Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C72×C49 · Schur multiplier (H2) = C72 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C343 | C72 |
| commutator, socle | C7 | C72×C49 |
| Frattini, cosocle | C49 | C73 |
| prime radical | H(3,𝔽7) | C49 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C343 ◁ G | |
| LECS | G ▷ C49 ▷ C7 ▷ C1 | |
231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 2401 |
| transitive degree | μt(G) = 2401 |
A minimal faithful permutation representation — 3 generators on 2401 points:
(1,2,6,17,42,92,176,4,8,19,44,94,178,304,14,25,50,100,184,310,481,38,63,113,197,323,494,704,87,137,221,347,518,728,966,170,254,380,551,761,999,1247,296,422,593,803,1041,1289,1527,5,9,20,45,95,179,305,15,26,51,101,185,311,482,39,64,114,198,324,495,705,88,138,222,348,519,729,967,171,255,381,552,762,1000,1248,297,423,594,804,1042,1290,1528,471,642,852,1090,1338,1576,1786,16,27,52,102,186,312,483,40,65,115,199,325,496,706,89,139,223,349,520,730,968,172,256,382,553,763,1001,1249,298,424,595,805,1043,1291,1529,472,643,853,1091,1339,1577,1787,689,899,1137,1385,1623,1833,2001,41,66,116,200,326,497,707,90,140,224,350,521,731,969,173,257,383,554,764,1002,1250,299,425,596,806,1044,1292,1530,473,644,854,1092,1340,1578,1788,690,900,1138,1386,1624,1834,2002,942,1180,1428,1666,1876,2044,2166,91,141,225,351,522,732,970,174,258,384,555,765,1003,1251,300,426,597,807,1045,1293,1531,474,645,855,1093,1341,1579,1789,691,901,1139,1387,1625,1835,2003,943,1181,1429,1667,1877,2045,2167,1216,1464,1702,1912,2080,2202,2282,175,259,385,556,766,1004,1252,301,427,598,808,1046,1294,1532,475,646,856,1094,1342,1580,1790,692,902,1140,1388,1626,1836,2004,944,1182,1430,1668,1878,2046,2168,1217,1465,1703,1913,2081,2203,2283,1489,1727,1937,2105,2227,2307,2352,302,428,599,809,1047,1295,1533,476,647,857,1095,1343,1581,1791,693,903,1141,1389,1627,1837,2005,945,1183,1431,1669,1879,2047,2169,1218,1466,1704,1914,2082,2204,2284,1490,1728,1938,2106,2228,2308,2353,1743,1953,2121,2243,2323,2368,2389)(3,7,18,43,93,177,303,12,23,48,98,182,308,479,35,60,110,194,320,491,701,83,133,217,343,514,724,962,166,250,376,547,757,995,1243,292,418,589,799,1037,1285,1523,467,638,848,1086,1334,1572,1782,13,24,49,99,183,309,480,36,61,111,195,321,492,702,84,134,218,344,515,725,963,167,251,377,548,758,996,1244,293,419,590,800,1038,1286,1524,468,639,849,1087,1335,1573,1783,686,896,1134,1382,1620,1830,1998,37,62,112,196,322,493,703,85,135,219,345,516,726,964,168,252,378,549,759,997,1245,294,420,591,801,1039,1287,1525,469,640,850,1088,1336,1574,1784,687,897,1135,1383,1621,1831,1999,940,1178,1426,1664,1874,2042,2164,86,136,220,346,517,727,965,169,253,379,550,760,998,1246,295,421,592,802,1040,1288,1526,470,641,851,1089,1337,1575,1785,688,898,1136,1384,1622,1832,2000,941,1179,1427,1665,1875,2043,2165,1215,1463,1701,1911,2079,2201,2281,80,130,214,340,511,721,959,163,247,373,544,754,992,1240,289,415,586,796,1034,1282,1520,464,635,845,1083,1331,1569,1779,683,893,1131,1379,1617,1827,1995,938,1176,1424,1662,1872,2040,2162,1214,1462,1700,1910,2078,2200,2280,33,58,108,192,318,489,699,81,131,215,341,512,722,960,164,248,374,545,755,993,1241,290,416,587,797,1035,1283,1521,465,636,846,1084,1332,1570,1780,684,894,1132,1380,1618,1828,1996,939,1177,1425,1663,1873,2041,2163,11,22,47,97,181,307,478,34,59,109,193,319,490,700,82,132,216,342,513,723,961,165,249,375,546,756,994,1242,291,417,588,798,1036,1284,1522,466,637,847,1085,1333,1571,1781,685,895,1133,1381,1619,1829,1997)(10,21,46,96,180,306,477,30,55,105,189,315,486,696,74,124,208,334,505,715,953,156,240,366,537,747,985,1233,281,407,578,788,1026,1274,1512,455,626,836,1074,1322,1560,1770,674,884,1122,1370,1608,1818,1986,31,56,106,190,316,487,697,75,125,209,335,506,716,954,157,241,367,538,748,986,1234,282,408,579,789,1027,1275,1513,456,627,837,1075,1323,1561,1771,675,885,1123,1371,1609,1819,1987,930,1168,1416,1654,1864,2032,2154,76,126,210,336,507,717,955,158,242,368,539,749,987,1235,283,409,580,790,1028,1276,1514,457,628,838,1076,1324,1562,1772,676,886,1124,1372,1610,1820,1988,931,1169,1417,1655,1865,2033,2155,1208,1456,1694,1904,2072,2194,2274,72,122,206,332,503,713,951,154,238,364,535,745,983,1231,279,405,576,786,1024,1272,1510,453,624,834,1072,1320,1558,1768,672,882,1120,1368,1606,1816,1984,928,1166,1414,1652,1862,2030,2152,1207,1455,1693,1903,2071,2193,2273,78,128,212,338,509,719,957,161,245,371,542,752,990,1238,287,413,584,794,1032,1280,1518,462,633,843,1081,1329,1567,1777,681,891,1129,1377,1615,1825,1993,936,1174,1422,1660,1870,2038,2160,1213,1461,1699,1909,2077,2199,2279,29,54,104,188,314,485,695,73,123,207,333,504,714,952,155,239,365,536,746,984,1232,280,406,577,787,1025,1273,1511,454,625,835,1073,1321,1559,1769,673,883,1121,1369,1607,1817,1985,929,1167,1415,1653,1863,2031,2153,32,57,107,191,317,488,698,79,129,213,339,510,720,958,162,246,372,543,753,991,1239,288,414,585,795,1033,1281,1519,463,634,844,1082,1330,1568,1778,682,892,1130,1378,1616,1826,1994,937,1175,1423,1661,1871,2039,2161)(28,53,103,187,313,484,694,69,119,203,329,500,710,948,148,232,358,529,739,977,1225,272,398,569,779,1017,1265,1503,445,616,826,1064,1312,1550,1760,664,874,1112,1360,1598,1808,1976,920,1158,1406,1644,1854,2022,2144,70,120,204,330,501,711,949,149,233,359,530,740,978,1226,273,399,570,780,1018,1266,1504,446,617,827,1065,1313,1551,1761,665,875,1113,1361,1599,1809,1977,921,1159,1407,1645,1855,2023,2145,1200,1448,1686,1896,2064,2186,2266,150,234,360,531,741,979,1227,274,400,571,781,1019,1267,1505,447,618,828,1066,1314,1552,1762,666,876,1114,1362,1600,1810,1978,922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41,1107,831,820,824)(569,825,827,832,1114,842,1119)(578,835,1120,837,844,1129,1124)(589,847,1132,1424,1427,1135,849)(601,858,860,864,870,877,887)(604,861,871,888,863,865,878)(609,866,889,1153,879,868,872)(616,873,875,880,1160,890,1165)(626,883,1166,885,892,1174,1169)(638,895,1177,1462,1463,1178,896)(649,904,906,910,916,923,932)(652,907,917,933,909,911,924)(657,912,934,1195,925,914,918)(664,919,921,926,1201,935,1206)(674,929,1207,930,937,1213,1208)(694,947,949,950,1227,956,1230)(696,952,1231,954,958,1238,1235)(701,961,1241,1520,1526,1245,963)(708,972,988,1261,980,974,975)(710,976,978,981,1267,989,1271)(715,984,1272,986,991,1280,1276)(724,994,1283,1569,1575,1287,996)(733,1006,1014,1029,1008,1009,1020)(735,1010,1030,1307,1021,1012,1015)(739,1016,1018,1022,1314,1031,1319)(747,1025,1320,1027,1033,1329,1324)(757,1036,1332,1617,1622,1336,1038)(767,1048,1050,1054,1060,1067,1077)(769,1051,1061,1078,1053,1055,1068)(773,1056,1079,1355,1069,1058,1062)(779,1063,1065,1070,1362,1080,1367)(788,1073,1368,1075,1082,1377,1372)(799,1085,1380,1662,1665,1383,1087)(811,1096,1098,1102,1108,1115,1125)(814,1099,1109,1126,1101,1103,1116)(819,1104,1127,1401,1117,1106,1110)(826,1111,1113,1118,1408,1128,1413)(836,1121,1414,1123,1130,1422,1417)(848,1133,1425,1700,1701,1426,1134)(859,1142,1144,1148,1154,1161,1170)(862,1145,1155,1171,1147,1149,1162)(867,1150,1172,1443,1163,1152,1156)(874,1157,1159,1164,1449,1173,1454)(884,1167,1455,1168,1175,1461,1456)(905,1184,1186,1190,1196,1202,1209)(908,1187,1197,1210,1189,1191,1203)(913,1192,1211,1477,1204,1194,1198)(920,1199,1200,1205,1481,1212,1485)(946,1220,1236,1499,1228,1222,1223)(948,1224,1226,1229,1505,1237,1509)(953,1232,1510,1234,1239,1518,1514)(962,1242,1521,1779,1785,1525,1244)(971,1254,1262,1277,1256,1257,1268)(973,1258,1278,1545,1269,1260,1263)(977,1264,1266,1270,1552,1279,1557)(985,1273,1558,1275,1281,1567,1562)(995,1284,1570,1827,1832,1574,1286)(1005,1296,1298,1302,1308,1315,1325)(1007,1299,1309,1326,1301,1303,1316)(1011,1304,1327,1593,1317,1306,1310)(1017,1311,1313,1318,1600,1328,1605)(1026,1321,1606,1323,1330,1615,1610)(1037,1333,1618,1872,1875,1621,1335)(1049,1344,1346,1350,1356,1363,1373)(1052,1347,1357,1374,1349,1351,1364)(1057,1352,1375,1639,1365,1354,1358)(1064,1359,1361,1366,1646,1376,1651)(1074,1369,1652,1371,1378,1660,1655)(1086,1381,1663,1910,1911,1664,1382)(1097,1390,1392,1396,1402,1409,1418)(1100,1393,1403,1419,1395,1397,1410)(1105,1398,1420,1681,1411,1400,1404)(1112,1405,1407,1412,1687,1421,1692)(1122,1415,1693,1416,1423,1699,1694)(1143,1432,1434,1438,1444,1450,1457)(1146,1435,1445,1458,1437,1439,1451)(1151,1440,1459,1715,1452,1442,1446)(1158,1447,1448,1453,1719,1460,1723)(1185,1467,1469,1473,1478,1482,1486)(1188,1470,1479,1487,1472,1474,1483)(1193,1475,1488,1736,1484,1476,1480)(1219,1492,1500,1515,1494,1495,1506)(1221,1496,1516,1755,1507,1498,1501)(1225,1502,1504,1508,1762,1517,1767)(1233,1511,1768,1513,1519,1777,1772)(1243,1522,1780,1995,2000,1784,1524)(1253,1534,1536,1540,1546,1553,1563)(1255,1537,1547,1564,1539,1541,1554)(1259,1542,1565,1803,1555,1544,1548)(1265,1549,1551,1556,1810,1566,1815)(1274,1559,1816,1561,1568,1825,1820)(1285,1571,1828,2040,2043,1831,1573)(1297,1582,1584,1588,1594,1601,1611)(1300,1585,1595,1612,1587,1589,1602)(1305,1590,1613,1849,1603,1592,1596)(1312,1597,1599,1604,1856,1614,1861)(1322,1607,1862,1609,1616,1870,1865)(1334,1619,1873,2078,2079,1874,1620)(1345,1628,1630,1634,1640,1647,1656)(1348,1631,1641,1657,1633,1635,1648)(1353,1636,1658,1891,1649,1638,1642)(1360,1643,1645,1650,1897,1659,1902)(1370,1653,1903,1654,1661,1909,1904)(1391,1670,1672,1676,1682,1688,1695)(1394,1673,1683,1696,1675,1677,1689)(1399,1678,1697,1925,1690,1680,1684)(1406,1685,1686,1691,1929,1698,1933)(1433,1705,1707,1711,1716,1720,1724)(1436,1708,1717,1725,1710,1712,1721)(1441,1713,1726,1946,1722,1714,1718)(1468,1729,1731,1734,1737,1739,1741)(1471,1732,1738,1742,1733,1735,1740)(1491,1744,1746,1750,1756,1763,1773)(1493,1747,1757,1774,1749,1751,1764)(1497,1752,1775,1971,1765,1754,1758)(1503,1759,1761,1766,1978,1776,1983)(1512,1769,1984,1771,1778,1993,1988)(1523,1781,1996,2162,2165,1999,1783)(1535,1792,1794,1798,1804,1811,1821)(1538,1795,1805,1822,1797,1799,1812)(1543,1800,1823,2017,1813,1802,1806)(1550,1807,1809,1814,2024,1824,2029)(1560,1817,2030,1819,1826,2038,2033)(1572,1829,2041,2200,2201,2042,1830)(1583,1838,1840,1844,1850,1857,1866)(1586,1841,1851,1867,1843,1845,1858)(1591,1846,1868,2059,1859,1848,1852)(1598,1853,1855,1860,2065,1869,2070)(1608,1863,2071,1864,1871,2077,2072)(1629,1880,1882,1886,1892,1898,1905)(1632,1883,1893,1906,1885,1887,1899)(1637,1888,1907,2093,1900,1890,1894)(1644,1895,1896,1901,2097,1908,2101)(1671,1915,1917,1921,1926,1930,1934)(1674,1918,1927,1935,1920,1922,1931)(1679,1923,1936,2114,1932,1924,1928)(1706,1939,1941,1944,1947,1949,1951)(1709,1942,1948,1952,1943,1945,1950)(1730,1954,1955,1956,1957,1958,1959)(1745,1960,1962,1966,1972,1979,1989)(1748,1963,1973,1990,1965,1967,1980)(1753,1968,1991,2139,1981,1970,1974)(1760,1975,1977,1982,2146,1992,2151)(1770,1985,2152,1987,1994,2160,2155)(1782,1997,2163,2280,2281,2164,1998)(1793,2006,2008,2012,2018,2025,2034)(1796,2009,2019,2035,2011,2013,2026)(1801,2014,2036,2181,2027,2016,2020)(1808,2021,2023,2028,2187,2037,2192)(1818,2031,2193,2032,2039,2199,2194)(1839,2048,2050,2054,2060,2066,2073)(1842,2051,2061,2074,2053,2055,2067)(1847,2056,2075,2215,2068,2058,2062)(1854,2063,2064,2069,2219,2076,2223)(1881,2083,2085,2089,2094,2098,2102)(1884,2086,2095,2103,2088,2090,2099)(1889,2091,2104,2236,2100,2092,2096)(1916,2107,2109,2112,2115,2117,2119)(1919,2110,2116,2120,2111,2113,2118)(1940,2122,2123,2124,2125,2126,2127)(1961,2128,2130,2134,2140,2147,2156)(1964,2131,2141,2157,2133,2135,2148)(1969,2136,2158,2261,2149,2138,2142)(1976,2143,2145,2150,2267,2159,2272)(1986,2153,2273,2154,2161,2279,2274)(2007,2170,2172,2176,2182,2188,2195)(2010,2173,2183,2196,2175,2177,2189)(2015,2178,2197,2295,2190,2180,2184)(2022,2185,2186,2191,2299,2198,2303)(2049,2205,2207,2211,2216,2220,2224)(2052,2208,2217,2225,2210,2212,2221)(2057,2213,2226,2316,2222,2214,2218)(2084,2229,2231,2234,2237,2239,2241)(2087,2232,2238,2242,2233,2235,2240)(2108,2244,2245,2246,2247,2248,2249)(2129,2250,2252,2256,2262,2268,2275)(2132,2253,2263,2276,2255,2257,2269)(2137,2258,2277,2340,2270,2260,2264)(2144,2265,2266,2271,2344,2278,2348)(2171,2285,2287,2291,2296,2300,2304)(2174,2288,2297,2305,2290,2292,2301)(2179,2293,2306,2361,2302,2294,2298)(2206,2309,2311,2314,2317,2319,2321)(2209,2312,2318,2322,2313,2315,2320)(2230,2324,2325,2326,2327,2328,2329)(2251,2330,2332,2336,2341,2345,2349)(2254,2333,2342,2350,2335,2337,2346)(2259,2338,2351,2382,2347,2339,2343)(2286,2354,2356,2359,2362,2364,2366)(2289,2357,2363,2367,2358,2360,2365)(2310,2369,2370,2371,2372,2373,2374)(2331,2375,2377,2380,2383,2385,2387)(2334,2378,2384,2388,2379,2381,2386)(2355,2390,2391,2392,2393,2394,2395)(2376,2396,2397,2398,2399,2400,2401)Polycyclic presentation:
7^5|a^7=d,d^7=e,c^b=ce2695 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 65);