D₂₇₃₂₅ #72290637

order	54650
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₇₃₂₅

Splittings

Fitting, derived, Hall	C₂₅×C₁₀₉₃ ⋊ C₂
Sylow	C₁₀₉₃ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₁₀₉₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₅×C₁₀₉₃	C₂
Frattini	C₅	D₅₄₆₅
socle	C₅×C₁₀₉₃	D₅
LCS	G ▷ C₂₅×C₁₀₉₃
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1118
transitive degree	μ_t(G) = 27325

A minimal faithful permutation representation — 2 generators on 1118 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,1118)(28,1117)(29,1116)(30,1115)(31,1114)(32,1113)(33,1112)(34,1111)(35,1110)(36,1109)(37,1108)(38,1107)(39,1106)(40,1105)(41,1104)(42,1103)(43,1102)(44,1101)(45,1100)(46,1099)(47,1098)(48,1097)(49,1096)(50,1095)(51,1094)(52,1093)(53,1092)(54,1091)(55,1090)(56,1089)(57,1088)(58,1087)(59,1086)(60,1085)(61,1084)(62,1083)(63,1082)(64,1081)(65,1080)(66,1079)(67,1078)(68,1077)(69,1076)(70,1075)(71,1074)(72,1073)(73,1072)(74,1071)(75,1070)(76,1069)(77,1068)(78,1067)(79,1066)(80,1065)(81,1064)(82,1063)(83,1062)(84,1061)(85,1060)(86,1059)(87,1058)(88,1057)(89,1056)(90,1055)(91,1054)(92,1053)(93,1052)(94,1051)(95,1050)(96,1049)(97,1048)(98,1047)(99,1046)(100,1045)(101,1044)(102,1043)(103,1042)(104,1041)(105,1040)(106,1039)(107,1038)(108,1037)(109,1036)(110,1035)(111,1034)(112,1033)(113,1032)(114,1031)(115,1030)(116,1029)(117,1028)(118,1027)(119,1026)(120,1025)(121,1024)(122,1023)(123,1022)(124,1021)(125,1020)(126,1019)(127,1018)(128,1017)(129,1016)(130,1015)(131,1014)(132,1013)(133,1012)(134,1011)(135,1010)(136,1009)(137,1008)(138,1007)(139,1006)(140,1005)(141,1004)(142,1003)(143,1002)(144,1001)(145,1000)(146,999)(147,998)(148,997)(149,996)(150,995)(151,994)(152,993)(153,992)(154,991)(155,990)(156,989)(157,988)(158,987)(159,986)(160,985)(161,984)(162,983)(163,982)(164,981)(165,980)(166,979)(167,978)(168,977)(169,976)(170,975)(171,974)(172,973)(173,972)(174,971)(175,970)(176,969)(177,968)(178,967)(179,966)(180,965)(181,964)(182,963)(183,962)(184,961)(185,960)(186,959)(187,958)(188,957)(189,956)(190,955)(191,954)(192,953)(193,952)(194,951)(195,950)(196,949)(197,948)(198,947)(199,946)(200,945)(201,944)(202,943)(203,942)(204,941)(205,940)(206,939)(207,938)(208,937)(209,936)(210,935)(211,934)(212,933)(213,932)(214,931)(215,930)(216,929)(217,928)(218,927)(219,926)(220,925)(221,924)(222,923)(223,922)(224,921)(225,920)(226,919)(227,918)(228,917)(229,916)(230,915)(231,914)(232,913)(233,912)(234,911)(235,910)(236,909)(237,908)(238,907)(239,906)(240,905)(241,904)(242,903)(243,902)(244,901)(245,900)(246,899)(247,898)(248,897)(249,896)(250,895)(251,894)(252,893)(253,892)(254,891)(255,890)(256,889)(257,888)(258,887)(259,886)(260,885)(261,884)(262,883)(263,882)(264,881)(265,880)(266,879)(267,878)(268,877)(269,876)(270,875)(271,874)(272,873)(273,872)(274,871)(275,870)(276,869)(277,868)(278,867)(279,866)(280,865)(281,864)(282,863)(283,862)(284,861)(285,860)(286,859)(287,858)(288,857)(289,856)(290,855)(291,854)(292,853)(293,852)(294,851)(295,850)(296,849)(297,848)(298,847)(299,846)(300,845)(301,844)(302,843)(303,842)(304,841)(305,840)(306,839)(307,838)(308,837)(309,836)(310,835)(311,834)(312,833)(313,832)(314,831)(315,830)(316,829)(317,828)(318,827)(319,826)(320,825)(321,824)(322,823)(323,822)(324,821)(325,820)(326,819)(327,818)(328,817)(329,816)(330,815)(331,814)(332,813)(333,812)(334,811)(335,810)(336,809)(337,808)(338,807)(339,806)(340,805)(341,804)(342,803)(343,802)(344,801)(345,800)(346,799)(347,798)(348,797)(349,796)(350,795)(351,794)(352,793)(353,792)(354,791)(355,790)(356,789)(357,788)(358,787)(359,786)(360,785)(361,784)(362,783)(363,782)(364,781)(365,780)(366,779)(367,778)(368,777)(369,776)(370,775)(371,774)(372,773)(373,772)(374,771)(375,770)(376,769)(377,768)(378,767)(379,766)(380,765)(381,764)(382,763)(383,762)(384,761)(385,760)(386,759)(387,758)(388,757)(389,756)(390,755)(391,754)(392,753)(393,752)(394,751)(395,750)(396,749)(397,748)(398,747)(399,746)(400,745)(401,744)(402,743)(403,742)(404,741)(405,740)(406,739)(407,738)(408,737)(409,736)(410,735)(411,734)(412,733)(413,732)(414,731)(415,730)(416,729)(417,728)(418,727)(419,726)(420,725)(421,724)(422,723)(423,722)(424,721)(425,720)(426,719)(427,718)(428,717)(429,716)(430,715)(431,714)(432,713)(433,712)(434,711)(435,710)(436,709)(437,708)(438,707)(439,706)(440,705)(441,704)(442,703)(443,702)(444,701)(445,700)(446,699)(447,698)(448,697)(449,696)(450,695)(451,694)(452,693)(453,692)(454,691)(455,690)(456,689)(457,688)(458,687)(459,686)(460,685)(461,684)(462,683)(463,682)(464,681)(465,680)(466,679)(467,678)(468,677)(469,676)(470,675)(471,674)(472,673)(473,672)(474,671)(475,670)(476,669)(477,668)(478,667)(479,666)(480,665)(481,664)(482,663)(483,662)(484,661)(485,660)(486,659)(487,658)(488,657)(489,656)(490,655)(491,654)(492,653)(493,652)(494,651)(495,650)(496,649)(497,648)(498,647)(499,646)(500,645)(501,644)(502,643)(503,642)(504,641)(505,640)(506,639)(507,638)(508,637)(509,636)(510,635)(511,634)(512,633)(513,632)(514,631)(515,630)(516,629)(517,628)(518,627)(519,626)(520,625)(521,624)(522,623)(523,622)(524,621)(525,620)(526,619)(527,618)(528,617)(529,616)(530,615)(531,614)(532,613)(533,612)(534,611)(535,610)(536,609)(537,608)(538,607)(539,606)(540,605)(541,604)(542,603)(543,602)(544,601)(545,600)(546,599)(547,598)(548,597)(549,596)(550,595)(551,594)(552,593)(553,592)(554,591)(555,590)(556,589)(557,588)(558,587)(559,586)(560,585)(561,584)(562,583)(563,582)(564,581)(565,580)(566,579)(567,578)(568,577)(569,576)(570,575)(571,574)(572,573)

Polycyclic presentation:

2,5,1093,5|b^5=d,b^a=b^4d^4,c^a=c^1092,d^a=d^4

Conjugacy classes & character table

13664 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, 1093, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.4^4);;
gap> SetConjugate(coll, 3, 1, F.3^1092);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27325 #72290637