Finite Group Databasebeta

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#43270035

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monolithic
• monomial
• stem

Aliases

No aliases for this group.

Subgroups

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 2197 points:

(1,2,5,13,31,68,137,244,357,477,610,756,915)(3,9,23,53,111,210,326,449,582,728,887,1059,1087)(4,8,16,34,71,140,247,360,480,613,759,918,1088)(6,17,41,90,180,296,425,560,707,867,916,1089,11)(7,20,49,106,203,323,448,581,727,886,1058,1242,1086)(10,26,59,122,160,280,403,538,685,848,932,1114,1268)(12,22,40,77,146,252,365,484,618,764,923,917,1091)(14,35,78,159,277,400,535,684,757,919,1092,66,48)(15,38,85,173,294,424,559,706,866,1040,1226,914,21)(18,45,99,192,309,433,567,714,873,1045,1090,1270,65)(19,44,93,149,138,248,358,481,619,771,941,1127,29)(24,56,116,215,329,386,521,668,781,958,1146,1269,28)(25,43,97,161,273,398,519,667,779,955,1112,1299,1267)(27,62,127,182,268,391,500,646,789,937,1120,1290,1428)(30,52,89,158,263,251,370,496,643,806,758,922,1098)(32,72,147,264,385,517,611,760,924,1099,355,322,172)(33,75,154,272,397,534,683,846,1024,1211,755,51,50)(36,81,166,287,412,548,696,856,920,1096,1277,474,321)(37,80,162,279,373,501,635,797,921,1095,1273,135,105)(39,87,70,144,260,382,514,663,828,1008,1196,609,109)(42,94,184,249,367,482,622,774,947,1093,1274,241,104)(46,101,195,288,414,522,669,830,964,1115,1271,1431,240)(47,54,113,148,267,362,488,630,766,934,1116,1272,134)(55,92,143,256,374,498,616,767,933,1113,1297,1085,58)(57,119,164,285,404,507,652,784,961,1106,1291,1430,133)(60,124,224,301,376,503,644,775,945,1132,1300,1429,64)(61,118,73,151,254,375,497,620,761,928,1105,1289,1427)(63,130,233,305,415,542,636,793,973,1147,1306,1448,1568)(67,110,179,157,276,261,383,515,664,612,763,930,1109)(69,141,253,371,478,614,765,931,1110,913,121,98,112)(74,150,266,387,520,621,762,927,1101,1281,475,447,214)(76,156,275,246,364,490,634,796,978,1166,356,324,88)(79,163,282,406,541,689,847,925,1102,1284,912,340,223)(82,168,269,379,508,651,773,935,1097,1279,1439,1083,460)(83,91,142,257,378,506,615,768,936,1100,1282,607,446)(84,107,205,255,377,505,642,805,943,1094,1276,242,202)(86,175,139,250,369,495,641,804,986,1175,476,208,108)(95,187,304,402,502,645,786,959,1135,1278,1437,752,445)(96,186,284,278,372,499,626,783,942,1128,1316,354,201)(100,152,270,390,491,631,790,929,1107,1275,1434,473,200)(102,197,314,428,540,654,815,965,1148,1319,1432,1572,472)(103,117,216,258,380,510,657,769,938,1122,1302,1433,353)(114,212,283,368,493,623,776,950,1117,1303,1436,606,320)(115,190,204,297,363,487,627,782,926,1104,1288,754,222)(120,220,302,408,504,649,807,951,1136,1307,1452,1571,352)(123,206,181,298,401,536,686,849,957,1111,1298,1266,126)(125,226,291,394,526,647,809,970,1118,1287,1449,1570,239)(128,230,310,381,512,638,788,939,1124,1312,1451,1569,132)(129,225,167,289,416,492,629,787,966,1103,1286,1447,1567)(131,236,307,431,543,656,819,994,1139,1320,1468,1584,1694)(136,209,178,176,145,262,384,516,479,617,770,940,1126)(153,174,295,245,361,485,625,780,956,1145,608,337,191)(155,274,399,359,483,624,778,954,1144,243,325,207,177)(165,211,327,366,486,628,785,963,1150,1296,1084,228,218)(169,292,409,524,658,821,962,1138,1280,1441,1583,1693,235)(170,185,281,389,523,666,829,948,1119,1285,1445,1265,342)(171,183,259,265,388,518,665,772,944,1130,1317,753,458)(188,306,392,527,653,816,969,1123,1310,1440,1581,1425,462)(189,193,299,413,539,687,850,1025,1129,1283,1443,911,580)(194,286,410,494,639,777,952,1140,1293,1438,1578,1082,579)(196,300,393,511,633,794,946,1121,1301,1435,1575,751,319)(198,316,421,545,650,812,987,1141,1324,1450,1573,1699,750)(199,219,290,418,549,670,831,1009,1153,1292,1453,1574,605)(213,312,417,537,648,808,967,1151,1304,1444,1580,1264,592)(217,331,405,525,637,791,971,1133,1309,1461,1577,910,444)(221,335,396,532,673,810,989,1158,1295,1457,1593,1698,604)(227,338,430,529,660,801,968,1134,1311,1464,1591,1697,471)(229,303,330,407,489,632,792,960,1131,1318,1459,1426,232)(231,293,422,552,690,799,974,1160,1334,1442,1585,1696,351)(234,343,411,546,671,795,976,1152,1326,1456,1590,1695,238)(237,348,451,568,677,825,991,1177,1321,1471,1608,1716,1806)(271,395,509,655,814,949,1108,1294,1446,1588,1566,345,308)(311,419,513,661,811,988,1125,1314,1455,1582,1708,1692,464)(313,427,544,672,800,972,1149,1322,1460,1579,1705,1424,726)(315,434,528,675,822,1001,1156,1313,1470,1576,1702,1081,443)(317,440,564,691,834,990,1178,1333,1475,1605,1700,1812,1080)(318,334,429,551,688,818,995,1154,1332,1465,1594,1701,909)(328,426,530,674,817,996,1157,1308,1462,1589,1710,1805,347)(332,437,550,640,802,953,1142,1328,1467,1587,1707,1565,594)(333,420,531,659,798,975,1137,1305,1454,1595,1704,1263,578)(336,423,557,679,824,983,1155,1323,1466,1586,1713,1811,908)(339,456,553,676,837,981,1161,1338,1472,1598,1718,1810,749)(341,432,565,697,832,979,1163,1336,1483,1609,1712,1809,603)(344,450,561,692,820,999,1183,1327,1477,1603,1717,1808,470)(346,435,554,680,835,997,1185,1340,1463,1597,1711,1807,350)(349,467,571,695,854,1010,1171,1341,1491,1622,1726,1822,1904)(436,562,698,803,984,1143,1330,1485,1602,1709,1821,1903,466)(438,533,681,823,980,1162,1325,1458,1600,1706,1818,1691,739)(439,556,662,826,998,1184,1315,1473,1592,1703,1815,1423,577)(441,574,700,843,1003,1172,1337,1480,1604,1724,1813,1911,1422)(442,454,547,6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Polycyclic presentation:

13^4|b^13=d,b^a=bc,c^a=cd

Conjugacy classes & character table

337 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28561, 9);