Finite Group DatabasebetaFinite Group Databasebeta| order | 60440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D30220 |
| Fitting | C4×C5×C1511 ⋊ C2 |
| Hall | C5×C1511 ⋊ D4 |
| Sylow | C1511 ⋊ D20 |
| Sylow | C5 ⋊ D6044 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7555 |
| commutator, socle, cosocle | C2×C5×C1511 | C22 |
| Fitting | C4×C5×C1511 | C2 |
| hypercenter | C4 | D7555 |
| hypocenter | C5×C1511 | D4 |
| LCS | G ▷ C2×C5×C1511 ▷ C5×C1511 | |
| UCS | C1 ◁ C2 ◁ C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1520 |
| transitive degree | μt(G) = 30220 |
A minimal faithful permutation representation — 2 generators on 1520 points:
(1,2,3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520);(2,4)(6,9)(7,8)(11,1520)(12,1519)(13,1518)(14,1517)(15,1516)(16,1515)(17,1514)(18,1513)(19,1512)(20,1511)(21,1510)(22,1509)(23,1508)(24,1507)(25,1506)(26,1505)(27,1504)(28,1503)(29,1502)(30,1501)(31,1500)(32,1499)(33,1498)(34,1497)(35,1496)(36,1495)(37,1494)(38,1493)(39,1492)(40,1491)(41,1490)(42,1489)(43,1488)(44,1487)(45,1486)(46,1485)(47,1484)(48,1483)(49,1482)(50,1481)(51,1480)(52,1479)(53,1478)(54,1477)(55,1476)(56,1475)(57,1474)(58,1473)(59,1472)(60,1471)(61,1470)(62,1469)(63,1468)(64,1467)(65,1466)(66,1465)(67,1464)(68,1463)(69,1462)(70,1461)(71,1460)(72,1459)(73,1458)(74,1457)(75,1456)(76,1455)(77,1454)(78,1453)(79,1452)(80,1451)(81,1450)(82,1449)(83,1448)(84,1447)(85,1446)(86,1445)(87,1444)(88,1443)(89,1442)(90,1441)(91,1440)(92,1439)(93,1438)(94,1437)(95,1436)(96,1435)(97,1434)(98,1433)(99,1432)(100,1431)(101,1430)(102,1429)(103,1428)(104,1427)(105,1426)(106,1425)(107,1424)(108,1423)(109,1422)(110,1421)(111,1420)(112,1419)(113,1418)(114,1417)(115,1416)(116,1415)(117,1414)(118,1413)(119,1412)(120,1411)(121,1410)(122,1409)(123,1408)(124,1407)(125,1406)(126,1405)(127,1404)(128,1403)(129,1402)(130,1401)(131,1400)(132,1399)(133,1398)(134,1397)(135,1396)(136,1395)(137,1394)(138,1393)(139,1392)(140,1391)(141,1390)(142,1389)(143,1388)(144,1387)(145,1386)(146,1385)(147,1384)(148,1383)(149,1382)(150,1381)(151,1380)(152,1379)(153,1378)(154,1377)(155,1376)(156,1375)(157,1374)(158,1373)(159,1372)(160,1371)(161,1370)(162,1369)(163,1368)(164,1367)(165,1366)(166,1365)(167,1364)(168,1363)(169,1362)(170,1361)(171,1360)(172,1359)(173,1358)(174,1357)(175,1356)(176,1355)(177,1354)(178,1353)(179,1352)(180,1351)(181,1350)(182,1349)(183,1348)(184,1347)(185,1346)(186,1345)(187,1344)(188,1343)(189,1342)(190,1341)(191,1340)(192,1339)(193,1338)(194,1337)(195,1336)(196,1335)(197,1334)(198,1333)(199,1332)(200,1331)(201,1330)(202,1329)(203,1328)(204,1327)(205,1326)(206,1325)(207,1324)(208,1323)(209,1322)(210,1321)(211,1320)(212,1319)(213,1318)(214,1317)(215,1316)(216,1315)(217,1314)(218,1313)(219,1312)(220,1311)(221,1310)(222,1309)(223,1308)(224,1307)(225,1306)(226,1305)(227,1304)(228,1303)(229,1302)(230,1301)(231,1300)(232,1299)(233,1298)(234,1297)(235,1296)(236,1295)(237,1294)(238,1293)(239,1292)(240,1291)(241,1290)(242,1289)(243,1288)(244,1287)(245,1286)(246,1285)(247,1284)(248,1283)(249,1282)(250,1281)(251,1280)(252,1279)(253,1278)(254,1277)(255,1276)(256,1275)(257,1274)(258,1273)(259,1272)(260,1271)(261,1270)(262,1269)(263,1268)(264,1267)(265,1266)(266,1265)(267,1264)(268,1263)(269,1262)(270,1261)(271,1260)(272,1259)(273,1258)(274,1257)(275,1256)(276,1255)(277,1254)(278,1253)(279,1252)(280,1251)(281,1250)(282,1249)(283,1248)(284,1247)(285,1246)(286,1245)(287,1244)(288,1243)(289,1242)(290,1241)(291,1240)(292,1239)(293,1238)(294,1237)(295,1236)(296,1235)(297,1234)(298,1233)(299,1232)(300,1231)(301,1230)(302,1229)(303,1228)(304,1227)(305,1226)(306,1225)(307,1224)(308,1223)(309,1222)(310,1221)(311,1220)(312,1219)(313,1218)(314,1217)(315,1216)(316,1215)(317,1214)(318,1213)(319,1212)(320,1211)(321,1210)(322,1209)(323,1208)(324,1207)(325,1206)(326,1205)(327,1204)(328,1203)(329,1202)(330,1201)(331,1200)(332,1199)(333,1198)(334,1197)(335,1196)(336,1195)(337,1194)(338,1193)(339,1192)(340,1191)(341,1190)(342,1189)(343,1188)(344,1187)(345,1186)(346,1185)(347,1184)(348,1183)(349,1182)(350,1181)(351,1180)(352,1179)(353,1178)(354,1177)(355,1176)(356,1175)(357,1174)(358,1173)(359,1172)(360,1171)(361,1170)(362,1169)(363,1168)(364,1167)(365,1166)(366,1165)(367,1164)(368,1163)(369,1162)(370,1161)(371,1160)(372,1159)(373,1158)(374,1157)(375,1156)(376,1155)(377,1154)(378,1153)(379,1152)(380,1151)(381,1150)(382,1149)(383,1148)(384,1147)(385,1146)(386,1145)(387,1144)(388,1143)(389,1142)(390,1141)(391,1140)(392,1139)(393,1138)(394,1137)(395,1136)(396,1135)(397,1134)(398,1133)(399,1132)(400,1131)(401,1130)(402,1129)(403,1128)(404,1127)(405,1126)(406,1125)(407,1124)(408,1123)(409,1122)(410,1121)(411,1120)(412,1119)(413,1118)(414,1117)(415,1116)(416,1115)(417,1114)(418,1113)(419,1112)(420,1111)(421,1110)(422,1109)(423,1108)(424,1107)(425,1106)(426,1105)(427,1104)(428,1103)(429,1102)(430,1101)(431,1100)(432,1099)(433,1098)(434,1097)(435,1096)(436,1095)(437,1094)(438,1093)(439,1092)(440,1091)(441,1090)(442,1089)(443,1088)(444,1087)(445,1086)(446,1085)(447,1084)(448,1083)(449,1082)(450,1081)(451,1080)(452,1079)(453,1078)(454,1077)(455,1076)(456,1075)(457,1074)(458,1073)(459,1072)(460,1071)(461,1070)(462,1069)(463,1068)(464,1067)(465,1066)(466,1065)(467,1064)(468,1063)(469,1062)(470,1061)(471,1060)(472,1059)(473,1058)(474,1057)(475,1056)(476,1055)(477,1054)(478,1053)(479,1052)(480,1051)(481,1050)(482,1049)(483,1048)(484,1047)(485,1046)(486,1045)(487,1044)(488,1043)(489,1042)(490,1041)(491,1040)(492,1039)(493,1038)(494,1037)(495,1036)(496,1035)(497,1034)(498,1033)(499,1032)(500,1031)(501,1030)(502,1029)(503,1028)(504,1027)(505,1026)(506,1025)(507,1024)(508,1023)(509,1022)(510,1021)(511,1020)(512,1019)(513,1018)(514,1017)(515,1016)(516,1015)(517,1014)(518,1013)(519,1012)(520,1011)(521,1010)(522,1009)(523,1008)(524,1007)(525,1006)(526,1005)(527,1004)(528,1003)(529,1002)(530,1001)(531,1000)(532,999)(533,998)(534,997)(535,996)(536,995)(537,994)(538,993)(539,992)(540,991)(541,990)(542,989)(543,988)(544,987)(545,986)(546,985)(547,984)(548,983)(549,982)(550,981)(551,980)(552,979)(553,978)(554,977)(555,976)(556,975)(557,974)(558,973)(559,972)(560,971)(561,970)(562,969)(563,968)(564,967)(565,966)(566,965)(567,964)(568,963)(569,962)(570,961)(571,960)(572,959)(573,958)(574,957)(575,956)(576,955)(577,954)(578,953)(579,952)(580,951)(581,950)(582,949)(583,948)(584,947)(585,946)(586,945)(587,944)(588,943)(589,942)(590,941)(591,940)(592,939)(593,938)(594,937)(595,936)(596,935)(597,934)(598,933)(599,932)(600,931)(601,930)(602,929)(603,928)(604,927)(605,926)(606,925)(607,924)(608,923)(609,922)(610,921)(611,920)(612,919)(613,918)(614,917)(615,916)(616,915)(617,914)(618,913)(619,912)(620,911)(621,910)(622,909)(623,908)(624,907)(625,906)(626,905)(627,904)(628,903)(629,902)(630,901)(631,900)(632,899)(633,898)(634,897)(635,896)(636,895)(637,894)(638,893)(639,892)(640,891)(641,890)(642,889)(643,888)(644,887)(645,886)(646,885)(647,884)(648,883)(649,882)(650,881)(651,880)(652,879)(653,878)(654,877)(655,876)(656,875)(657,874)(658,873)(659,872)(660,871)(661,870)(662,869)(663,868)(664,867)(665,866)(666,865)(667,864)(668,863)(669,862)(670,861)(671,860)(672,859)(673,858)(674,857)(675,856)(676,855)(677,854)(678,853)(679,852)(680,851)(681,850)(682,849)(683,848)(684,847)(685,846)(686,845)(687,844)(688,843)(689,842)(690,841)(691,840)(692,839)(693,838)(694,837)(695,836)(696,835)(697,834)(698,833)(699,832)(700,831)(701,830)(702,829)(703,828)(704,827)(705,826)(706,825)(707,824)(708,823)(709,822)(710,821)(711,820)(712,819)(713,818)(714,817)(715,816)(716,815)(717,814)(718,813)(719,812)(720,811)(721,810)(722,809)(723,808)(724,807)(725,806)(726,805)(727,804)(728,803)(729,802)(730,801)(731,800)(732,799)(733,798)(734,797)(735,796)(736,795)(737,794)(738,793)(739,792)(740,791)(741,790)(742,789)(743,788)(744,787)(745,786)(746,785)(747,784)(748,783)(749,782)(750,781)(751,780)(752,779)(753,778)(754,777)(755,776)(756,775)(757,774)(758,773)(759,772)(760,771)(761,770)(762,769)(763,768)(764,767)(765,766)Polycyclic presentation:
2^3,5,1511|b^2=c,b^a=bc,d^a=d^4,e^a=e^151015113 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 1511]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^1510);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);