Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C49 · Schur multiplier (H2) = C7 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C49 | H(3,𝔽7) |
| commutator | C72 | C7×C49 |
| Frattini, cosocle | C7×C49 | C72 |
| socle | C7 | C72⋊C49 |
| prime radical | H(3,𝔽7) | C49 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C49 ◁ C7×C49 ◁ G | |
| LECS | G ▷ C7×C49 ▷ C7 ▷ C1 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 2401 |
| transitive degree | μt(G) = 2401 |
A minimal faithful permutation representation — 2 generators on 2401 points:
(1,2,6,19,55,131,249,4,9,22,58,134,252,427,16,32,68,143,261,433,657,51,90,163,280,447,666,925,126,196,312,475,684,937,1214,243,357,518,722,961,1230,1501,418,575,775,1007,1260,1521,1762,5,10,23,59,135,253,426,17,33,69,144,262,434,656,52,91,164,281,448,667,924,127,197,313,476,685,938,1213,244,358,519,723,962,1231,1500,419,576,776,1008,1261,1522,1761,648,842,1068,1312,1556,1785,1986,18,34,70,145,263,251,430,53,92,165,282,449,432,663,128,198,314,477,686,665,934,245,359,520,724,963,936,1227,420,577,777,1009,1262,1229,1518,649,843,1069,1313,1557,1520,1782,916,1135,1374,1608,1819,1784,2005,54,93,66,160,133,258,442,129,199,161,309,260,443,678,246,360,310,515,446,679,954,421,578,516,772,683,955,1253,650,844,773,1065,960,1254,1549,917,1136,1066,1372,1259,1550,1813,1205,1434,1373,1663,1555,1814,2031,130,87,82,57,140,274,468,247,193,187,142,275,469,715,422,354,347,279,470,716,1000,651,572,564,474,717,1001,1305,918,839,830,721,1002,1306,1602,1206,1133,1124,1006,1307,1603,1862,1492,1433,1426,1311,1604,1863,2071,248,30,21,64,157,305,510,423,88,67,158,306,511,767,652,194,162,307,512,768,1060,919,355,311,513,769,1061,1367,1207,573,517,770,1062,1368,1660,1493,840,774,1063,1369,1661,1910,1753,1134,1067,1370,1662,1911,2105,424,8,28,80,182,341,557,653,31,83,183,342,558,823,920,89,188,343,559,824,1117,1208,195,348,560,825,1118,1419,1494,356,565,826,1119,1420,1703,1754,574,831,1120,1421,1704,1944,1979,841,1125,1422,1705,1945,2128)(3,11,35,94,132,254,435,14,38,96,200,256,437,669,48,102,202,361,440,672,941,122,210,363,579,676,946,1234,239,375,581,845,952,1242,1525,414,597,847,1137,1251,1536,1788,644,866,1139,1435,1547,1801,2009,15,29,84,148,257,431,662,49,86,189,290,441,664,933,123,192,349,490,677,935,1226,240,353,566,743,953,1228,1517,415,571,832,1032,1252,1519,1781,645,838,1126,1339,1548,1783,2004,913,1132,1427,1635,1812,2006,2168,50,7,24,56,136,264,450,124,26,72,138,267,452,688,241,77,169,272,458,692,966,416,179,321,466,702,972,1266,646,338,530,713,984,1273,1562,914,554,791,998,1287,1570,1825,1203,820,1085,1303,1585,1833,2040,125,27,65,139,259,444,680,242,78,159,273,445,681,957,417,180,308,467,682,958,1256,647,339,514,714,959,1257,1552,915,555,771,999,1258,1553,1816,1204,821,1064,1304,1554,1817,2033,1491,1115,1371,1601,1818,2034,2191,119,79,81,184,266,455,697,236,181,186,344,457,699,979,411,340,346,561,701,981,1282,641,556,563,827,983,1284,1580,910,822,829,1121,1286,1582,1843,1201,1116,1123,1423,1584,1845,2056,1490,1418,1425,1706,1847,2058,2210,46,37,20,60,146,283,478,120,101,62,149,285,480,725,237,209,155,291,483,729,1010,412,374,303,491,734,1015,1314,642,596,508,744,1021,1320,1609,911,865,765,1033,1327,1616,1866,1202,1158,1058,1340,1623,1873,2072,13,44,63,141,276,250,425,47,115,156,278,471,429,655,121,231,304,473,718,661,923,238,405,509,720,1003,932,1212,413,634,766,1005,1308,1225,1499,643,903,1059,1310,1605,1516,1760,912,1194,1366,1607,1864,1780,1985)(12,39,95,137,268,459,687,42,104,201,270,461,704,965,112,213,362,464,707,987,1265,228,379,580,711,991,1290,1561,402,602,846,996,1295,1588,1824,631,871,1138,1301,1593,1850,2039,900,1163,1436,1599,1855,2062,2196,43,100,152,271,456,698,956,113,208,300,465,700,980,1255,229,373,505,712,982,1283,1551,403,595,762,997,1285,1581,1815,632,864,1055,1302,1583,1844,2032,901,1157,1363,1600,1846,2057,2190,1192,1453,1658,1861,2059,2211,2296,114,74,71,166,284,479,726,230,176,168,316,482,728,1011,404,335,320,523,733,1014,1315,633,551,529,782,1020,1319,1610,902,817,790,1076,1326,1615,1867,1193,1112,1084,1381,1622,1872,2073,1484,1416,1389,1670,1879,2078,2220,110,36,97,147,286,428,658,226,99,204,288,485,660,927,400,207,367,488,737,931,1217,629,372,587,741,1025,1224,1505,898,594,855,1030,1332,1515,1767,1190,863,1148,1337,1628,1779,1991,1483,1156,1445,1633,1884,2003,2156,117,85,185,277,255,436,654,234,191,345,472,439,671,922,409,352,562,719,675,945,1211,639,570,828,1004,951,1241,1498,908,837,1122,1309,1250,1535,1759,1199,1131,1424,1606,1546,1800,1984,1489,1432,1707,1865,1811,2021,2151,41,25,61,150,292,492,668,111,75,153,294,494,745,940,227,177,301,497,748,1034,1233,401,336,506,752,1038,1341,1524,630,552,763,1043,1346,1636,1787,899,818,1056,1351,1641,1890,2008,1191,1113,1364,1646,1895,2091,2170,45,76,154,289,265,451,689,118,178,302,489,454,691,968,235,337,507,742,696,971,1268,410,553,764,1031,978,1272,1564,640,819,1057,1338,1281,1569,1827,909,1114,1365,1634,1579,1832,2042,1200,1417,1659,1889,1842,2047,2198)(40,98,151,295,453,693,926,107,206,297,499,695,974,1216,220,371,502,755,977,1275,1504,393,593,759,1047,1280,1572,1766,621,862,1052,1355,1578,1835,1990,890,1155,1360,1650,1841,2049,2155,1182,1452,1655,1903,2055,2204,2267,108,190,298,269,460,690,921,221,351,503,463,706,970,1210,394,569,760,710,990,1271,1497,622,836,1053,995,1294,1568,1758,891,1130,1361,1300,1592,1831,1983,1183,1431,1656,1598,1854,2046,2150,1476,1711,1908,1860,2066,2202,2263,222,73,170,322,531,703,939,395,173,324,533,792,986,1232,623,331,536,795,1086,1289,1523,892,546,799,1090,1390,1587,1786,1184,811,1094,1394,1678,1849,2007,1477,1106,13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presentation:
7^5|a^7=d,d^7=e,b^a=bc,c^a=ce,c^b=ce679 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 26);