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Aliases
Splittings
Subgroups
6 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 1480 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)(718,784,852)(721,787,849)(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)(781,847,789)(783,850,786)(785,854,925)(788,857,922)(791,860,919)(794,863,916)(797,866,913)(800,869,910)(803,872,907)(806,875,904)(809,878,901)(812,881,898)(815,884,895)(818,887,892)(821,890,889)(824,893,886)(827,896,883)(830,899,880)(833,902,877)(836,905,874)(839,908,871)(842,911,868)(845,914,865)(848,917,862)(851,920,859)(853,923,856)(855,927,975)(858,929,997)(861,932,994)(864,935,991)(867,938,988)(870,941,985)(873,944,982)(876,947,979)(879,950,976)(882,953,973)(885,956,970)(888,959,967)(891,962,964)(894,965,961)(897,968,958)(900,971,955)(903,974,952)(906,977,949)(909,980,946)(912,983,943)(915,986,940)(918,989,937)(921,992,934)(924,995,931)(926,998,928)(930,1002,1045)(933,1004,1067)(936,1007,1064)(939,1010,1061)(942,1013,1058)(945,1016,1055)(948,1019,1052)(951,1022,1049)(954,1025,1046)(957,1027,1044)(960,1030,1041)(963,1033,1038)(966,1036,1035)(969,1039,1032)(972,1042,1029)(978,1047,1024)(981,1050,1021)(984,1053,1018)(987,1056,1015)(990,1059,1012)(993,1062,1009)(996,1065,1006)(999,1068,1003)(1000,1048,1001)(1005,1072,1111)(1008,1074,1131)(1011,1077,1128)(1014,1080,1125)(1017,1083,1122)(1020,1086,1119)(1023,1089,1116)(1026,1092,1113)(1028,1094,1112)(1031,1096,1110)(1034,1099,1107)(1037,1102,1104)(1040,1105,1101)(1043,1108,1098)(1051,1114,1091)(1054,1117,1088)(1057,1120,1085)(1060,1123,1082)(1063,1126,1079)(1066,1129,1076)(1069,1132,1073)(1070,1115,1071)(1075,1136,1171)(1078,1138,1189)(1081,1141,1186)(1084,1144,1183)(1087,1147,1180)(1090,1150,1177)(1093,1153,1174)(1095,1155,1173)(1097,1157,1172)(1100,1159,1170)(1103,1162,1167)(1106,1165,1164)(1109,1168,1161)(1118,1175,1152)(1121,1178,1149)(1124,1181,1146)(1127,1184,1143)(1130,1187,1140)(1133,1190,1137)(1134,1176,1135)(1139,1194,1225)(1142,1196,1241)(1145,1199,1238)(1148,1202,1235)(1151,1205,1232)(1154,1208,1229)(1156,1210,1228)(1158,1212,1227)(1160,1214,1226)(1163,1216,1224)(1166,1219,1221)(1169,1222,1218)(1179,1230,1207)(1182,1233,1204)(1185,1236,1201)(1188,1239,1198)(1191,1242,1195)(1192,1231,1193)(1197,1246,1273)(1200,1248,1287)(1203,1251,1284)(1206,1254,1281)(1209,1257,1278)(1211,1259,1277)(1213,1261,1276)(1215,1263,1275)(1217,1265,1274)(1220,1267,1272)(1223,1270,1269)(1234,1279,1256)(1237,1282,1253)(1240,1285,1250)(1243,1288,1247)(1244,1280,1245)(1249,1292,1315)(1252,1294,1327)(1255,1297,1324)(1258,1300,1321)(1260,1302,1320)(1262,1304,1319)(1264,1306,1318)(1266,1308,1317)(1268,1310,1316)(1271,1312,1314)(1283,1322,1299)(1286,1325,1296)(1289,1328,1293)(1290,1323,1291)(1295,1332,1351)(1298,1334,1361)(1301,1337,1358)(1303,1339,1357)(1305,1341,1356)(1307,1343,1355)(1309,1345,1354)(1311,1347,1353)(1313,1349,1352)(1326,1359,1336)(1329,1362,1333)(1330,1360,1331)(1335,1366,1382)(1338,1368,1389)(1340,1370,1388)(1342,1372,1387)(1344,1374,1386)(1346,1376,1385)(1348,1378,1384)(1350,1380,1383)(1363,1390,1367)(1364,1391,1365)(1369,1394,1407)(1371,1395,1413)(1373,1397,1412)(1375,1399,1411)(1377,1401,1410)(1379,1403,1409)(1381,1405,1408)(1392,1414,1393)(1396,1415,1426)(1398,1416,1431)(1400,1418,1430)(1402,1420,1429)(1404,1422,1428)(1406,1424,1427)(1417,1432,1441)(1419,1433,1445)(1421,1435,1444)(1423,1437,1443)(1425,1439,1442)(1434,1446,1453)(1436,1447,1456)(1438,1449,1455)(1440,1451,1454)(1448,1457,1462)(1450,1458,1464)(1452,1460,1463)(1459,1465,1468)(1461,1466,1469)(1467,1470,1471)(1472,1473,1475,1474,1476,1478,1477,1479,1480);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,428,480,535,593,654,718,785,855,899,759,631,515,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,444,496,551,609,670,734,801,871,943,1015,1082,1143,1198,1247,1291,1279,1178,1053,908,768,640,524,420,328,247,284,324,367,413,462,514,569,627,688,752,819,889,961,1032,1098,1072,929,784,853,924,996,1066,1130,1188,1240,1286,1326,1360,1288,1187,1062,917,777,649,532,587,645,706,770,837,907,979,1049,1113,1173,1227,1275,1317,1353,1383,1366,1294,1199,1080,938,793,659,537,427,478,531,586,644,705,769,836,906,978,1048,925,997,1067,1131,1189,1241,1287,1327,1361,1389,1413,1431,1445,1456,1464,1469,1465,1447,1418,1372,1302,1208,1089,947,802,668,546,436,338,252,178,211,246,283,323,366,412,461,513,568,626,687,751,818,888,960,1031,1097,1158,1213,1262,1305,1342,1373,1398,1417,1424,1378,1308,1214,1096,956,811,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,733,800,870,942,1014,1081,1142,1197,1222,1105,965,820,686,564,454,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,429,481,536,594,655,719,786,856,928,1001,974,829,695,573,463,365,279,205,143,92,114,139,167,198,232,269,309,352,398,447,499,554,612,673,737,804,874,946,1018,1085,1146,1201,1250,1293,1331,1322,1233,1120,983,838,704,582,472,374,287,327,370,416,465,517,572,630,691,755,822,892,964,1035,1101,1161,1136,1004,857,717,783,851,921,993,1063,1127,1185,1237,1283,1323,1242,1129,992,847,713,590,648,709,773,840,910,982,1052,1116,1174,1228,1276,1318,1354,1384,1408,1394,1334,1251,1144,1013,866,726,598,482,378,426,476,528,583,641,702,766,833,903,975,1045,1111,1171,1225,1273,1315,1351,1382,1407,1426,1441,1453,1462,1468,1471,1470,1458,1435,1399,1341,1259,1153,1022,875,735,607,491,387,295,215,147,177,209,243,280,320,363,409,458,510,565,623,684,748,815,885,957,1028,1095,1156,1211,1260,1303,1340,1371,1396,1405,1347,1265,1159,1030,884,744,616,500,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,440,492,547,605,666,730,797,867,939,1011,1078,1139,1168,1039,893,753,625,509,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,539,597,658,722,789,859,931,1003,1071,1047,902,762,634,518,414,322,242,174,117,142,170,201,235,272,312,355,401,450,502,557,615,676,740,807,877,949,1021,1088,1149,1204,1253,1296,1333,1365,1359,1282,1181,1056,911,771,643,527,423,330,373,419,468,520,575,633,694,758,825,895,967,1038,1104,1164,1218,1194,1074,932,787,653,716,781,848,918,990,1060,1124,1182,1234,1280,1190,1065,920,780,651,712,776,843,913,985,1055,1119,1177,1229,1277,1319,1355,1385,1409,1427,1415,1368,1297,1202,1083,941,796,662,540,430,332,377,424,473,525,580,638,699,763,830,900,972,1043,1109,1169,1223,1271,1313,1350,1381,1406,1425,1440,1452,1461,1467,1466,1449,1420,1374,1304,1210,1092,950,805,671,549,439,341,255,181,119,146,175,206,240,277,317,360,406,455,507,562,620,681,745,812,882,954,1026,1093,1154,1209,1258,1301,1338,1369,1380,1310,1216,1099,959,814,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,727,794,864,936,1008,1075,1108,968,823,689,567,457,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,435,487,542,600,661,725,792,862,934,1006,1073,1135,1114,977,832,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,743,810,880,952,1024,1091,1152,1207,1256,1299,1336,1367,1393,1390,1325,1236,1123,986,841,707,585,475,376,422,471,523,578,636,697,761,828,898,970,1041,1107,1167,1221,1269,1246,1138,1007,860,720,592,652,714,778,845,915,987,1057,1121,1179,1231,1132,995,850,715,779,846,916,988,1058,1122,1180,1232,1278,1320,1356,1386,1410,1428,1442,1432,1395,1337,1254,1147,1016,869,729,601,485,381,289,331,375,421,470,522,577,635,696,760,827,897,969,1040,1106,1166,1220,1268,1311,1348,1379,1404,1423,1438,1450,1459,1460,1437,1401,1343,1261,1155,1025,878,738,610,494,390,298,218,150,94,118,144,172,203,237,274,314,357,403,452,504,559,617,678,742,809,879,951,1023,1090,1151,1206,1255,1298,1335,1349,1267,1162,1033,887,747,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,724,791,861,933,1005,1042,896,756,628,512,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,728,795,865,937,1009,1076,1137,1193,1175,1050,905,765,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,746,813,883,955,927,1000,1070,1134,1192,1244,1290,1330,1364,1392,1414,1362,1285,1184,1059,914,774,646,530,425,474,526,581,639,700,764,831,901,973,1044,1110,1170,1224,1272,1314,1292,1196,1077,935,790,656,534,591,650,711,775,842,912,984,1054,1118,1176,1068,923,782,849,919,991,1061,1125,1183,1235,1281,1321,1357,1387,1411,1429,1443,1454,1446,1416,1370,1300,1205,1086,944,799,665,543,433,335,249,288,329,372,418,467,519,574,632,693,757,824,894,966,1037,1103,1163,1217,1266,1309,1346,1377,1402,1421,1436,1448,1451,1422,1376,1306,1212,1094,953,808,674,552,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,501,556,614,675,739,806,876,948,1020,1087,1148,1203,1252,1295,1312,1219,1102,962,817,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,721,788,858,930,971,826,692,570,460,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,441,493,548,606,667,731,798,868,940,1012,1079,1140,1195,1245,1230,1117,980,835,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,749,816,886,958,1029,1002,854,926,999,1069,1133,1191,1243,1289,1329,1363,1391,1328,1239,1126,989,844,710,588,477,529,584,642,703,767,834,904,976,1046,1112,1172,1226,1274,1316,1352,1332,1248,1141,1010,863,723,595,479,533,589,647,708,772,839,909,981,1051,1115,998,852,922,994,1064,1128,1186,1238,1284,1324,1358,1388,1412,1430,1444,1455,1463,1457,1433,1397,1339,1257,1150,1019,872,732,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,754,821,891,963,1034,1100,1160,1215,1264,1307,1344,1375,1400,1419,1434,1439,1403,1345,1263,1157,1027,881,741,613,497,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,446,498,553,611,672,736,803,873,945,1017,1084,1145,1200,1249,1270,1165,1036,890,750,622,506,402,310,230,162,106,62,30,10)
Polycyclic presentation:
3^2,1471|a^3=b,c^a=c^1219
Conjugacy classes & character table
1479 conjugacy classes. The character table is too large to display.
Relationships
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13239, 1);
Finite Group Databasebeta