Dic₁₄₃₃₃ #72290944

order	57332
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₃₃

Splittings

derived, Hall	C₁₁×C₁₃₀₃ ⋊ C₄
Sylow	C₁₃₀₃ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₁₃₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₃₃₃
commutator, hypocenter	C₁₁×C₁₃₀₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₃₀₃	C₂
LCS	G ▷ C₁₁×C₁₃₀₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1318
transitive degree	μ_t(G) = 57332

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

Polycyclic presentation:

2^2,11,1303|a^2=b,c^a=c^10,d^a=d^1302

Conjugacy classes & character table

14336 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d");;
gap> coll := SingleCollector(F, [2, 2, 11, 1303]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^1302);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14333 #72290944