Finite Group Databasebeta

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M₄₀₉₆ #72210198

• p-group
• nilpotent
• metacyclic
• supersolvable
• metabelian
• solvable
• monomial

Aliases

Subgroups

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 2048 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,1711,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,1729,1730,1731,1732,1733,1734,1735,1736,1737,1738,1739,1740,1741,1742,1743,1744,1745,1746,1747,1748,1749,1750,1751,1752,1753,1754,1755,1756,1757,1758,1759,1760,1761,1762,1763,1764,1765,1766,1767,1768,1769,1770,1771,1772,1773,1774,1775,1776,1777,1778,1779,1780,1781,1782,1783,1784,1785,1786,1787,1788,1789,1790,1791,1792,1793,1794,1795,1796,1797,1798,1799,1800,1801,1802,1803,1804,1805,1806,1807,1808,1809,1810,1811,1812,1813,1814,1815,1816,1817,1818,1819,1820,1821,1822,1823,1824,1825,1826,1827,1828,1829,1830,1831,1832,1833,1834,1835,1836,1837,1838,1839,1840,1841,1842,1843,1844,1845,1846,1847,1848,1849,1850,1851,1852,1853,1854,1855,1856,1857,1858,1859,1860,1861,1862,1863,1864,1865,1866,1867,1868,1869,1870,1871,1872,1873,1874,1875,1876,1877,1878,1879,1880,1881,1882,1883,1884,1885,1886,1887,1888,1889,1890,1891,1892,1893,1894,1895,1896,1897,1898,1899,1900,1901,1902,1903,1904,1905,1906,1907,1908,1909,1910,1911,1912,1913,1914,1915,1916,1917,1918,1919,1920,1921,1922,1923,1924,1925,1926,1927,1928,1929,1930,1931,1932,1933,1934,1935,1936,1937,1938,1939,1940,1941,1942,1943,1944,1945,1946,1947,1948,1949,1950,1951,1952,1953,1954,1955,1956,1957,1958,1959,1960,1961,1962,1963,1964,1965,1966,1967,1968,1969,1970,1971,1972,1973,1974,1975,1976,1977,1978,1979,1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024,2025,2026,2027,2028,2029,2030,2031,2032,2033,2034,2035,2036,2037,2038,2039,2040,2041,2042,2043,2044,2045,2046,2047,2048);(2,1026)(4,1028)(6,1030)(8,1032)(10,1034)(12,1036)(14,1038)(16,1040)(18,1042)(20,1044)(22,1046)(24,1048)(26,1050)(28,1052)(30,1054)(32,1056)(34,1058)(36,1060)(38,1062)(40,1064)(42,1066)(44,1068)(46,1070)(48,1072)(50,1074)(52,1076)(54,1078)(56,1080)(58,1082)(60,1084)(62,1086)(64,1088)(66,1090)(68,1092)(70,1094)(72,1096)(74,1098)(76,1100)(78,1102)(80,1104)(82,1106)(84,1108)(86,1110)(88,1112)(90,1114)(92,1116)(94,1118)(96,1120)(98,1122)(100,1124)(102,1126)(104,1128)(106,1130)(108,1132)(110,1134)(112,1136)(114,1138)(116,1140)(118,1142)(120,1144)(122,1146)(124,1148)(126,1150)(128,1152)(130,1154)(132,1156)(134,1158)(136,1160)(138,1162)(140,1164)(142,1166)(144,1168)(146,1170)(148,1172)(150,1174)(152,1176)(154,1178)(156,1180)(158,1182)(160,1184)(162,1186)(164,1188)(166,1190)(168,1192)(170,1194)(172,1196)(174,1198)(176,1200)(178,1202)(180,1204)(182,1206)(184,1208)(186,1210)(188,1212)(190,1214)(192,1216)(194,1218)(196,1220)(198,1222)(200,1224)(202,1226)(204,1228)(206,1230)(208,1232)(210,1234)(212,1236)(214,1238)(216,1240)(218,1242)(220,1244)(222,1246)(224,1248)(226,1250)(228,1252)(230,1254)(232,1256)(234,1258)(236,1260)(238,1262)(240,1264)(242,1266)(244,1268)(246,1270)(248,1272)(250,1274)(252,1276)(254,1278)(256,1280)(258,1282)(260,1284)(262,1286)(264,1288)(266,1290)(268,1292)(270,1294)(272,1296)(274,1298)(276,1300)(278,1302)(280,1304)(282,1306)(284,1308)(286,1310)(288,1312)(290,1314)(292,1316)(294,1318)(296,1320)(298,1322)(300,1324)(302,1326)(304,1328)(306,1330)(308,1332)(310,1334)(312,1336)(314,1338)(316,1340)(318,1342)(320,1344)(322,1346)(324,1348)(326,1350)(328,1352)(330,1354)(332,1356)(334,1358)(336,1360)(338,1362)(340,1364)(342,1366)(344,1368)(346,1370)(348,1372)(350,1374)(352,1376)(354,1378)(356,1380)(358,1382)(360,1384)(362,1386)(364,1388)(366,1390)(368,1392)(370,1394)(372,1396)(374,1398)(376,1400)(378,1402)(380,1404)(382,1406)(384,1408)(386,1410)(388,1412)(390,1414)(392,1416)(394,1418)(396,1420)(398,1422)(400,1424)(402,1426)(404,1428)(406,1430)(408,1432)(410,1434)(412,1436)(414,1438)(416,1440)(418,1442)(420,1444)(422,1446)(424,1448)(426,1450)(428,1452)(430,1454)(432,1456)(434,1458)(436,1460)(438,1462)(440,1464)(442,1466)(444,1468)(446,1470)(448,1472)(450,1474)(452,1476)(454,1478)(456,1480)(458,1482)(460,1484)(462,1486)(464,1488)(466,1490)(468,1492)(470,1494)(472,1496)(474,1498)(476,1500)(478,1502)(480,1504)(482,1506)(484,1508)(486,1510)(488,1512)(490,1514)(492,1516)(494,1518)(496,1520)(498,1522)(500,1524)(502,1526)(504,1528)(506,1530)(508,1532)(510,1534)(512,1536)(514,1538)(516,1540)(518,1542)(520,1544)(522,1546)(524,1548)(526,1550)(528,1552)(530,1554)(532,1556)(534,1558)(536,1560)(538,1562)(540,1564)(542,1566)(544,1568)(546,1570)(548,1572)(550,1574)(552,1576)(554,1578)(556,1580)(558,1582)(560,1584)(562,1586)(564,1588)(566,1590)(568,1592)(570,1594)(572,1596)(574,1598)(576,1600)(578,1602)(580,1604)(582,1606)(584,1608)(586,1610)(588,1612)(590,1614)(592,1616)(594,1618)(596,1620)(598,1622)(600,1624)(602,1626)(604,1628)(606,1630)(608,1632)(610,1634)(612,1636)(614,1638)(616,1640)(618,1642)(620,1644)(622,1646)(624,1648)(626,1650)(628,1652)(630,1654)(632,1656)(634,1658)(636,1660)(638,1662)(640,1664)(642,1666)(644,1668)(646,1670)(648,1672)(650,1674)(652,1676)(654,1678)(656,1680)(658,1682)(660,1684)(662,1686)(664,1688)(666,1690)(668,1692)(670,1694)(672,1696)(674,1698)(676,1700)(678,1702)(680,1704)(682,1706)(684,1708)(686,1710)(688,1712)(690,1714)(692,1716)(694,1718)(696,1720)(698,1722)(700,1724)(702,1726)(704,1728)(706,1730)(708,1732)(710,1734)(712,1736)(714,1738)(716,1740)(718,1742)(720,1744)(722,1746)(724,1748)(726,1750)(728,1752)(730,1754)(732,1756)(734,1758)(736,1760)(738,1762)(740,1764)(742,1766)(744,1768)(746,1770)(748,1772)(750,1774)(752,1776)(754,1778)(756,1780)(758,1782)(760,1784)(762,1786)(764,1788)(766,1790)(768,1792)(770,1794)(772,1796)(774,1798)(776,1800)(778,1802)(780,1804)(782,1806)(784,1808)(786,1810)(788,1812)(790,1814)(792,1816)(794,1818)(796,1820)(798,1822)(800,1824)(802,1826)(804,1828)(806,1830)(808,1832)(810,1834)(812,1836)(814,1838)(816,1840)(818,1842)(820,1844)(822,1846)(824,1848)(826,1850)(828,1852)(830,1854)(832,1856)(834,1858)(836,1860)(838,1862)(840,1864)(842,1866)(844,1868)(846,1870)(848,1872)(850,1874)(852,1876)(854,1878)(856,1880)(858,1882)(860,1884)(862,1886)(864,1888)(866,1890)(868,1892)(870,1894)(872,1896)(874,1898)(876,1900)(878,1902)(880,1904)(882,1906)(884,1908)(886,1910)(888,1912)(890,1914)(892,1916)(894,1918)(896,1920)(898,1922)(900,1924)(902,1926)(904,1928)(906,1930)(908,1932)(910,1934)(912,1936)(914,1938)(916,1940)(918,1942)(920,1944)(922,1946)(924,1948)(926,1950)(928,1952)(930,1954)(932,1956)(934,1958)(936,1960)(938,1962)(940,1964)(942,1966)(944,1968)(946,1970)(948,1972)(950,1974)(952,1976)(954,1978)(956,1980)(958,1982)(960,1984)(962,1986)(964,1988)(966,1990)(968,1992)(970,1994)(972,1996)(974,1998)(976,2000)(978,2002)(980,2004)(982,2006)(984,2008)(986,2010)(988,2012)(990,2014)(992,2016)(994,2018)(996,2020)(998,2022)(1000,2024)(1002,2026)(1004,2028)(1006,2030)(1008,2032)(1010,2034)(1012,2036)(1014,2038)(1016,2040)(1018,2042)(1020,2044)(1022,2046)(1024,2048)

Polycyclic presentation:

2^12|b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,i^2=j,j^2=k,k^2=l,b^a=bl

Conjugacy classes & character table

2560 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);