Dic₁₃₅₄₁ #72289639

order	54164
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₄₁

Splittings

derived, Hall	C₁₁×C₁₂₃₁ ⋊ C₄
Sylow	C₁₂₃₁ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₁₂₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₅₄₁
commutator, hypocenter	C₁₁×C₁₂₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₂₃₁	C₂
LCS	G ▷ C₁₁×C₁₂₃₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1246
transitive degree	μ_t(G) = 54164

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

Polycyclic presentation:

2^2,11,1231|a^2=b,c^a=c^10,d^a=d^1230

Conjugacy classes & character table

13544 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, 1231]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^1230);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13541 #72289639