Dic₁₄₃₃₉ #72290962

order	57356
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₃₉

Splittings

derived, Hall	C₁₃×C₁₁₀₃ ⋊ C₄
Sylow	C₁₁₀₃ ⋊ Dic₁₃
Sylow	C₁₃ ⋊ Dic₁₁₀₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₃₃₉
commutator, hypocenter	C₁₃×C₁₁₀₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₁₁₀₃	C₂
LCS	G ▷ C₁₃×C₁₁₀₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1120
transitive degree	μ_t(G) = 57356

A minimal faithful permutation representation — 2 generators on 1120 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,1120)(20,1119)(21,1118)(22,1117)(23,1116)(24,1115)(25,1114)(26,1113)(27,1112)(28,1111)(29,1110)(30,1109)(31,1108)(32,1107)(33,1106)(34,1105)(35,1104)(36,1103)(37,1102)(38,1101)(39,1100)(40,1099)(41,1098)(42,1097)(43,1096)(44,1095)(45,1094)(46,1093)(47,1092)(48,1091)(49,1090)(50,1089)(51,1088)(52,1087)(53,1086)(54,1085)(55,1084)(56,1083)(57,1082)(58,1081)(59,1080)(60,1079)(61,1078)(62,1077)(63,1076)(64,1075)(65,1074)(66,1073)(67,1072)(68,1071)(69,1070)(70,1069)(71,1068)(72,1067)(73,1066)(74,1065)(75,1064)(76,1063)(77,1062)(78,1061)(79,1060)(80,1059)(81,1058)(82,1057)(83,1056)(84,1055)(85,1054)(86,1053)(87,1052)(88,1051)(89,1050)(90,1049)(91,1048)(92,1047)(93,1046)(94,1045)(95,1044)(96,1043)(97,1042)(98,1041)(99,1040)(100,1039)(101,1038)(102,1037)(103,1036)(104,1035)(105,1034)(106,1033)(107,1032)(108,1031)(109,1030)(110,1029)(111,1028)(112,1027)(113,1026)(114,1025)(115,1024)(116,1023)(117,1022)(118,1021)(119,1020)(120,1019)(121,1018)(122,1017)(123,1016)(124,1015)(125,1014)(126,1013)(127,1012)(128,1011)(129,1010)(130,1009)(131,1008)(132,1007)(133,1006)(134,1005)(135,1004)(136,1003)(137,1002)(138,1001)(139,1000)(140,999)(141,998)(142,997)(143,996)(144,995)(145,994)(146,993)(147,992)(148,991)(149,990)(150,989)(151,988)(152,987)(153,986)(154,985)(155,984)(156,983)(157,982)(158,981)(159,980)(160,979)(161,978)(162,977)(163,976)(164,975)(165,974)(166,973)(167,972)(168,971)(169,970)(170,969)(171,968)(172,967)(173,966)(174,965)(175,964)(176,963)(177,962)(178,961)(179,960)(180,959)(181,958)(182,957)(183,956)(184,955)(185,954)(186,953)(187,952)(188,951)(189,950)(190,949)(191,948)(192,947)(193,946)(194,945)(195,944)(196,943)(197,942)(198,941)(199,940)(200,939)(201,938)(202,937)(203,936)(204,935)(205,934)(206,933)(207,932)(208,931)(209,930)(210,929)(211,928)(212,927)(213,926)(214,925)(215,924)(216,923)(217,922)(218,921)(219,920)(220,919)(221,918)(222,917)(223,916)(224,915)(225,914)(226,913)(227,912)(228,911)(229,910)(230,909)(231,908)(232,907)(233,906)(234,905)(235,904)(236,903)(237,902)(238,901)(239,900)(240,899)(241,898)(242,897)(243,896)(244,895)(245,894)(246,893)(247,892)(248,891)(249,890)(250,889)(251,888)(252,887)(253,886)(254,885)(255,884)(256,883)(257,882)(258,881)(259,880)(260,879)(261,878)(262,877)(263,876)(264,875)(265,874)(266,873)(267,872)(268,871)(269,870)(270,869)(271,868)(272,867)(273,866)(274,865)(275,864)(276,863)(277,862)(278,861)(279,860)(280,859)(281,858)(282,857)(283,856)(284,855)(285,854)(286,853)(287,852)(288,851)(289,850)(290,849)(291,848)(292,847)(293,846)(294,845)(295,844)(296,843)(297,842)(298,841)(299,840)(300,839)(301,838)(302,837)(303,836)(304,835)(305,834)(306,833)(307,832)(308,831)(309,830)(310,829)(311,828)(312,827)(313,826)(314,825)(315,824)(316,823)(317,822)(318,821)(319,820)(320,819)(321,818)(322,817)(323,816)(324,815)(325,814)(326,813)(327,812)(328,811)(329,810)(330,809)(331,808)(332,807)(333,806)(334,805)(335,804)(336,803)(337,802)(338,801)(339,800)(340,799)(341,798)(342,797)(343,796)(344,795)(345,794)(346,793)(347,792)(348,791)(349,790)(350,789)(351,788)(352,787)(353,786)(354,785)(355,784)(356,783)(357,782)(358,781)(359,780)(360,779)(361,778)(362,777)(363,776)(364,775)(365,774)(366,773)(367,772)(368,771)(369,770)(370,769)(371,768)(372,767)(373,766)(374,765)(375,764)(376,763)(377,762)(378,761)(379,760)(380,759)(381,758)(382,757)(383,756)(384,755)(385,754)(386,753)(387,752)(388,751)(389,750)(390,749)(391,748)(392,747)(393,746)(394,745)(395,744)(396,743)(397,742)(398,741)(399,740)(400,739)(401,738)(402,737)(403,736)(404,735)(405,734)(406,733)(407,732)(408,731)(409,730)(410,729)(411,728)(412,727)(413,726)(414,725)(415,724)(416,723)(417,722)(418,721)(419,720)(420,719)(421,718)(422,717)(423,716)(424,715)(425,714)(426,713)(427,712)(428,711)(429,710)(430,709)(431,708)(432,707)(433,706)(434,705)(435,704)(436,703)(437,702)(438,701)(439,700)(440,699)(441,698)(442,697)(443,696)(444,695)(445,694)(446,693)(447,692)(448,691)(449,690)(450,689)(451,688)(452,687)(453,686)(454,685)(455,684)(456,683)(457,682)(458,681)(459,680)(460,679)(461,678)(462,677)(463,676)(464,675)(465,674)(466,673)(467,672)(468,671)(469,670)(470,669)(471,668)(472,667)(473,666)(474,665)(475,664)(476,663)(477,662)(478,661)(479,660)(480,659)(481,658)(482,657)(483,656)(484,655)(485,654)(486,653)(487,652)(488,651)(489,650)(490,649)(491,648)(492,647)(493,646)(494,645)(495,644)(496,643)(497,642)(498,641)(499,640)(500,639)(501,638)(502,637)(503,636)(504,635)(505,634)(506,633)(507,632)(508,631)(509,630)(510,629)(511,628)(512,627)(513,626)(514,625)(515,624)(516,623)(517,622)(518,621)(519,620)(520,619)(521,618)(522,617)(523,616)(524,615)(525,614)(526,613)(527,612)(528,611)(529,610)(530,609)(531,608)(532,607)(533,606)(534,605)(535,604)(536,603)(537,602)(538,601)(539,600)(540,599)(541,598)(542,597)(543,596)(544,595)(545,594)(546,593)(547,592)(548,591)(549,590)(550,589)(551,588)(552,587)(553,586)(554,585)(555,584)(556,583)(557,582)(558,581)(559,580)(560,579)(561,578)(562,577)(563,576)(564,575)(565,574)(566,573)(567,572)(568,571)(569,570)

Polycyclic presentation:

2^2,13,1103|a^2=b,c^a=c^12,d^a=d^1102

Conjugacy classes & character table

14342 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, 13, 1103]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^12);;
gap> SetConjugate(coll, 4, 1, F.4^1102);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14339 #72290962