Dic₄₉₇₁ #34090994

order	19884
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₉₇₁
unambiguous semidirect product	C₃⋊Dic₁₆₅₇

Splittings

derived, Hall	C₃×C₁₆₅₇ ⋊ C₄
Sylow	C₁₆₅₇ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₆₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₉₇₁
commutator, hypocenter	C₃×C₁₆₅₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₆₅₇	C₂
LCS	G ▷ C₃×C₁₆₅₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1664
transitive degree	μ_t(G) = 19884

A minimal faithful permutation representation — 2 generators on 1664 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,1664);(1,3,2,4)(6,7)(9,1664)(10,1663)(11,1662)(12,1661)(13,1660)(14,1659)(15,1658)(16,1657)(17,1656)(18,1655)(19,1654)(20,1653)(21,1652)(22,1651)(23,1650)(24,1649)(25,1648)(26,1647)(27,1646)(28,1645)(29,1644)(30,1643)(31,1642)(32,1641)(33,1640)(34,1639)(35,1638)(36,1637)(37,1636)(38,1635)(39,1634)(40,1633)(41,1632)(42,1631)(43,1630)(44,1629)(45,1628)(46,1627)(47,1626)(48,1625)(49,1624)(50,1623)(51,1622)(52,1621)(53,1620)(54,1619)(55,1618)(56,1617)(57,1616)(58,1615)(59,1614)(60,1613)(61,1612)(62,1611)(63,1610)(64,1609)(65,1608)(66,1607)(67,1606)(68,1605)(69,1604)(70,1603)(71,1602)(72,1601)(73,1600)(74,1599)(75,1598)(76,1597)(77,1596)(78,1595)(79,1594)(80,1593)(81,1592)(82,1591)(83,1590)(84,1589)(85,1588)(86,1587)(87,1586)(88,1585)(89,1584)(90,1583)(91,1582)(92,1581)(93,1580)(94,1579)(95,1578)(96,1577)(97,1576)(98,1575)(99,1574)(100,1573)(101,1572)(102,1571)(103,1570)(104,1569)(105,1568)(106,1567)(107,1566)(108,1565)(109,1564)(110,1563)(111,1562)(112,1561)(113,1560)(114,1559)(115,1558)(116,1557)(117,1556)(118,1555)(119,1554)(120,1553)(121,1552)(122,1551)(123,1550)(124,1549)(125,1548)(126,1547)(127,1546)(128,1545)(129,1544)(130,1543)(131,1542)(132,1541)(133,1540)(134,1539)(135,1538)(136,1537)(137,1536)(138,1535)(139,1534)(140,1533)(141,1532)(142,1531)(143,1530)(144,1529)(145,1528)(146,1527)(147,1526)(148,1525)(149,1524)(150,1523)(151,1522)(152,1521)(153,1520)(154,1519)(155,1518)(156,1517)(157,1516)(158,1515)(159,1514)(160,1513)(161,1512)(162,1511)(163,1510)(164,1509)(165,1508)(166,1507)(167,1506)(168,1505)(169,1504)(170,1503)(171,1502)(172,1501)(173,1500)(174,1499)(175,1498)(176,1497)(177,1496)(178,1495)(179,1494)(180,1493)(181,1492)(182,1491)(183,1490)(184,1489)(185,1488)(186,1487)(187,1486)(188,1485)(189,1484)(190,1483)(191,1482)(192,1481)(193,1480)(194,1479)(195,1478)(196,1477)(197,1476)(198,1475)(199,1474)(200,1473)(201,1472)(202,1471)(203,1470)(204,1469)(205,1468)(206,1467)(207,1466)(208,1465)(209,1464)(210,1463)(211,1462)(212,1461)(213,1460)(214,1459)(215,1458)(216,1457)(217,1456)(218,1455)(219,1454)(220,1453)(221,1452)(222,1451)(223,1450)(224,1449)(225,1448)(226,1447)(227,1446)(228,1445)(229,1444)(230,1443)(231,1442)(232,1441)(233,1440)(234,1439)(235,1438)(236,1437)(237,1436)(238,1435)(239,1434)(240,1433)(241,1432)(242,1431)(243,1430)(244,1429)(245,1428)(246,1427)(247,1426)(248,1425)(249,1424)(250,1423)(251,1422)(252,1421)(253,1420)(254,1419)(255,1418)(256,1417)(257,1416)(258,1415)(259,1414)(260,1413)(261,1412)(262,1411)(263,1410)(264,1409)(265,1408)(266,1407)(267,1406)(268,1405)(269,1404)(270,1403)(271,1402)(272,1401)(273,1400)(274,1399)(275,1398)(276,1397)(277,1396)(278,1395)(279,1394)(280,1393)(281,1392)(282,1391)(283,1390)(284,1389)(285,1388)(286,1387)(287,1386)(288,1385)(289,1384)(290,1383)(291,1382)(292,1381)(293,1380)(294,1379)(295,1378)(296,1377)(297,1376)(298,1375)(299,1374)(300,1373)(301,1372)(302,1371)(303,1370)(304,1369)(305,1368)(306,1367)(307,1366)(308,1365)(309,1364)(310,1363)(311,1362)(312,1361)(313,1360)(314,1359)(315,1358)(316,1357)(317,1356)(318,1355)(319,1354)(320,1353)(321,1352)(322,1351)(323,1350)(324,1349)(325,1348)(326,1347)(327,1346)(328,1345)(329,1344)(330,1343)(331,1342)(332,1341)(333,1340)(334,1339)(335,1338)(336,1337)(337,1336)(338,1335)(339,1334)(340,1333)(341,1332)(342,1331)(343,1330)(344,1329)(345,1328)(346,1327)(347,1326)(348,1325)(349,1324)(350,1323)(351,1322)(352,1321)(353,1320)(354,1319)(355,1318)(356,1317)(357,1316)(358,1315)(359,1314)(360,1313)(361,1312)(362,1311)(363,1310)(364,1309)(365,1308)(366,1307)(367,1306)(368,1305)(369,1304)(370,1303)(371,1302)(372,1301)(373,1300)(374,1299)(375,1298)(376,1297)(377,1296)(378,1295)(379,1294)(380,1293)(381,1292)(382,1291)(383,1290)(384,1289)(385,1288)(386,1287)(387,1286)(388,1285)(389,1284)(390,1283)(391,1282)(392,1281)(393,1280)(394,1279)(395,1278)(396,1277)(397,1276)(398,1275)(399,1274)(400,1273)(401,1272)(402,1271)(403,1270)(404,1269)(405,1268)(406,1267)(407,1266)(408,1265)(409,1264)(410,1263)(411,1262)(412,1261)(413,1260)(414,1259)(415,1258)(416,1257)(417,1256)(418,1255)(419,1254)(420,1253)(421,1252)(422,1251)(423,1250)(424,1249)(425,1248)(426,1247)(427,1246)(428,1245)(429,1244)(430,1243)(431,1242)(432,1241)(433,1240)(434,1239)(435,1238)(436,1237)(437,1236)(438,1235)(439,1234)(440,1233)(441,1232)(442,1231)(443,1230)(444,1229)(445,1228)(446,1227)(447,1226)(448,1225)(449,1224)(450,1223)(451,1222)(452,1221)(453,1220)(454,1219)(455,1218)(456,1217)(457,1216)(458,1215)(459,1214)(460,1213)(461,1212)(462,1211)(463,1210)(464,1209)(465,1208)(466,1207)(467,1206)(468,1205)(469,1204)(470,1203)(471,1202)(472,1201)(473,1200)(474,1199)(475,1198)(476,1197)(477,1196)(478,1195)(479,1194)(480,1193)(481,1192)(482,1191)(483,1190)(484,1189)(485,1188)(486,1187)(487,1186)(488,1185)(489,1184)(490,1183)(491,1182)(492,1181)(493,1180)(494,1179)(495,1178)(496,1177)(497,1176)(498,1175)(499,1174)(500,1173)(501,1172)(502,1171)(503,1170)(504,1169)(505,1168)(506,1167)(507,1166)(508,1165)(509,1164)(510,1163)(511,1162)(512,1161)(513,1160)(514,1159)(515,1158)(516,1157)(517,1156)(518,1155)(519,1154)(520,1153)(521,1152)(522,1151)(523,1150)(524,1149)(525,1148)(526,1147)(527,1146)(528,1145)(529,1144)(530,1143)(531,1142)(532,1141)(533,1140)(534,1139)(535,1138)(536,1137)(537,1136)(538,1135)(539,1134)(540,1133)(541,1132)(542,1131)(543,1130)(544,1129)(545,1128)(546,1127)(547,1126)(548,1125)(549,1124)(550,1123)(551,1122)(552,1121)(553,1120)(554,1119)(555,1118)(556,1117)(557,1116)(558,1115)(559,1114)(560,1113)(561,1112)(562,1111)(563,1110)(564,1109)(565,1108)(566,1107)(567,1106)(568,1105)(569,1104)(570,1103)(571,1102)(572,1101)(573,1100)(574,1099)(575,1098)(576,1097)(577,1096)(578,1095)(579,1094)(580,1093)(581,1092)(582,1091)(583,1090)(584,1089)(585,1088)(586,1087)(587,1086)(588,1085)(589,1084)(590,1083)(591,1082)(592,1081)(593,1080)(594,1079)(595,1078)(596,1077)(597,1076)(598,1075)(599,1074)(600,1073)(601,1072)(602,1071)(603,1070)(604,1069)(605,1068)(606,1067)(607,1066)(608,1065)(609,1064)(610,1063)(611,1062)(612,1061)(613,1060)(614,1059)(615,1058)(616,1057)(617,1056)(618,1055)(619,1054)(620,1053)(621,1052)(622,1051)(623,1050)(624,1049)(625,1048)(626,1047)(627,1046)(628,1045)(629,1044)(630,1043)(631,1042)(632,1041)(633,1040)(634,1039)(635,1038)(636,1037)(637,1036)(638,1035)(639,1034)(640,1033)(641,1032)(642,1031)(643,1030)(644,1029)(645,1028)(646,1027)(647,1026)(648,1025)(649,1024)(650,1023)(651,1022)(652,1021)(653,1020)(654,1019)(655,1018)(656,1017)(657,1016)(658,1015)(659,1014)(660,1013)(661,1012)(662,1011)(663,1010)(664,1009)(665,1008)(666,1007)(667,1006)(668,1005)(669,1004)(670,1003)(671,1002)(672,1001)(673,1000)(674,999)(675,998)(676,997)(677,996)(678,995)(679,994)(680,993)(681,992)(682,991)(683,990)(684,989)(685,988)(686,987)(687,986)(688,985)(689,984)(690,983)(691,982)(692,981)(693,980)(694,979)(695,978)(696,977)(697,976)(698,975)(699,974)(700,973)(701,972)(702,971)(703,970)(704,969)(705,968)(706,967)(707,966)(708,965)(709,964)(710,963)(711,962)(712,961)(713,960)(714,959)(715,958)(716,957)(717,956)(718,955)(719,954)(720,953)(721,952)(722,951)(723,950)(724,949)(725,948)(726,947)(727,946)(728,945)(729,944)(730,943)(731,942)(732,941)(733,940)(734,939)(735,938)(736,937)(737,936)(738,935)(739,934)(740,933)(741,932)(742,931)(743,930)(744,929)(745,928)(746,927)(747,926)(748,925)(749,924)(750,923)(751,922)(752,921)(753,920)(754,919)(755,918)(756,917)(757,916)(758,915)(759,914)(760,913)(761,912)(762,911)(763,910)(764,909)(765,908)(766,907)(767,906)(768,905)(769,904)(770,903)(771,902)(772,901)(773,900)(774,899)(775,898)(776,897)(777,896)(778,895)(779,894)(780,893)(781,892)(782,891)(783,890)(784,889)(785,888)(786,887)(787,886)(788,885)(789,884)(790,883)(791,882)(792,881)(793,880)(794,879)(795,878)(796,877)(797,876)(798,875)(799,874)(800,873)(801,872)(802,871)(803,870)(804,869)(805,868)(806,867)(807,866)(808,865)(809,864)(810,863)(811,862)(812,861)(813,860)(814,859)(815,858)(816,857)(817,856)(818,855)(819,854)(820,853)(821,852)(822,851)(823,850)(824,849)(825,848)(826,847)(827,846)(828,845)(829,844)(830,843)(831,842)(832,841)(833,840)(834,839)(835,838)(836,837)

Polycyclic presentation:

2,3,1657,2|a^2=d,b^a=b^2,c^a=c^1656

Conjugacy classes & character table

4974 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₉₄₂, Dic₁₄₉₁₃
maximal normal subgroup of	Dic₉₉₄₂
maximal quotient of	Dic₁₄₉₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19884, 6);

Dic4971 #34090994