Dic₁₃₅₃₈ #72289633

order	54152
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₅₃₈

Splittings

Hall	C₇×C₉₆₇ ⋊ Q₈
Sylow	C₉₆₇ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₁₉₃₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₇₆₉
commutator, socle, prime radical, cosocle	C₂×C₇×C₉₆₇	C₂²
Fitting	C₄×C₇×C₉₆₇	C₂
hypercenter	C₄	D₆₇₆₉
hypocenter	C₇×C₉₆₇	Q₈
LCS	G ▷ C₂×C₇×C₉₆₇ ▷ C₇×C₉₆₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 982
transitive degree	μ_t(G) = 54152

A minimal faithful permutation representation — 2 generators on 982 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,982)(18,981)(19,980)(20,979)(21,978)(22,977)(23,976)(24,975)(25,974)(26,973)(27,972)(28,971)(29,970)(30,969)(31,968)(32,967)(33,966)(34,965)(35,964)(36,963)(37,962)(38,961)(39,960)(40,959)(41,958)(42,957)(43,956)(44,955)(45,954)(46,953)(47,952)(48,951)(49,950)(50,949)(51,948)(52,947)(53,946)(54,945)(55,944)(56,943)(57,942)(58,941)(59,940)(60,939)(61,938)(62,937)(63,936)(64,935)(65,934)(66,933)(67,932)(68,931)(69,930)(70,929)(71,928)(72,927)(73,926)(74,925)(75,924)(76,923)(77,922)(78,921)(79,920)(80,919)(81,918)(82,917)(83,916)(84,915)(85,914)(86,913)(87,912)(88,911)(89,910)(90,909)(91,908)(92,907)(93,906)(94,905)(95,904)(96,903)(97,902)(98,901)(99,900)(100,899)(101,898)(102,897)(103,896)(104,895)(105,894)(106,893)(107,892)(108,891)(109,890)(110,889)(111,888)(112,887)(113,886)(114,885)(115,884)(116,883)(117,882)(118,881)(119,880)(120,879)(121,878)(122,877)(123,876)(124,875)(125,874)(126,873)(127,872)(128,871)(129,870)(130,869)(131,868)(132,867)(133,866)(134,865)(135,864)(136,863)(137,862)(138,861)(139,860)(140,859)(141,858)(142,857)(143,856)(144,855)(145,854)(146,853)(147,852)(148,851)(149,850)(150,849)(151,848)(152,847)(153,846)(154,845)(155,844)(156,843)(157,842)(158,841)(159,840)(160,839)(161,838)(162,837)(163,836)(164,835)(165,834)(166,833)(167,832)(168,831)(169,830)(170,829)(171,828)(172,827)(173,826)(174,825)(175,824)(176,823)(177,822)(178,821)(179,820)(180,819)(181,818)(182,817)(183,816)(184,815)(185,814)(186,813)(187,812)(188,811)(189,810)(190,809)(191,808)(192,807)(193,806)(194,805)(195,804)(196,803)(197,802)(198,801)(199,800)(200,799)(201,798)(202,797)(203,796)(204,795)(205,794)(206,793)(207,792)(208,791)(209,790)(210,789)(211,788)(212,787)(213,786)(214,785)(215,784)(216,783)(217,782)(218,781)(219,780)(220,779)(221,778)(222,777)(223,776)(224,775)(225,774)(226,773)(227,772)(228,771)(229,770)(230,769)(231,768)(232,767)(233,766)(234,765)(235,764)(236,763)(237,762)(238,761)(239,760)(240,759)(241,758)(242,757)(243,756)(244,755)(245,754)(246,753)(247,752)(248,751)(249,750)(250,749)(251,748)(252,747)(253,746)(254,745)(255,744)(256,743)(257,742)(258,741)(259,740)(260,739)(261,738)(262,737)(263,736)(264,735)(265,734)(266,733)(267,732)(268,731)(269,730)(270,729)(271,728)(272,727)(273,726)(274,725)(275,724)(276,723)(277,722)(278,721)(279,720)(280,719)(281,718)(282,717)(283,716)(284,715)(285,714)(286,713)(287,712)(288,711)(289,710)(290,709)(291,708)(292,707)(293,706)(294,705)(295,704)(296,703)(297,702)(298,701)(299,700)(300,699)(301,698)(302,697)(303,696)(304,695)(305,694)(306,693)(307,692)(308,691)(309,690)(310,689)(311,688)(312,687)(313,686)(314,685)(315,684)(316,683)(317,682)(318,681)(319,680)(320,679)(321,678)(322,677)(323,676)(324,675)(325,674)(326,673)(327,672)(328,671)(329,670)(330,669)(331,668)(332,667)(333,666)(334,665)(335,664)(336,663)(337,662)(338,661)(339,660)(340,659)(341,658)(342,657)(343,656)(344,655)(345,654)(346,653)(347,652)(348,651)(349,650)(350,649)(351,648)(352,647)(353,646)(354,645)(355,644)(356,643)(357,642)(358,641)(359,640)(360,639)(361,638)(362,637)(363,636)(364,635)(365,634)(366,633)(367,632)(368,631)(369,630)(370,629)(371,628)(372,627)(373,626)(374,625)(375,624)(376,623)(377,622)(378,621)(379,620)(380,619)(381,618)(382,617)(383,616)(384,615)(385,614)(386,613)(387,612)(388,611)(389,610)(390,609)(391,608)(392,607)(393,606)(394,605)(395,604)(396,603)(397,602)(398,601)(399,600)(400,599)(401,598)(402,597)(403,596)(404,595)(405,594)(406,593)(407,592)(408,591)(409,590)(410,589)(411,588)(412,587)(413,586)(414,585)(415,584)(416,583)(417,582)(418,581)(419,580)(420,579)(421,578)(422,577)(423,576)(424,575)(425,574)(426,573)(427,572)(428,571)(429,570)(430,569)(431,568)(432,567)(433,566)(434,565)(435,564)(436,563)(437,562)(438,561)(439,560)(440,559)(441,558)(442,557)(443,556)(444,555)(445,554)(446,553)(447,552)(448,551)(449,550)(450,549)(451,548)(452,547)(453,546)(454,545)(455,544)(456,543)(457,542)(458,541)(459,540)(460,539)(461,538)(462,537)(463,536)(464,535)(465,534)(466,533)(467,532)(468,531)(469,530)(470,529)(471,528)(472,527)(473,526)(474,525)(475,524)(476,523)(477,522)(478,521)(479,520)(480,519)(481,518)(482,517)(483,516)(484,515)(485,514)(486,513)(487,512)(488,511)(489,510)(490,509)(491,508)(492,507)(493,506)(494,505)(495,504)(496,503)(497,502)(498,501)(499,500)

Polycyclic presentation:

2^3,7,967|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^966

Conjugacy classes & character table

13541 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 967]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^966);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13538 #72289633