Dic₁₃₄₇₆ #72289421

order	53904
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₄₇₆

Splittings

Hall	C₃×C₁₁₂₃ ⋊ Q₁₆
Sylow	C₁₁₂₃ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₄₄₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₄₇₆
commutator, cosocle	C₃×C₄×C₁₁₂₃	C₂²
Frattini	C₄	C₂×D₃₃₆₉
Fitting	C₃×C₈×C₁₁₂₃	C₂
socle, prime radical	C₂×C₃×C₁₁₂₃	D₄
hypercenter	C₈	D₃₃₆₉
hypocenter	C₃×C₁₁₂₃	Q₁₆
LCS	G ▷ C₃×C₄×C₁₁₂₃ ▷ C₂×C₃×C₁₁₂₃ ▷ C₃×C₁₁₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1142
transitive degree	μ_t(G) = 53904

A minimal faithful permutation representation — 2 generators on 1142 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,1142)(22,1141)(23,1140)(24,1139)(25,1138)(26,1137)(27,1136)(28,1135)(29,1134)(30,1133)(31,1132)(32,1131)(33,1130)(34,1129)(35,1128)(36,1127)(37,1126)(38,1125)(39,1124)(40,1123)(41,1122)(42,1121)(43,1120)(44,1119)(45,1118)(46,1117)(47,1116)(48,1115)(49,1114)(50,1113)(51,1112)(52,1111)(53,1110)(54,1109)(55,1108)(56,1107)(57,1106)(58,1105)(59,1104)(60,1103)(61,1102)(62,1101)(63,1100)(64,1099)(65,1098)(66,1097)(67,1096)(68,1095)(69,1094)(70,1093)(71,1092)(72,1091)(73,1090)(74,1089)(75,1088)(76,1087)(77,1086)(78,1085)(79,1084)(80,1083)(81,1082)(82,1081)(83,1080)(84,1079)(85,1078)(86,1077)(87,1076)(88,1075)(89,1074)(90,1073)(91,1072)(92,1071)(93,1070)(94,1069)(95,1068)(96,1067)(97,1066)(98,1065)(99,1064)(100,1063)(101,1062)(102,1061)(103,1060)(104,1059)(105,1058)(106,1057)(107,1056)(108,1055)(109,1054)(110,1053)(111,1052)(112,1051)(113,1050)(114,1049)(115,1048)(116,1047)(117,1046)(118,1045)(119,1044)(120,1043)(121,1042)(122,1041)(123,1040)(124,1039)(125,1038)(126,1037)(127,1036)(128,1035)(129,1034)(130,1033)(131,1032)(132,1031)(133,1030)(134,1029)(135,1028)(136,1027)(137,1026)(138,1025)(139,1024)(140,1023)(141,1022)(142,1021)(143,1020)(144,1019)(145,1018)(146,1017)(147,1016)(148,1015)(149,1014)(150,1013)(151,1012)(152,1011)(153,1010)(154,1009)(155,1008)(156,1007)(157,1006)(158,1005)(159,1004)(160,1003)(161,1002)(162,1001)(163,1000)(164,999)(165,998)(166,997)(167,996)(168,995)(169,994)(170,993)(171,992)(172,991)(173,990)(174,989)(175,988)(176,987)(177,986)(178,985)(179,984)(180,983)(181,982)(182,981)(183,980)(184,979)(185,978)(186,977)(187,976)(188,975)(189,974)(190,973)(191,972)(192,971)(193,970)(194,969)(195,968)(196,967)(197,966)(198,965)(199,964)(200,963)(201,962)(202,961)(203,960)(204,959)(205,958)(206,957)(207,956)(208,955)(209,954)(210,953)(211,952)(212,951)(213,950)(214,949)(215,948)(216,947)(217,946)(218,945)(219,944)(220,943)(221,942)(222,941)(223,940)(224,939)(225,938)(226,937)(227,936)(228,935)(229,934)(230,933)(231,932)(232,931)(233,930)(234,929)(235,928)(236,927)(237,926)(238,925)(239,924)(240,923)(241,922)(242,921)(243,920)(244,919)(245,918)(246,917)(247,916)(248,915)(249,914)(250,913)(251,912)(252,911)(253,910)(254,909)(255,908)(256,907)(257,906)(258,905)(259,904)(260,903)(261,902)(262,901)(263,900)(264,899)(265,898)(266,897)(267,896)(268,895)(269,894)(270,893)(271,892)(272,891)(273,890)(274,889)(275,888)(276,887)(277,886)(278,885)(279,884)(280,883)(281,882)(282,881)(283,880)(284,879)(285,878)(286,877)(287,876)(288,875)(289,874)(290,873)(291,872)(292,871)(293,870)(294,869)(295,868)(296,867)(297,866)(298,865)(299,864)(300,863)(301,862)(302,861)(303,860)(304,859)(305,858)(306,857)(307,856)(308,855)(309,854)(310,853)(311,852)(312,851)(313,850)(314,849)(315,848)(316,847)(317,846)(318,845)(319,844)(320,843)(321,842)(322,841)(323,840)(324,839)(325,838)(326,837)(327,836)(328,835)(329,834)(330,833)(331,832)(332,831)(333,830)(334,829)(335,828)(336,827)(337,826)(338,825)(339,824)(340,823)(341,822)(342,821)(343,820)(344,819)(345,818)(346,817)(347,816)(348,815)(349,814)(350,813)(351,812)(352,811)(353,810)(354,809)(355,808)(356,807)(357,806)(358,805)(359,804)(360,803)(361,802)(362,801)(363,800)(364,799)(365,798)(366,797)(367,796)(368,795)(369,794)(370,793)(371,792)(372,791)(373,790)(374,789)(375,788)(376,787)(377,786)(378,785)(379,784)(380,783)(381,782)(382,781)(383,780)(384,779)(385,778)(386,777)(387,776)(388,775)(389,774)(390,773)(391,772)(392,771)(393,770)(394,769)(395,768)(396,767)(397,766)(398,765)(399,764)(400,763)(401,762)(402,761)(403,760)(404,759)(405,758)(406,757)(407,756)(408,755)(409,754)(410,753)(411,752)(412,751)(413,750)(414,749)(415,748)(416,747)(417,746)(418,745)(419,744)(420,743)(421,742)(422,741)(423,740)(424,739)(425,738)(426,737)(427,736)(428,735)(429,734)(430,733)(431,732)(432,731)(433,730)(434,729)(435,728)(436,727)(437,726)(438,725)(439,724)(440,723)(441,722)(442,721)(443,720)(444,719)(445,718)(446,717)(447,716)(448,715)(449,714)(450,713)(451,712)(452,711)(453,710)(454,709)(455,708)(456,707)(457,706)(458,705)(459,704)(460,703)(461,702)(462,701)(463,700)(464,699)(465,698)(466,697)(467,696)(468,695)(469,694)(470,693)(471,692)(472,691)(473,690)(474,689)(475,688)(476,687)(477,686)(478,685)(479,684)(480,683)(481,682)(482,681)(483,680)(484,679)(485,678)(486,677)(487,676)(488,675)(489,674)(490,673)(491,672)(492,671)(493,670)(494,669)(495,668)(496,667)(497,666)(498,665)(499,664)(500,663)(501,662)(502,661)(503,660)(504,659)(505,658)(506,657)(507,656)(508,655)(509,654)(510,653)(511,652)(512,651)(513,650)(514,649)(515,648)(516,647)(517,646)(518,645)(519,644)(520,643)(521,642)(522,641)(523,640)(524,639)(525,638)(526,637)(527,636)(528,635)(529,634)(530,633)(531,632)(532,631)(533,630)(534,629)(535,628)(536,627)(537,626)(538,625)(539,624)(540,623)(541,622)(542,621)(543,620)(544,619)(545,618)(546,617)(547,616)(548,615)(549,614)(550,613)(551,612)(552,611)(553,610)(554,609)(555,608)(556,607)(557,606)(558,605)(559,604)(560,603)(561,602)(562,601)(563,600)(564,599)(565,598)(566,597)(567,596)(568,595)(569,594)(570,593)(571,592)(572,591)(573,590)(574,589)(575,588)(576,587)(577,586)(578,585)(579,584)(580,583)(581,582)

Polycyclic presentation:

2^4,3,1123|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^1122

Conjugacy classes & character table

13479 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", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 1123]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^1122);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13476 #72289421