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Aliases
Splittings
Subgroups
18 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 1534 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,330)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(290,332,376)(293,335,373)(296,338,370)(299,341,367)(302,344,364)(305,347,361)(308,350,358)(311,353,355)(314,356,352)(317,359,349)(320,362,346)(323,365,343)(326,368,340)(329,371,337)(331,374,334)(333,378,425)(336,381,422)(339,384,419)(342,387,416)(345,390,413)(348,393,410)(351,396,407)(354,399,404)(357,402,401)(360,405,398)(363,408,395)(366,411,392)(369,414,389)(372,417,386)(375,420,383)(377,423,380)(379,427,477)(382,430,474)(385,433,471)(388,436,468)(391,439,465)(394,442,462)(397,445,459)(400,448,456)(403,451,453)(406,454,450)(409,457,447)(412,460,444)(415,463,441)(418,466,438)(421,469,435)(424,472,432)(426,475,429)(428,479,532)(431,482,529)(434,485,526)(437,488,523)(440,491,520)(443,494,517)(446,497,514)(449,500,511)(452,503,508)(455,506,505)(458,509,502)(461,512,499)(464,515,496)(467,518,493)(470,521,490)(473,524,487)(476,527,484)(478,530,481)(480,534,590)(483,537,587)(486,540,584)(489,543,581)(492,546,578)(495,549,575)(498,552,572)(501,555,569)(504,558,566)(507,561,563)(510,564,560)(513,567,557)(516,570,554)(519,573,551)(522,576,548)(525,579,545)(528,582,542)(531,585,539)(533,588,536)(535,592,651)(538,595,648)(541,598,645)(544,601,642)(547,604,639)(550,607,636)(553,610,633)(556,613,630)(559,616,627)(562,619,624)(565,622,621)(568,625,618)(571,628,615)(574,631,612)(577,634,609)(580,637,606)(583,640,603)(586,643,600)(589,646,597)(591,649,594)(593,653,715)(596,656,712)(599,659,709)(602,662,706)(605,665,703)(608,668,700)(611,671,697)(614,674,694)(617,677,691)(620,680,688)(623,683,685)(626,686,682)(629,689,679)(632,692,676)(635,695,673)(638,698,670)(641,701,667)(644,704,664)(647,707,661)(650,710,658)(652,713,655)(654,717,782)(657,720,779)(660,723,776)(663,726,773)(666,729,770)(669,732,767)(672,735,764)(675,738,761)(678,741,758)(681,744,755)(684,747,752)(687,750,749)(690,753,746)(693,756,743)(696,759,740)(699,762,737)(702,765,734)(705,768,731)(708,771,728)(711,774,725)(714,777,722)(716,780,719)(718,784,851)(721,787,848)(724,790,846)(727,793,843)(730,796,840)(733,799,837)(736,802,834)(739,805,831)(742,808,828)(745,811,825)(748,814,822)(751,817,819)(754,820,816)(757,823,813)(760,826,810)(763,829,807)(766,832,804)(769,835,801)(772,838,798)(775,841,795)(778,844,792)(781,785,789)(783,849,786)(788,854,917)(791,857,915)(794,860,913)(797,863,910)(800,866,907)(803,869,904)(806,872,901)(809,875,898)(812,878,895)(815,881,892)(818,884,889)(821,887,886)(824,890,883)(827,893,880)(830,896,877)(833,899,874)(836,902,871)(839,905,868)(842,908,865)(845,911,862)(847,855,859)(850,916,856)(852,918,853)(858,921,979)(861,923,978)(864,926,976)(867,929,973)(870,932,970)(873,935,967)(876,938,964)(879,941,961)(882,944,958)(885,947,955)(888,950,952)(891,953,949)(894,956,946)(897,959,943)(900,962,940)(903,965,937)(906,968,934)(909,971,931)(912,974,928)(914,922,925)(919,980,920)(924,981,1036)(927,983,1035)(930,986,1033)(933,989,1030)(936,992,1027)(939,995,1024)(942,998,1021)(945,1001,1018)(948,1004,1015)(951,1007,1012)(954,1010,1009)(957,1013,1006)(960,1016,1003)(963,1019,1000)(966,1022,997)(969,1025,994)(972,1028,991)(975,1031,988)(977,982,985)(984,1037,1089)(987,1039,1088)(990,1042,1086)(993,1045,1083)(996,1048,1080)(999,1051,1077)(1002,1054,1074)(1005,1057,1071)(1008,1060,1068)(1011,1063,1065)(1014,1066,1062)(1017,1069,1059)(1020,1072,1056)(1023,1075,1053)(1026,1078,1050)(1029,1081,1047)(1032,1084,1044)(1034,1038,1041)(1040,1090,1139)(1043,1092,1138)(1046,1095,1136)(1049,1098,1133)(1052,1101,1130)(1055,1104,1127)(1058,1107,1124)(1061,1110,1121)(1064,1113,1118)(1067,1116,1115)(1070,1119,1112)(1073,1122,1109)(1076,1125,1106)(1079,1128,1103)(1082,1131,1100)(1085,1134,1097)(1087,1091,1094)(1093,1140,1186)(1096,1142,1185)(1099,1145,1183)(1102,1148,1180)(1105,1151,1177)(1108,1154,1174)(1111,1157,1171)(1114,1160,1168)(1117,1163,1165)(1120,1166,1162)(1123,1169,1159)(1126,1172,1156)(1129,1175,1153)(1132,1178,1150)(1135,1181,1147)(1137,1141,1144)(1143,1187,1230)(1146,1189,1229)(1149,1192,1227)(1152,1195,1224)(1155,1198,1221)(1158,1201,1218)(1161,1204,1215)(1164,1207,1212)(1167,1210,1209)(1170,1213,1206)(1173,1216,1203)(1176,1219,1200)(1179,1222,1197)(1182,1225,1194)(1184,1188,1191)(1190,1231,1271)(1193,1233,1270)(1196,1236,1268)(1199,1239,1265)(1202,1242,1262)(1205,1245,1259)(1208,1248,1256)(1211,1251,1253)(1214,1254,1250)(1217,1257,1247)(1220,1260,1244)(1223,1263,1241)(1226,1266,1238)(1228,1232,1235)(1234,1272,1309)(1237,1274,1308)(1240,1277,1306)(1243,1280,1303)(1246,1283,1300)(1249,1286,1297)(1252,1289,1294)(1255,1292,1291)(1258,1295,1288)(1261,1298,1285)(1264,1301,1282)(1267,1304,1279)(1269,1273,1276)(1275,1310,1344)(1278,1312,1343)(1281,1315,1341)(1284,1318,1338)(1287,1321,1335)(1290,1324,1332)(1293,1327,1329)(1296,1330,1326)(1299,1333,1323)(1302,1336,1320)(1305,1339,1317)(1307,1311,1314)(1313,1345,1376)(1316,1347,1375)(1319,1350,1373)(1322,1353,1370)(1325,1356,1367)(1328,1359,1364)(1331,1362,1361)(1334,1365,1358)(1337,1368,1355)(1340,1371,1352)(1342,1346,1349)(1348,1377,1405)(1351,1379,1404)(1354,1382,1402)(1357,1385,1399)(1360,1388,1396)(1363,1391,1393)(1366,1394,1390)(1369,1397,1387)(1372,1400,1384)(1374,1378,1381)(1380,1406,1431)(1383,1408,1430)(1386,1411,1428)(1389,1414,1425)(1392,1417,1422)(1395,1420,1419)(1398,1423,1416)(1401,1426,1413)(1403,1407,1410)(1409,1432,1454)(1412,1434,1453)(1415,1437,1451)(1418,1440,1448)(1421,1443,1445)(1424,1446,1442)(1427,1449,1439)(1429,1433,1436)(1435,1455,1474)(1438,1457,1473)(1441,1460,1471)(1444,1463,1468)(1447,1466,1465)(1450,1469,1462)(1452,1456,1459)(1458,1475,1491)(1461,1477,1490)(1464,1480,1488)(1467,1483,1485)(1470,1486,1482)(1472,1476,1479)(1478,1492,1505)(1481,1494,1504)(1484,1497,1502)(1487,1500,1499)(1489,1493,1496)(1495,1506,1516)(1498,1508,1515)(1501,1511,1513)(1503,1507,1510)(1509,1517,1524)(1512,1519,1523)(1514,1518,1521)(1520,1525,1529)(1522,1526,1528)(1527,1530,1531)(1533,1534);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,428,480,535,593,654,718,785,713,590,648,709,773,840,907,970,1027,1080,1130,1177,1221,1262,1300,1335,1367,1396,1422,1445,1465,1482,1496,1475,1434,1382,1318,1242,1154,1054,941,808,674,552,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,501,556,614,675,739,806,873,936,993,1046,1096,1143,1188,1134,1028,905,768,640,524,420,328,247,284,324,367,413,462,514,569,627,688,752,819,886,949,1006,1059,1109,1156,1200,1241,1279,1314,1272,1189,1095,989,866,729,601,485,381,289,331,375,421,470,522,577,635,696,760,827,894,957,1014,1067,1117,1164,1208,1249,1287,1322,1354,1383,1409,1433,1400,1336,1260,1172,1072,959,826,692,570,460,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,441,493,548,606,667,731,798,865,928,985,921,787,653,716,781,847,914,977,1034,1087,1137,1184,1228,1269,1307,1342,1374,1403,1429,1452,1472,1489,1503,1514,1522,1527,1530,1519,1497,1463,1417,1359,1289,1207,1113,1007,884,747,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,724,791,858,922,844,710,588,477,529,584,642,703,767,834,901,964,1021,1074,1124,1171,1215,1256,1294,1329,1361,1390,1416,1439,1459,1432,1379,1315,1239,1151,1051,938,805,671,549,439,341,255,181,119,146,175,206,240,277,317,360,406,455,507,562,620,681,745,812,879,942,999,1052,1102,1149,1193,1234,1273,1225,1131,1025,902,765,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,746,813,880,943,1000,1053,1103,1150,1194,1235,1187,1092,986,863,726,598,482,378,426,476,528,583,641,702,766,833,900,963,1020,1073,1123,1170,1214,1255,1293,1328,1360,1389,1415,1438,1458,1476,1449,1397,1333,1257,1169,1069,956,823,689,567,457,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,435,487,542,600,661,725,792,859,784,852,919,918,782,848,915,978,1035,1088,1138,1185,1229,1270,1308,1343,1375,1404,1430,1453,1473,1490,1504,1515,1523,1528,1517,1494,1460,1414,1356,1286,1204,1110,1004,881,744,616,500,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,440,492,547,605,666,730,797,864,927,984,1038,974,841,707,585,475,376,422,471,523,578,636,697,761,828,895,958,1015,1068,1118,1165,1209,1250,1288,1323,1355,1384,1410,1377,1312,1236,1148,1048,935,802,668,546,436,338,252,178,211,246,283,323,366,412,461,513,568,626,687,751,818,885,948,1005,1058,1108,1155,1199,1240,1278,1313,1346,1304,1222,1128,1022,899,762,634,518,414,322,242,174,117,142,170,201,235,272,312,355,401,450,502,557,615,676,740,807,874,937,994,1047,1097,1144,1090,983,860,723,595,479,533,589,647,708,772,839,906,969,1026,1079,1129,1176,1220,1261,1299,1334,1366,1395,1421,1444,1464,1481,1495,1507,1486,1446,1394,1330,1254,1166,1066,953,820,686,564,454,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,429,481,536,594,655,719,786,853,916,780,651,712,776,843,910,973,1030,1083,1133,1180,1224,1265,1303,1338,1370,1399,1425,1448,1468,1485,1499,1510,1492,1457,1411,1353,1283,1201,1107,1001,878,741,613,497,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,446,498,553,611,672,736,803,870,933,990,1043,1093,1141,1084,971,838,704,582,472,374,287,327,370,416,465,517,572,630,691,755,822,889,952,1009,1062,1112,1159,1203,1244,1282,1317,1349,1310,1233,1145,1045,932,799,665,543,433,335,249,288,329,372,418,467,519,574,632,693,757,824,891,954,1011,1064,1114,1161,1205,1246,1284,1319,1351,1380,1407,1371,1301,1219,1125,1019,896,759,631,515,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,444,496,551,609,670,734,801,868,931,988,1041,981,857,720,592,652,714,778,845,912,975,1032,1085,1135,1182,1226,1267,1305,1340,1372,1401,1427,1450,1470,1487,1501,1512,1520,1526,1511,1483,1443,1391,1327,1251,1163,1063,950,817,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,721,788,855,777,649,532,587,645,706,770,837,904,967,1024,1077,1127,1174,1218,1259,1297,1332,1364,1393,1419,1442,1462,1479,1455,1408,1350,1280,1198,1104,998,875,738,610,494,390,298,218,150,94,118,144,172,203,237,274,314,357,403,452,504,559,617,678,742,809,876,939,996,1049,1099,1146,1190,1232,1181,1081,968,835,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,749,816,883,946,1003,1056,1106,1153,1197,1238,1276,1231,1142,1042,929,796,662,540,430,332,377,424,473,525,580,638,699,763,830,897,960,1017,1070,1120,1167,1211,1252,1290,1325,1357,1386,1412,1435,1456,1426,1368,1298,1216,1122,1016,893,756,628,512,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,728,795,862,925,854,717,783,850,851,917,979,1036,1089,1139,1186,1230,1271,1309,1344,1376,1405,1431,1454,1474,1491,1505,1516,1524,1529,1531,1525,1508,1480,1440,1388,1324,1248,1160,1060,947,814,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,727,794,861,924,982,911,774,646,530,425,474,526,581,639,700,764,831,898,961,1018,1071,1121,1168,1212,1253,1291,1326,1358,1387,1413,1436,1406,1347,1277,1195,1101,995,872,735,607,491,387,295,215,147,177,209,243,280,320,363,409,458,510,565,623,684,748,815,882,945,1002,1055,1105,1152,1196,1237,1275,1311,1266,1178,1078,965,832,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,743,810,877,940,997,1050,1100,1147,1191,1140,1039,926,793,659,537,427,478,531,586,644,705,769,836,903,966,1023,1076,1126,1173,1217,1258,1296,1331,1363,1392,1418,1441,1461,1478,1493,1469,1423,1365,1295,1213,1119,1013,890,753,625,509,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,539,597,658,722,789,856,920,980,849,715,779,846,913,976,1033,1086,1136,1183,1227,1268,1306,1341,1373,1402,1428,1451,1471,1488,1502,1513,1521,1506,1477,1437,1385,1321,1245,1157,1057,944,811,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,733,800,867,930,987,1040,1091,1031,908,771,643,527,423,330,373,419,468,520,575,633,694,758,825,892,955,1012,1065,1115,1162,1206,1247,1285,1320,1352,1381,1345,1274,1192,1098,992,869,732,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,754,821,888,951,1008,1061,1111,1158,1202,1243,1281,1316,1348,1378,1339,1263,1175,1075,962,829,695,573,463,365,279,205,143,92,114,139,167,198,232,269,309,352,398,447,499,554,612,673,737,804,871,934,991,1044,1094,1037,923,790,656,534,591,650,711,775,842,909,972,1029,1082,1132,1179,1223,1264,1302,1337,1369,1398,1424,1447,1467,1484,1498,1509,1518,1500,1466,1420,1362,1292,1210,1116,1010,887,750,622,506,402,310,230,162,106,62,30,10)(1532,1533,1534)
Conjugacy classes & character table
1539 conjugacy classes. The character table is too large to display.
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27558, 12);
Finite Group Databasebeta