Dic₅₁₉₉ #35209485

order	20796
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁₉₉
unambiguous semidirect product	C₃⋊Dic₁₇₃₃

Splittings

derived, Hall	C₃×C₁₇₃₃ ⋊ C₄
Sylow	C₁₇₃₃ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₇₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁₉₉
commutator, hypocenter	C₃×C₁₇₃₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₇₃₃	C₂
LCS	G ▷ C₃×C₁₇₃₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1740
transitive degree	μ_t(G) = 20796

A minimal faithful permutation representation — 2 generators on 1740 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,1711,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,1729,1730,1731,1732,1733,1734,1735,1736,1737,1738,1739,1740);(1,3,2,4)(6,7)(9,1740)(10,1739)(11,1738)(12,1737)(13,1736)(14,1735)(15,1734)(16,1733)(17,1732)(18,1731)(19,1730)(20,1729)(21,1728)(22,1727)(23,1726)(24,1725)(25,1724)(26,1723)(27,1722)(28,1721)(29,1720)(30,1719)(31,1718)(32,1717)(33,1716)(34,1715)(35,1714)(36,1713)(37,1712)(38,1711)(39,1710)(40,1709)(41,1708)(42,1707)(43,1706)(44,1705)(45,1704)(46,1703)(47,1702)(48,1701)(49,1700)(50,1699)(51,1698)(52,1697)(53,1696)(54,1695)(55,1694)(56,1693)(57,1692)(58,1691)(59,1690)(60,1689)(61,1688)(62,1687)(63,1686)(64,1685)(65,1684)(66,1683)(67,1682)(68,1681)(69,1680)(70,1679)(71,1678)(72,1677)(73,1676)(74,1675)(75,1674)(76,1673)(77,1672)(78,1671)(79,1670)(80,1669)(81,1668)(82,1667)(83,1666)(84,1665)(85,1664)(86,1663)(87,1662)(88,1661)(89,1660)(90,1659)(91,1658)(92,1657)(93,1656)(94,1655)(95,1654)(96,1653)(97,1652)(98,1651)(99,1650)(100,1649)(101,1648)(102,1647)(103,1646)(104,1645)(105,1644)(106,1643)(107,1642)(108,1641)(109,1640)(110,1639)(111,1638)(112,1637)(113,1636)(114,1635)(115,1634)(116,1633)(117,1632)(118,1631)(119,1630)(120,1629)(121,1628)(122,1627)(123,1626)(124,1625)(125,1624)(126,1623)(127,1622)(128,1621)(129,1620)(130,1619)(131,1618)(132,1617)(133,1616)(134,1615)(135,1614)(136,1613)(137,1612)(138,1611)(139,1610)(140,1609)(141,1608)(142,1607)(143,1606)(144,1605)(145,1604)(146,1603)(147,1602)(148,1601)(149,1600)(150,1599)(151,1598)(152,1597)(153,1596)(154,1595)(155,1594)(156,1593)(157,1592)(158,1591)(159,1590)(160,1589)(161,1588)(162,1587)(163,1586)(164,1585)(165,1584)(166,1583)(167,1582)(168,1581)(169,1580)(170,1579)(171,1578)(172,1577)(173,1576)(174,1575)(175,1574)(176,1573)(177,1572)(178,1571)(179,1570)(180,1569)(181,1568)(182,1567)(183,1566)(184,1565)(185,1564)(186,1563)(187,1562)(188,1561)(189,1560)(190,1559)(191,1558)(192,1557)(193,1556)(194,1555)(195,1554)(196,1553)(197,1552)(198,1551)(199,1550)(200,1549)(201,1548)(202,1547)(203,1546)(204,1545)(205,1544)(206,1543)(207,1542)(208,1541)(209,1540)(210,1539)(211,1538)(212,1537)(213,1536)(214,1535)(215,1534)(216,1533)(217,1532)(218,1531)(219,1530)(220,1529)(221,1528)(222,1527)(223,1526)(224,1525)(225,1524)(226,1523)(227,1522)(228,1521)(229,1520)(230,1519)(231,1518)(232,1517)(233,1516)(234,1515)(235,1514)(236,1513)(237,1512)(238,1511)(239,1510)(240,1509)(241,1508)(242,1507)(243,1506)(244,1505)(245,1504)(246,1503)(247,1502)(248,1501)(249,1500)(250,1499)(251,1498)(252,1497)(253,1496)(254,1495)(255,1494)(256,1493)(257,1492)(258,1491)(259,1490)(260,1489)(261,1488)(262,1487)(263,1486)(264,1485)(265,1484)(266,1483)(267,1482)(268,1481)(269,1480)(270,1479)(271,1478)(272,1477)(273,1476)(274,1475)(275,1474)(276,1473)(277,1472)(278,1471)(279,1470)(280,1469)(281,1468)(282,1467)(283,1466)(284,1465)(285,1464)(286,1463)(287,1462)(288,1461)(289,1460)(290,1459)(291,1458)(292,1457)(293,1456)(294,1455)(295,1454)(296,1453)(297,1452)(298,1451)(299,1450)(300,1449)(301,1448)(302,1447)(303,1446)(304,1445)(305,1444)(306,1443)(307,1442)(308,1441)(309,1440)(310,1439)(311,1438)(312,1437)(313,1436)(314,1435)(315,1434)(316,1433)(317,1432)(318,1431)(319,1430)(320,1429)(321,1428)(322,1427)(323,1426)(324,1425)(325,1424)(326,1423)(327,1422)(328,1421)(329,1420)(330,1419)(331,1418)(332,1417)(333,1416)(334,1415)(335,1414)(336,1413)(337,1412)(338,1411)(339,1410)(340,1409)(341,1408)(342,1407)(343,1406)(344,1405)(345,1404)(346,1403)(347,1402)(348,1401)(349,1400)(350,1399)(351,1398)(352,1397)(353,1396)(354,1395)(355,1394)(356,1393)(357,1392)(358,1391)(359,1390)(360,1389)(361,1388)(362,1387)(363,1386)(364,1385)(365,1384)(366,1383)(367,1382)(368,1381)(369,1380)(370,1379)(371,1378)(372,1377)(373,1376)(374,1375)(375,1374)(376,1373)(377,1372)(378,1371)(379,1370)(380,1369)(381,1368)(382,1367)(383,1366)(384,1365)(385,1364)(386,1363)(387,1362)(388,1361)(389,1360)(390,1359)(391,1358)(392,1357)(393,1356)(394,1355)(395,1354)(396,1353)(397,1352)(398,1351)(399,1350)(400,1349)(401,1348)(402,1347)(403,1346)(404,1345)(405,1344)(406,1343)(407,1342)(408,1341)(409,1340)(410,1339)(411,1338)(412,1337)(413,1336)(414,1335)(415,1334)(416,1333)(417,1332)(418,1331)(419,1330)(420,1329)(421,1328)(422,1327)(423,1326)(424,1325)(425,1324)(426,1323)(427,1322)(428,1321)(429,1320)(430,1319)(431,1318)(432,1317)(433,1316)(434,1315)(435,1314)(436,1313)(437,1312)(438,1311)(439,1310)(440,1309)(441,1308)(442,1307)(443,1306)(444,1305)(445,1304)(446,1303)(447,1302)(448,1301)(449,1300)(450,1299)(451,1298)(452,1297)(453,1296)(454,1295)(455,1294)(456,1293)(457,1292)(458,1291)(459,1290)(460,1289)(461,1288)(462,1287)(463,1286)(464,1285)(465,1284)(466,1283)(467,1282)(468,1281)(469,1280)(470,1279)(471,1278)(472,1277)(473,1276)(474,1275)(475,1274)(476,1273)(477,1272)(478,1271)(479,1270)(480,1269)(481,1268)(482,1267)(483,1266)(484,1265)(485,1264)(486,1263)(487,1262)(488,1261)(489,1260)(490,1259)(491,1258)(492,1257)(493,1256)(494,1255)(495,1254)(496,1253)(497,1252)(498,1251)(499,1250)(500,1249)(501,1248)(502,1247)(503,1246)(504,1245)(505,1244)(506,1243)(507,1242)(508,1241)(509,1240)(510,1239)(511,1238)(512,1237)(513,1236)(514,1235)(515,1234)(516,1233)(517,1232)(518,1231)(519,1230)(520,1229)(521,1228)(522,1227)(523,1226)(524,1225)(525,1224)(526,1223)(527,1222)(528,1221)(529,1220)(530,1219)(531,1218)(532,1217)(533,1216)(534,1215)(535,1214)(536,1213)(537,1212)(538,1211)(539,1210)(540,1209)(541,1208)(542,1207)(543,1206)(544,1205)(545,1204)(546,1203)(547,1202)(548,1201)(549,1200)(550,1199)(551,1198)(552,1197)(553,1196)(554,1195)(555,1194)(556,1193)(557,1192)(558,1191)(559,1190)(560,1189)(561,1188)(562,1187)(563,1186)(564,1185)(565,1184)(566,1183)(567,1182)(568,1181)(569,1180)(570,1179)(571,1178)(572,1177)(573,1176)(574,1175)(575,1174)(576,1173)(577,1172)(578,1171)(579,1170)(580,1169)(581,1168)(582,1167)(583,1166)(584,1165)(585,1164)(586,1163)(587,1162)(588,1161)(589,1160)(590,1159)(591,1158)(592,1157)(593,1156)(594,1155)(595,1154)(596,1153)(597,1152)(598,1151)(599,1150)(600,1149)(601,1148)(602,1147)(603,1146)(604,1145)(605,1144)(606,1143)(607,1142)(608,1141)(609,1140)(610,1139)(611,1138)(612,1137)(613,1136)(614,1135)(615,1134)(616,1133)(617,1132)(618,1131)(619,1130)(620,1129)(621,1128)(622,1127)(623,1126)(624,1125)(625,1124)(626,1123)(627,1122)(628,1121)(629,1120)(630,1119)(631,1118)(632,1117)(633,1116)(634,1115)(635,1114)(636,1113)(637,1112)(638,1111)(639,1110)(640,1109)(641,1108)(642,1107)(643,1106)(644,1105)(645,1104)(646,1103)(647,1102)(648,1101)(649,1100)(650,1099)(651,1098)(652,1097)(653,1096)(654,1095)(655,1094)(656,1093)(657,1092)(658,1091)(659,1090)(660,1089)(661,1088)(662,1087)(663,1086)(664,1085)(665,1084)(666,1083)(667,1082)(668,1081)(669,1080)(670,1079)(671,1078)(672,1077)(673,1076)(674,1075)(675,1074)(676,1073)(677,1072)(678,1071)(679,1070)(680,1069)(681,1068)(682,1067)(683,1066)(684,1065)(685,1064)(686,1063)(687,1062)(688,1061)(689,1060)(690,1059)(691,1058)(692,1057)(693,1056)(694,1055)(695,1054)(696,1053)(697,1052)(698,1051)(699,1050)(700,1049)(701,1048)(702,1047)(703,1046)(704,1045)(705,1044)(706,1043)(707,1042)(708,1041)(709,1040)(710,1039)(711,1038)(712,1037)(713,1036)(714,1035)(715,1034)(716,1033)(717,1032)(718,1031)(719,1030)(720,1029)(721,1028)(722,1027)(723,1026)(724,1025)(725,1024)(726,1023)(727,1022)(728,1021)(729,1020)(730,1019)(731,1018)(732,1017)(733,1016)(734,1015)(735,1014)(736,1013)(737,1012)(738,1011)(739,1010)(740,1009)(741,1008)(742,1007)(743,1006)(744,1005)(745,1004)(746,1003)(747,1002)(748,1001)(749,1000)(750,999)(751,998)(752,997)(753,996)(754,995)(755,994)(756,993)(757,992)(758,991)(759,990)(760,989)(761,988)(762,987)(763,986)(764,985)(765,984)(766,983)(767,982)(768,981)(769,980)(770,979)(771,978)(772,977)(773,976)(774,975)(775,974)(776,973)(777,972)(778,971)(779,970)(780,969)(781,968)(782,967)(783,966)(784,965)(785,964)(786,963)(787,962)(788,961)(789,960)(790,959)(791,958)(792,957)(793,956)(794,955)(795,954)(796,953)(797,952)(798,951)(799,950)(800,949)(801,948)(802,947)(803,946)(804,945)(805,944)(806,943)(807,942)(808,941)(809,940)(810,939)(811,938)(812,937)(813,936)(814,935)(815,934)(816,933)(817,932)(818,931)(819,930)(820,929)(821,928)(822,927)(823,926)(824,925)(825,924)(826,923)(827,922)(828,921)(829,920)(830,919)(831,918)(832,917)(833,916)(834,915)(835,914)(836,913)(837,912)(838,911)(839,910)(840,909)(841,908)(842,907)(843,906)(844,905)(845,904)(846,903)(847,902)(848,901)(849,900)(850,899)(851,898)(852,897)(853,896)(854,895)(855,894)(856,893)(857,892)(858,891)(859,890)(860,889)(861,888)(862,887)(863,886)(864,885)(865,884)(866,883)(867,882)(868,881)(869,880)(870,879)(871,878)(872,877)(873,876)(874,875)

Polycyclic presentation:

2,3,1733,2|a^2=d,b^a=b^2,c^a=c^1732

Conjugacy classes & character table

5202 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₃₉₈, Dic₁₅₅₉₇
maximal normal subgroup of	Dic₁₀₃₉₈
maximal quotient of	Dic₁₅₅₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20796, 4);

Dic5199 #35209485