Dic₄₂₈₁ #30686136

order	17124
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₂₈₁
unambiguous semidirect product	C₃⋊Dic₁₄₂₇
unambiguous semidirect product	C₁₄₂₇⋊Dic₃

Splittings

derived, Hall	C₃×C₁₄₂₇ ⋊ C₄
Sylow	C₁₄₂₇ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₄₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₂₈₁
commutator, hypocenter	C₃×C₁₄₂₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₄₂₇	C₂
LCS	G ▷ C₃×C₁₄₂₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1434
transitive degree	μ_t(G) = 17124

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

Polycyclic presentation:

2,3,1427,2|a^2=d,b^a=b^2,c^a=c^1426

Conjugacy classes & character table

4284 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₅₆₂, Dic₁₂₈₄₃
maximal normal subgroup of	Dic₈₅₆₂
maximal quotient of	Dic₁₂₈₄₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17124, 4);

Dic4281 #30686136