D₂₆₁₀₉ #72289322

order	52218
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₁₀₉

Splittings

Fitting, derived, Hall	C₂₇×C₉₆₇ ⋊ C₂
Sylow	C₉₆₇ ⋊ D₂₇
Sylow	C₂₇ ⋊ D₉₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₇×C₉₆₇	C₂
Frattini	C₉	D₂₉₀₁
socle	C₃×C₉₆₇	D₉
LCS	G ▷ C₂₇×C₉₆₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 994
transitive degree	μ_t(G) = 26109

A minimal faithful permutation representation — 2 generators on 994 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,994)(30,993)(31,992)(32,991)(33,990)(34,989)(35,988)(36,987)(37,986)(38,985)(39,984)(40,983)(41,982)(42,981)(43,980)(44,979)(45,978)(46,977)(47,976)(48,975)(49,974)(50,973)(51,972)(52,971)(53,970)(54,969)(55,968)(56,967)(57,966)(58,965)(59,964)(60,963)(61,962)(62,961)(63,960)(64,959)(65,958)(66,957)(67,956)(68,955)(69,954)(70,953)(71,952)(72,951)(73,950)(74,949)(75,948)(76,947)(77,946)(78,945)(79,944)(80,943)(81,942)(82,941)(83,940)(84,939)(85,938)(86,937)(87,936)(88,935)(89,934)(90,933)(91,932)(92,931)(93,930)(94,929)(95,928)(96,927)(97,926)(98,925)(99,924)(100,923)(101,922)(102,921)(103,920)(104,919)(105,918)(106,917)(107,916)(108,915)(109,914)(110,913)(111,912)(112,911)(113,910)(114,909)(115,908)(116,907)(117,906)(118,905)(119,904)(120,903)(121,902)(122,901)(123,900)(124,899)(125,898)(126,897)(127,896)(128,895)(129,894)(130,893)(131,892)(132,891)(133,890)(134,889)(135,888)(136,887)(137,886)(138,885)(139,884)(140,883)(141,882)(142,881)(143,880)(144,879)(145,878)(146,877)(147,876)(148,875)(149,874)(150,873)(151,872)(152,871)(153,870)(154,869)(155,868)(156,867)(157,866)(158,865)(159,864)(160,863)(161,862)(162,861)(163,860)(164,859)(165,858)(166,857)(167,856)(168,855)(169,854)(170,853)(171,852)(172,851)(173,850)(174,849)(175,848)(176,847)(177,846)(178,845)(179,844)(180,843)(181,842)(182,841)(183,840)(184,839)(185,838)(186,837)(187,836)(188,835)(189,834)(190,833)(191,832)(192,831)(193,830)(194,829)(195,828)(196,827)(197,826)(198,825)(199,824)(200,823)(201,822)(202,821)(203,820)(204,819)(205,818)(206,817)(207,816)(208,815)(209,814)(210,813)(211,812)(212,811)(213,810)(214,809)(215,808)(216,807)(217,806)(218,805)(219,804)(220,803)(221,802)(222,801)(223,800)(224,799)(225,798)(226,797)(227,796)(228,795)(229,794)(230,793)(231,792)(232,791)(233,790)(234,789)(235,788)(236,787)(237,786)(238,785)(239,784)(240,783)(241,782)(242,781)(243,780)(244,779)(245,778)(246,777)(247,776)(248,775)(249,774)(250,773)(251,772)(252,771)(253,770)(254,769)(255,768)(256,767)(257,766)(258,765)(259,764)(260,763)(261,762)(262,761)(263,760)(264,759)(265,758)(266,757)(267,756)(268,755)(269,754)(270,753)(271,752)(272,751)(273,750)(274,749)(275,748)(276,747)(277,746)(278,745)(279,744)(280,743)(281,742)(282,741)(283,740)(284,739)(285,738)(286,737)(287,736)(288,735)(289,734)(290,733)(291,732)(292,731)(293,730)(294,729)(295,728)(296,727)(297,726)(298,725)(299,724)(300,723)(301,722)(302,721)(303,720)(304,719)(305,718)(306,717)(307,716)(308,715)(309,714)(310,713)(311,712)(312,711)(313,710)(314,709)(315,708)(316,707)(317,706)(318,705)(319,704)(320,703)(321,702)(322,701)(323,700)(324,699)(325,698)(326,697)(327,696)(328,695)(329,694)(330,693)(331,692)(332,691)(333,690)(334,689)(335,688)(336,687)(337,686)(338,685)(339,684)(340,683)(341,682)(342,681)(343,680)(344,679)(345,678)(346,677)(347,676)(348,675)(349,674)(350,673)(351,672)(352,671)(353,670)(354,669)(355,668)(356,667)(357,666)(358,665)(359,664)(360,663)(361,662)(362,661)(363,660)(364,659)(365,658)(366,657)(367,656)(368,655)(369,654)(370,653)(371,652)(372,651)(373,650)(374,649)(375,648)(376,647)(377,646)(378,645)(379,644)(380,643)(381,642)(382,641)(383,640)(384,639)(385,638)(386,637)(387,636)(388,635)(389,634)(390,633)(391,632)(392,631)(393,630)(394,629)(395,628)(396,627)(397,626)(398,625)(399,624)(400,623)(401,622)(402,621)(403,620)(404,619)(405,618)(406,617)(407,616)(408,615)(409,614)(410,613)(411,612)(412,611)(413,610)(414,609)(415,608)(416,607)(417,606)(418,605)(419,604)(420,603)(421,602)(422,601)(423,600)(424,599)(425,598)(426,597)(427,596)(428,595)(429,594)(430,593)(431,592)(432,591)(433,590)(434,589)(435,588)(436,587)(437,586)(438,585)(439,584)(440,583)(441,582)(442,581)(443,580)(444,579)(445,578)(446,577)(447,576)(448,575)(449,574)(450,573)(451,572)(452,571)(453,570)(454,569)(455,568)(456,567)(457,566)(458,565)(459,564)(460,563)(461,562)(462,561)(463,560)(464,559)(465,558)(466,557)(467,556)(468,555)(469,554)(470,553)(471,552)(472,551)(473,550)(474,549)(475,548)(476,547)(477,546)(478,545)(479,544)(480,543)(481,542)(482,541)(483,540)(484,539)(485,538)(486,537)(487,536)(488,535)(489,534)(490,533)(491,532)(492,531)(493,530)(494,529)(495,528)(496,527)(497,526)(498,525)(499,524)(500,523)(501,522)(502,521)(503,520)(504,519)(505,518)(506,517)(507,516)(508,515)(509,514)(510,513)(511,512)

Polycyclic presentation:

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

Conjugacy classes & character table

13056 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, 3, 967, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^966);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26109 #72289322