,159,318,589,1006,858,723,1176,1661,1974,1568,1417,1662,1829,1478,280,538,590,1008,1524,1910,1670,1183,725,407,203,100,45,19,42,91,182,363,637,770,1248,1274,1130,863,1016,1520,1001,157,118,238,474,852,1367,1826,1400,885,498,256,28,61,132,268,512,910,1424,1810,1530,1021,607,328,464,229,461,832,405,718,1168,1658,1821,1772,1923,1909,1518,668,574,307,284,544,962,1485,1813,1194,1017,600,325,163,81,160,321,594,898,343,630,1053,786,1269,1218,1499,978,555,73,147,298,561,984,1505,1871,1941,1586,1078,647,90,179,357,656,1086,1598,1451,1869,1920,1536,1034,615,1029,622,1042,1153,201,400,673,1100,1617,1294,1767,1380,864,482,245,120,243,480,862,1378,1527,730,1188,1159,399,401,200,398,713,708,503,680,1108,1626,1957,1825,746,1216,1699,1011,499,556,977,1495,1895,1782,1314,811,1312,1687,1204,209,417,744,1212,1692,1282,932,1449,1223,1702,1548,1041,1545,1180,1132,696,98,196,391,705,1147,1655,1915,1288,1412,897,161,263,505,896,1411,1843,1083,1230,1709,1754,1275,788,438,785,1268,381,691,1117,1635,1965,1968,1362,416,741,1209,1691,913,847,1363,1824,1873,1459,937,1323,1788,1777,1769,447,804,1302,1773,1971,789,524,927,1444,1865,1200,1552,1562,1056,631,342,43,93,186,370,677,1105,1624,1780,1866,1167,928,523,925,1443,1748,1441,1860,1718,1984,1998,1795,662,822,1327,1792,1587,1940,1638,1705,1225,755,422,206,411,731,1190,1457,467,843,1359,1822,1599,1468,1095,1612,1953,1597,831,1339,1803,1989,1649,436,271,519,917,1435,1125,1640,1945,1592,1082,608,351,646,1074,1581,1243,1722,1723,1242,1721,1881,950,947,1469,1879,1246,1727,1950,1986,2008,1911,1494,356,603,1019,1346,1806,1925,1737,1988,1740,1256,776,431,771,1250,1732,1336,230,463,837,1350,1796,1472,659,1092,1607,1070,568,991,1514,1710,1229,214,427,763,1235,1716,690,1121,1583,1914,1610,826,657,1088,1600,1876,765,1238,1437,1745,1990,1511,527,933,1161,1657,812,1313,1781,1994,1926,1563,973,175,347,638,833,1341,1554,1196,1679,1715,1580,1270,784,1215,1697,1293,799,111,225,453,815,1318,945,1464,1874,1900,1407,300,563,986,1509,1537,1205,1686,1978,1951,1438,378,688,377,687,1118,442,793,1283,1757,1463,426,760,1232,1713,1845,983,267,509,904,1416,449,807,1306,1775,1993,1113,881,386,701,1139,1184,1669,1045,877,1391,1226,1704,1760,1278,1681,1199,735,412,49,106,215,430,769,1247,1730,1892,1491,892,146,295,558,981,1502,1695,1972,1955,1608,919,188,373,682,1111,1163,204,406,721,1172,942,211,421,752,1220,1382,552,124,250,490,872,1387,1551,1290,1574,1857,676,1102,712,1156,1656,626,1048,620,492,875,1390,1833,1889,1575,1069,642,348,174,20,44,96,192,382,694,1128,970,551,290,70,69,143,289,550,969,1489,1557,1050,518,92,183,365,670,604,85,168,334,614,1027,1191,839,1353,1815,1404,286,310,577,664,1096,1614,1043,1547,1068,1573,1927,1846,1164,992,567,233,468,333,249,487,869,1385,1392,878,494,251,123,56,7,15,34,74,150,304,571,849,472,235,30,65,139,283,543,865,1133,1644,1566,777,634,1058,748,418,208,23,50,109,221,443,796,462,834,1344,1654,1377,593,533,276,530,941,1458,1421,640,1065,1571,1827,1370,855,1004,959,1013,164,121,244,481,469,232,113,51,22,9,3)Polycyclic presentation:
2,5,2011|c^a=c^2010,c^b=c^1328211 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20110, 1);