D₂₅₉₇₅ #72289089

order	51950
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₅₉₇₅

Splittings

Fitting, derived, Hall	C₂₅×C₁₀₃₉ ⋊ C₂
Sylow	C₁₀₃₉ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₁₀₃₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₅×C₁₀₃₉	C₂
Frattini	C₅	D₅₁₉₅
socle	C₅×C₁₀₃₉	D₅
LCS	G ▷ C₂₅×C₁₀₃₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1064
transitive degree	μ_t(G) = 25975

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

Polycyclic presentation:

2,5,1039,5|b^5=d,b^a=b^4d^4,c^a=c^1038,d^a=d^4

Conjugacy classes & character table

12989 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, 1039, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.4^4);;
gap> SetConjugate(coll, 3, 1, F.3^1038);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25975 #72289089