D₂₆₅₂₅ #72289876

order	53050
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₅₂₅

Splittings

Fitting, derived, Hall	C₂₅×C₁₀₆₁ ⋊ C₂
Sylow	C₁₀₆₁ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₁₀₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₅×C₁₀₆₁	C₂
Frattini	C₅	D₅₃₀₅
socle	C₅×C₁₀₆₁	D₅
LCS	G ▷ C₂₅×C₁₀₆₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1086
transitive degree	μ_t(G) = 26525

A minimal faithful permutation representation — 2 generators on 1086 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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,1086)(28,1085)(29,1084)(30,1083)(31,1082)(32,1081)(33,1080)(34,1079)(35,1078)(36,1077)(37,1076)(38,1075)(39,1074)(40,1073)(41,1072)(42,1071)(43,1070)(44,1069)(45,1068)(46,1067)(47,1066)(48,1065)(49,1064)(50,1063)(51,1062)(52,1061)(53,1060)(54,1059)(55,1058)(56,1057)(57,1056)(58,1055)(59,1054)(60,1053)(61,1052)(62,1051)(63,1050)(64,1049)(65,1048)(66,1047)(67,1046)(68,1045)(69,1044)(70,1043)(71,1042)(72,1041)(73,1040)(74,1039)(75,1038)(76,1037)(77,1036)(78,1035)(79,1034)(80,1033)(81,1032)(82,1031)(83,1030)(84,1029)(85,1028)(86,1027)(87,1026)(88,1025)(89,1024)(90,1023)(91,1022)(92,1021)(93,1020)(94,1019)(95,1018)(96,1017)(97,1016)(98,1015)(99,1014)(100,1013)(101,1012)(102,1011)(103,1010)(104,1009)(105,1008)(106,1007)(107,1006)(108,1005)(109,1004)(110,1003)(111,1002)(112,1001)(113,1000)(114,999)(115,998)(116,997)(117,996)(118,995)(119,994)(120,993)(121,992)(122,991)(123,990)(124,989)(125,988)(126,987)(127,986)(128,985)(129,984)(130,983)(131,982)(132,981)(133,980)(134,979)(135,978)(136,977)(137,976)(138,975)(139,974)(140,973)(141,972)(142,971)(143,970)(144,969)(145,968)(146,967)(147,966)(148,965)(149,964)(150,963)(151,962)(152,961)(153,960)(154,959)(155,958)(156,957)(157,956)(158,955)(159,954)(160,953)(161,952)(162,951)(163,950)(164,949)(165,948)(166,947)(167,946)(168,945)(169,944)(170,943)(171,942)(172,941)(173,940)(174,939)(175,938)(176,937)(177,936)(178,935)(179,934)(180,933)(181,932)(182,931)(183,930)(184,929)(185,928)(186,927)(187,926)(188,925)(189,924)(190,923)(191,922)(192,921)(193,920)(194,919)(195,918)(196,917)(197,916)(198,915)(199,914)(200,913)(201,912)(202,911)(203,910)(204,909)(205,908)(206,907)(207,906)(208,905)(209,904)(210,903)(211,902)(212,901)(213,900)(214,899)(215,898)(216,897)(217,896)(218,895)(219,894)(220,893)(221,892)(222,891)(223,890)(224,889)(225,888)(226,887)(227,886)(228,885)(229,884)(230,883)(231,882)(232,881)(233,880)(234,879)(235,878)(236,877)(237,876)(238,875)(239,874)(240,873)(241,872)(242,871)(243,870)(244,869)(245,868)(246,867)(247,866)(248,865)(249,864)(250,863)(251,862)(252,861)(253,860)(254,859)(255,858)(256,857)(257,856)(258,855)(259,854)(260,853)(261,852)(262,851)(263,850)(264,849)(265,848)(266,847)(267,846)(268,845)(269,844)(270,843)(271,842)(272,841)(273,840)(274,839)(275,838)(276,837)(277,836)(278,835)(279,834)(280,833)(281,832)(282,831)(283,830)(284,829)(285,828)(286,827)(287,826)(288,825)(289,824)(290,823)(291,822)(292,821)(293,820)(294,819)(295,818)(296,817)(297,816)(298,815)(299,814)(300,813)(301,812)(302,811)(303,810)(304,809)(305,808)(306,807)(307,806)(308,805)(309,804)(310,803)(311,802)(312,801)(313,800)(314,799)(315,798)(316,797)(317,796)(318,795)(319,794)(320,793)(321,792)(322,791)(323,790)(324,789)(325,788)(326,787)(327,786)(328,785)(329,784)(330,783)(331,782)(332,781)(333,780)(334,779)(335,778)(336,777)(337,776)(338,775)(339,774)(340,773)(341,772)(342,771)(343,770)(344,769)(345,768)(346,767)(347,766)(348,765)(349,764)(350,763)(351,762)(352,761)(353,760)(354,759)(355,758)(356,757)(357,756)(358,755)(359,754)(360,753)(361,752)(362,751)(363,750)(364,749)(365,748)(366,747)(367,746)(368,745)(369,744)(370,743)(371,742)(372,741)(373,740)(374,739)(375,738)(376,737)(377,736)(378,735)(379,734)(380,733)(381,732)(382,731)(383,730)(384,729)(385,728)(386,727)(387,726)(388,725)(389,724)(390,723)(391,722)(392,721)(393,720)(394,719)(395,718)(396,717)(397,716)(398,715)(399,714)(400,713)(401,712)(402,711)(403,710)(404,709)(405,708)(406,707)(407,706)(408,705)(409,704)(410,703)(411,702)(412,701)(413,700)(414,699)(415,698)(416,697)(417,696)(418,695)(419,694)(420,693)(421,692)(422,691)(423,690)(424,689)(425,688)(426,687)(427,686)(428,685)(429,684)(430,683)(431,682)(432,681)(433,680)(434,679)(435,678)(436,677)(437,676)(438,675)(439,674)(440,673)(441,672)(442,671)(443,670)(444,669)(445,668)(446,667)(447,666)(448,665)(449,664)(450,663)(451,662)(452,661)(453,660)(454,659)(455,658)(456,657)(457,656)(458,655)(459,654)(460,653)(461,652)(462,651)(463,650)(464,649)(465,648)(466,647)(467,646)(468,645)(469,644)(470,643)(471,642)(472,641)(473,640)(474,639)(475,638)(476,637)(477,636)(478,635)(479,634)(480,633)(481,632)(482,631)(483,630)(484,629)(485,628)(486,627)(487,626)(488,625)(489,624)(490,623)(491,622)(492,621)(493,620)(494,619)(495,618)(496,617)(497,616)(498,615)(499,614)(500,613)(501,612)(502,611)(503,610)(504,609)(505,608)(506,607)(507,606)(508,605)(509,604)(510,603)(511,602)(512,601)(513,600)(514,599)(515,598)(516,597)(517,596)(518,595)(519,594)(520,593)(521,592)(522,591)(523,590)(524,589)(525,588)(526,587)(527,586)(528,585)(529,584)(530,583)(531,582)(532,581)(533,580)(534,579)(535,578)(536,577)(537,576)(538,575)(539,574)(540,573)(541,572)(542,571)(543,570)(544,569)(545,568)(546,567)(547,566)(548,565)(549,564)(550,563)(551,562)(552,561)(553,560)(554,559)(555,558)(556,557)

Polycyclic presentation:

2,5,1061,5|b^5=d,b^a=b^4d^4,c^a=c^1060,d^a=d^4

Conjugacy classes & character table

13264 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, 5, 1061, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.4^4);;
gap> SetConjugate(coll, 3, 1, F.3^1060);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26525 #72289876