Finite Group Databasebeta

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#27166117

• metacyclic
• supersolvable
• metabelian
• solvable
• monolithic
• monomial
• stem
• Frobenius

Aliases

No aliases for this group.

Splittings

Subgroups

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1609 points:

(2,3,4,7,5,8,10,16)(6,12,13,28,14,29,31,63)(9,19,20,45,21,46,48,37)(11,24,25,56,26,57,23,54)(15,33,34,75,35,76,78,157)(17,38,39,85,40,18,42,43)(22,50,51,107,52,108,110,132)(27,59,60,125,61,126,70,148)(30,66,67,139,68,140,142,84)(32,71,72,150,73,151,55,117)(36,80,81,169,82,170,172,316)(41,87,88,182,89,65,136,137)(44,93,94,64,95,133,191,134)(47,98,99,199,100,200,102,116)(49,103,104,207,105,120,91,186)(53,112,113,222,114,223,225,292)(58,121,122,188,123,97,197,92)(62,128,129,243,130,244,164,307)(69,144,145,270,146,271,273,250)(74,153,154,285,155,286,149,277)(77,160,161,298,162,299,301,176)(79,165,166,309,167,310,118,230)(83,174,175,189,127,241,242,346)(86,178,179,119,180,232,177,233)(90,184,185,341,147,159,275,296)(96,193,194,158,195,293,354,294)(101,202,203,368,204,369,217,276)(106,208,209,377,210,240,256,427)(109,213,214,383,215,384,265,228)(111,218,219,391,220,280,187,344)(115,212,227,257,124,237,238,422)(131,246,247,434,248,435,322,554)(135,252,253,444,254,138,258,259)(141,261,262,407,263,457,206,229)(143,266,267,464,268,235,190,348)(152,281,282,351,283,198,361,192)(156,288,289,497,290,498,308,528)(163,303,304,521,305,522,524,440)(168,312,313,536,314,537,278,480)(171,318,319,496,320,546,548,329)(173,323,324,556,325,557,231,410)(181,332,333,239,334,424,350,425)(183,337,338,279,339,482,330,483)(196,356,357,317,358,543,602,544)(201,364,365,585,366,363,360,343)(205,371,372,614,373,615,400,527)(211,378,379,622,380,431,477,680)(216,386,387,631,388,632,473,478)(221,392,393,635,394,492,450,703)(224,396,397,490,398,640,516,406)(226,401,402,647,403,531,345,587)(234,413,414,336,415,251,441,442)(236,418,419,376,420,412,331,569)(245,432,433,589,359,511,604,347)(249,437,438,657,439,688,429,675)(255,446,447,700,448,297,508,509)(260,453,454,295,455,505,708,506)(264,459,460,717,461,718,390,479)(269,465,466,725,467,428,451,673)(272,469,470,449,471,702,463,408)(274,474,381,623,475,485,349,592)(284,487,488,706,489,382,624,452)(287,493,494,594,495,327,563,352)(291,500,501,766,502,767,555,845)(300,512,513,654,514,784,375,409)(302,517,518,790,519,417,353,596)(306,504,526,772,416,659,660,581)(311,532,533,599,534,362,607,355)(315,539,540,822,541,823,529,806)(321,550,551,821,552,840,842,690)(326,559,560,852,561,853,481,743)(328,565,370,613,566,862,411,626)(335,571,572,430,573,677,598,678)(340,577,578,491,579,755,593,756)(342,582,389,530,583,808,568,809)(367,610,611,847,588,612,405,652)(374,617,618,745,570,867,423,666)(385,627,628,886,629,712,605,586)(395,636,637,933,638,761,739,1036)(399,642,643,815,644,940,799,740)(404,648,649,944,650,817,704,1007)(421,662,663,621,664,668,691,920)(426,669,670,695,671,443,692,693)(436,684,685,888,686,832,985,590)(445,696,697,484,698,746,575,747)(456,710,711,545,687,829,987,830)(458,713,714,890,715,609,606,591)(462,720,721,1021,722,1022,646,805)(468,726,727,1025,728,681,779,977)(472,731,732,778,733,1030,735,741)(476,736,639,934,737,758,705,1008)(486,750,751,620,752,658,576,874)(499,762,763,894,764,783,1064,595)(503,769,770,963,771,1072,676,970)(507,775,776,1079,777,510,780,781)(515,786,787,1038,788,1091,634,742)(520,791,792,1098,793,674,707,968)(523,795,796,951,797,1103,724,655)(525,800,801,1109,802,749,597,897)(535,812,813,1011,814,625,924,709)(538,818,819,900,820,564,860,600)(542,825,826,1135,827,1136,846,1161)(547,833,834,953,835,1144,619,656)(549,765,837,1066,838,661,601,902)(553,774,844,1077,748,1042,1043,881)(558,848,849,905,850,608,911,603)(562,855,856,1169,857,1170,807,1117)(567,864,633,930,865,1179,744,1006)(574,869,870,683,871,982,904,983)(580,877,878,760,879,1060,899,1061)(584,883,734,816,884,1126,893,1127)(616,915,916,1178,912,935,907,863)(630,927,928,1166,929,1018,653,949)(641,936,937,1194,938,1086,908,887)(645,811,942,1122,919,1123,957,882)(651,945,946,1232,947,1130,1037,1304)(665,958,959,922,898,972,804,1114)(667,960,723,932,961,1053,991,1223)(672,965,966,1001,952,773,1074,1075)(679,973,974,1010,975,694,992,993)(682,979,872,1097,980,955,868,1183)(689,988,956,876,989,759,1058,889)(699,997,998,757,999,1054,995,1055)(701,1002,1003,810,1004,1119,873,1120)(716,1015,1016,1246,1017,914,923,967)(719,1019,1020,1195,918,926,859,891)(729,1026,1027,1014,909,892,730,1028)(738,1033,948,1233,1034,1063,1081,1324)(753,1045,1046,921,1047,964,996,1216)(754,1049,1050,917,1051,1041,875,1185)(768,1068,1069,1198,1070,1143,1335,895)(782,1083,1084,831,990,1141,1240,1142)(785,1087,1088,1200,1089,913,910,896)(789,1093,1094,1357,1095,1358,943,1116)(794,1099,1100,1362,1101,978,1082,1257)(798,1105,1106,1256,1107,1366,1032,1039)(803,1110,1111,1368,1112,1057,1009,1289)(824,1131,1132,1204,1133,1085,1350,901)(828,1138,1139,1251,1140,1390,971,1249)(836,1146,1147,1305,1148,1396,931,1040)(839,1071,1150,1337,1151,969,1012,1248)(841,1153,1154,1236,1155,1400,1024,954)(843,1134,1157,1384,1158,1044,903,1207)(851,1163,1164,1292,1165,925,1125,1013)(854,1121,1167,1210,1168,861,1080,906)(858,1172,1173,1415,1174,1416,1162,1411)(866,1180,941,1230,1181,1419,1118,1364)(880,1187,1188,1067,1189,1332,1209,1333)(885,1191,1108,1129,1192,1380,1203,1381)(939,1227,1228,1412,1229,1354,950,1234)(962,1241,1096,1225,1242,1320,1115,1370)(976,1253,1254,1346,1255,1076,1340,1341)(981,1259,1190,1361,1260,1243,1271,1450)(984,1262,1263,1270,1264,994,1272,1273)(986,1266,1267,1331,1268,1238,1184,1421)(1000,1277,1278,1062,1279,1321,1290,1322)(1005,1283,1284,1128,1285,1377,1274,1378)(1023,1295,1296,1428,1297,1226,1326,1196)(1029,1300,1035,1303,1235,1294,1219,1288)(1031,1301,1102,1363,1222,1287,1176,1197)(1048,1307,1308,1218,1208,1252,1160,1409)(1052,1309,1310,1224,1311,1313,1275,1444)(1056,1314,1315,1281,1237,1078,1343,1344)(1059,1317,1199,1073,1318,1244,1276,1338)(1065,1327,1328,1360,1329,1239,1186,1422)(1090,1352,1353,1454,1182,1215,1220,1247)(1092,1355,1356,1429,1299,1221,1177,1201)(1104,1365,1250,1431,1175,1293,1212,1202)(1113,1369,1231,1371,1149,1325,1348,1498)(1124,1372,1373,1217,1374,1245,1282,1420)(1137,1386,1387,1433,1388,1392,1541,1205)(1145,1298,1393,1435,1394,1214,1213,1206)(1152,1339,1398,1453,1399,1258,1349,1459)(1156,1402,1403,1487,1404,1549,1302,1306)(1159,1389,1406,1543,1407,1316,1291,1483)(1171,1375,1413,1439,1414,1351,1458,1211)(1193,1425,1405,1385,1426,1538,1438,1539)(1261,1460,1427,1529,1461,1456,1513,1567)(1265,1463,1464,1449,1424,1342,1383,1514)(1269,1465,1466,1512,1467,1451,1470,1563)(1280,1472,1473,1334,1474,1505,1469,1506)(1286,1477,1478,1382,1479,1534,1482,1535)(1312,1489,1490,1446,1484,1495,1410,1553)(1319,1492,1434,1391,1493,1457,1515,1544)(1323,1496,1497,1519,1488,1345,1516,1517)(1330,1500,1501,1528,1502,1452,1471,1564)(1336,1508,1509,1504,1510,1448,1423,1557)(1347,1436,1397,1379,1520,1533,1475,1443)(1359,1524,1525,1554,1526,1447,1523,1430)(1367,1530,1462,1559,1486,1481,1418,1432)(1376,1455,1395,1445,1531,1491,1476,1442)(1401,1527,1494,1560,1547,1485,1441,1437)(1408,1545,1551,1570,1552,1499,1522,1581)(1417,1532,1555,1548,1556,1546,1580,1440)(1468,1571,1572,1566,1573,1568,1537,1598)(1480,1576,1577,1540,1578,1599,1574,1600)(1503,1582,1583,1596,1584,1569,1593,1601)(1507,1586,1587,1562,1558,1518,1542,1594)(1511,1588,1589,1592,1590,1565,1575,1603)(1521,1561,1550,1536,1585,1597,1595,1579)(1591,1607,1608,1604,1606,1605,1602,1609);(1,2,6,15,36,83,126,240,428,674,969,1070,767,1008,1288,1146,1397,1344,1075,509,397,161,302,520,794,1102,1020,494,759,1059,1319,1494,1393,849,642,941,856,540,289,129,34,79,168,315,542,828,966,1245,1453,1566,1604,1608,1587,1541,1489,1277,1471,1508,1327,1058,750,441,691,991,722,461,263,140,174,327,564,861,1120,809,230,409,655,952,671,425,148,276,478,740,814,534,310,531,622,922,1218,1414,1170,1412,944,720,1023,1298,1087,713,364,98,201,367,402,219,104,39,86,181,335,574,872,677,972,1251,1456,1568,1564,1450,968,1247,1153,1401,1548,1439,1210,905,602,904,1209,1438,1404,1155,840,710,1012,1291,1484,1311,1051,624,923,1219,1034,737,475,273,473,735,1032,1302,1486,1366,1117,1371,1425,1558,1506,1333,845,960,577,876,888,589,346,188,85,177,330,568,388,215,108,212,382,625,925,1185,867,344,586,864,1003,697,442,137,257,169,317,545,831,1126,1021,497,760,1062,1323,1487,1305,852,731,1031,1147,787,460,203,51,111,221,395,639,466,253,445,699,1000,1280,1466,1263,1207,1437,1561,1577,1473,1273,711,1013,844,1160,1410,1554,1429,1200,790,1097,1360,1463,1253,965,659,193,353,597,898,664,420,197,360,605,908,1017,715,457,712,1014,1293,1485,1526,1358,1304,945,539,824,1134,800,474,218,71,152,284,490,383,199,56,119,234,416,305,162,76,159,297,510,782,989,688,348,591,891,1196,1310,1046,746,1041,685,986,1269,1468,1534,1525,1132,1383,1537,1578,1479,1285,1004,508,779,1081,947,650,403,225,400,646,943,1231,1100,776,1080,1347,1521,1584,1502,1329,1064,1326,1283,1399,1151,838,548,706,444,695,995,1274,1363,1026,623,246,436,687,455,252,87,12,30,69,147,89,40,16,37,84,176,329,495,286,492,758,1057,1316,1388,1136,1324,1227,786,1092,1154,796,470,214,67,143,269,468,729,714,282,486,753,1048,1157,837,543,303,523,798,1108,878,572,194,355,600,901,1205,1434,1315,1491,1337,1512,1592,1607,1582,1492,1307,992,1271,1189,879,579,339,136,256,450,704,929,629,384,626,926,1221,1443,1378,1127,480,742,1039,1255,975,678,307,527,582,882,1165,850,557,847,377,621,921,1217,1419,1194,647,371,616,848,532,281,121,19,47,101,205,374,419,232,412,657,955,1238,1328,1060,1320,1135,1385,1540,1601,1567,1257,1230,795,1104,1174,857,561,325,172,322,555,846,1162,1353,813,1125,1376,1532,1386,1068,684,318,547,836,1149,1112,802,524,799,1079,1346,1519,1595,1549,1411,1368,1187,1423,1465,1259,970,662,332,439,248,130,61,26,10,23,55,118,231,411,369,612,914,1215,1442,1444,1223,703,949,559,854,1077,772,296,222,75,158,295,507,778,717,285,491,757,1056,1236,953,556,386,633,560,313,154,60,13,32,74,156,291,503,670,964,1244,1452,1565,1589,1505,1483,1555,1516,1593,1594,1514,1084,942,526,804,1115,1095,788,514,299,511,783,1085,1351,1374,1123,607,910,1212,1139,770,438,175,328,567,866,1182,1089,784,1086,1025,1138,1391,1545,1369,1033,636,288,499,765,517,266,103,24,58,124,114,52,21,8,18,44,96,196,359,244,431,681,978,1258,1338,1072,673,967,833,1145,1395,1373,1119,916,433,682,981,1261,1462,1356,819,1083,1349,1522,1524,1295,915,396,641,939,946,637,379,185,342,584,885,1193,1427,1254,1458,1570,1576,1606,1590,1510,1335,1309,997,1276,1198,894,594,351,191,350,593,893,733,471,271,80,171,321,553,747,483,117,229,408,448,254,134,63,132,250,440,690,820,537,817,933,1225,1446,1556,1416,1300,927,459,719,834,513,262,99,20,49,106,211,381,267,133,251,443,694,994,1267,982,897,1202,1432,1478,1278,993,454,709,551,843,1159,1408,1533,1435,1066,1331,1504,1586,1496,1314,1042,453,707,1009,1242,961,618,227,405,653,950,1235,1027,702,1006,1287,1481,1579,1535,1381,806,1116,1191,1424,1264,983,554,666,337,576,875,1178,862,585,207,376,620,919,644,398,223,112,224,399,645,874,569,186,343,565,863,1050,751,482,745,434,683,984,1265,1380,1357,822,1129,1382,1536,1530,1301,930,469,730,1029,1228,928,611,238,423,667,962,1111,792,505,774,1078,1345,1518,1557,1421,902,1206,1436,1552,1407,1158,842,1011,708,1010,1290,1482,1559,1431,1109,869,1184,1262,973,669,413,93,190,349,380,210,105,48,102,206,375,619,918,718,1018,1294,1172,1417,1553,1370,1036,648,312,538,821,521,270,107,28,64,135,255,449,407,150,279,484,748,552,320,170,144,272,472,734,578,333,94,192,352,595,895,1199,1054,1313,1069,1336,1511,1591,1599,1581,1413,1340,1513,1426,1192,884,583,275,477,739,1037,1229,938,640,935,1226,1447,1551,1398,1141,860,1177,1418,1461,1260,980,604,907,1002,1282,1476,1394,1144,1246,725,769,1073,1339,1099,726,378,128,245,358,195,95,38,3,9,22,53,115,123,57,120,235,417,661,764,498,761,1063,1325,1499,1544,1390,977,936,512,785,1090,1181,865,566,242,429,676,971,1250,1088,533,811,1124,1375,1131,762,432,160,300,515,789,1096,959,660,957,1240,1449,1562,1583,1497,1580,1543,1472,1575,1588,1500,1317,1045,692,771,502,290,155,73,31,70,149,278,481,744,632,410,656,954,1237,1055,756,277,479,741,777,506,294,157,292,341,581,881,1168,853,1166,635,932,1224,1445,1352,1015,610,202,370,324,166,72,25,4,11,27,62,131,249,424,668,963,1243,1451,1509,1332,1289,1415,1402,1550,1517,1341,781,643,304,525,803,1113,1148,835,546,832,1143,1392,1546,1531,1164,924,1220,1173,826,501,247,81,173,326,562,858,1175,1103,1364,1362,1529,1596,1600,1539,1161,1241,877,1186,1266,979,675,418,178,331,570,373,204,100,46,97,198,362,608,912,615,587,887,1180,1420,1216,920,427,652,323,558,851,1122,808,614,243,430,679,976,1256,1038,536,816,1128,1379,1355,1019,613,213,385,630,649,393,209,88,183,340,580,880,1190,974,1252,1457,1569,1605,1603,1563,1248,1455,1343,1515,1433,1204,900,599,354,598,899,1203,1107,797,522,356,601,903,1208,1047,752,361,606,909,728,467,268,142,265,463,724,1024,1299,1091,1354,1233,825,1137,1389,1110,736,392,153,287,496,298,139,45,7,17,41,90,146,68,29,65,138,260,456,686,435,592,892,1197,1284,998,693,259,452,319,549,839,1152,1377,1296,763,1065,1330,1503,1585,1520,1167,780,1082,1348,1094,721,372,113,226,404,651,948,727,447,701,1005,1286,1480,1572,1464,1350,1523,1477,1573,1467,1268,985,1049,696,996,1275,1428,1195,890,464,437,689,990,775,446,184,33,77,163,306,415,233,54,116,228,406,489,283,151,280,485,749,1044,1133,823,1130,1232,1093,1359,1527,1365,1028,627,261,458,716,937,628,365,122,236,421,665,801,518,293,504,773,1076,1342,1422,1183,596,896,1201,1430,1490,1308,1043,1292,987,1270,1469,1574,1597,1560,1384,1272,1470,1474,1279,999,698,258,451,705,638,394,220,110,217,390,634,931,1222,1030,743,1040,1306,1488,1322,1061,528,805,883,1142,830,544,316,422,182,336,575,873,1179,886,391,617,917,1163,812,487,237,50,109,216,389,338,179,43,92,189,347,590,889,755,1053,766,1067,1334,1507,1538,1498,1169,1105,1367,1403,1106,732,387,145,274,476,738,1035,1016,488,754,1052,1312,1406,1150,829,550,841,1156,1405,1188,870,357,603,906,1211,1440,1547,1400,1454,1098,1361,1528,1571,1460,1249,958,571,868,871,573,334,180,42,91,187,345,588,366,200,363,609,913,1214,1297,1022,1007,1234,855,1171,1409,1114,680,401,165,311,535,815,631,368,125,239,426,672,951,654,309,530,810,1121,818,493,241,66,141,264,462,723,663,414,658,956,1239,1448,1501,1321,1495,1387,1542,1602,1609,1598,1459,1372,1074,1140,827,541,314,167,78,164,308,529,807,1118,940,911,1213,1441,1493,1318,988,563,859,1176,1101,793,519,301,516,700,1001,1281,1475,1396,1303,934,500,768,1071,791,465,208,59,127,82,35,14,5)

Polycyclic presentation:

2^3,1609|a^2=b,b^2=c,d^a=d^979,d^b=d^1086,d^c=d^1608

Conjugacy classes & character table

209 conjugacy classes — view the full character table.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12872, 15);