Finite Group DatabasebetaFinite Group Databasebeta| order | 58184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic14546 |
| Hall | C7×C1039 ⋊ Q8 |
| Sylow | C1039 ⋊ Dic14 |
| Sylow | C7 ⋊ Dic2078 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7273 |
| commutator, socle, prime radical, cosocle | C2×C7×C1039 | C22 |
| Fitting | C4×C7×C1039 | C2 |
| hypercenter | C4 | D7273 |
| hypocenter | C7×C1039 | Q8 |
| LCS | G ▷ C2×C7×C1039 ▷ C7×C1039 | |
| UCS | C1 ◁ C2 ◁ C4 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1054 |
| transitive degree | μt(G) = 58184 |
A minimal faithful permutation representation — 2 generators on 1054 points:
(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,1054)(18,1053)(19,1052)(20,1051)(21,1050)(22,1049)(23,1048)(24,1047)(25,1046)(26,1045)(27,1044)(28,1043)(29,1042)(30,1041)(31,1040)(32,1039)(33,1038)(34,1037)(35,1036)(36,1035)(37,1034)(38,1033)(39,1032)(40,1031)(41,1030)(42,1029)(43,1028)(44,1027)(45,1026)(46,1025)(47,1024)(48,1023)(49,1022)(50,1021)(51,1020)(52,1019)(53,1018)(54,1017)(55,1016)(56,1015)(57,1014)(58,1013)(59,1012)(60,1011)(61,1010)(62,1009)(63,1008)(64,1007)(65,1006)(66,1005)(67,1004)(68,1003)(69,1002)(70,1001)(71,1000)(72,999)(73,998)(74,997)(75,996)(76,995)(77,994)(78,993)(79,992)(80,991)(81,990)(82,989)(83,988)(84,987)(85,986)(86,985)(87,984)(88,983)(89,982)(90,981)(91,980)(92,979)(93,978)(94,977)(95,976)(96,975)(97,974)(98,973)(99,972)(100,971)(101,970)(102,969)(103,968)(104,967)(105,966)(106,965)(107,964)(108,963)(109,962)(110,961)(111,960)(112,959)(113,958)(114,957)(115,956)(116,955)(117,954)(118,953)(119,952)(120,951)(121,950)(122,949)(123,948)(124,947)(125,946)(126,945)(127,944)(128,943)(129,942)(130,941)(131,940)(132,939)(133,938)(134,937)(135,936)(136,935)(137,934)(138,933)(139,932)(140,931)(141,930)(142,929)(143,928)(144,927)(145,926)(146,925)(147,924)(148,923)(149,922)(150,921)(151,920)(152,919)(153,918)(154,917)(155,916)(156,915)(157,914)(158,913)(159,912)(160,911)(161,910)(162,909)(163,908)(164,907)(165,906)(166,905)(167,904)(168,903)(169,902)(170,901)(171,900)(172,899)(173,898)(174,897)(175,896)(176,895)(177,894)(178,893)(179,892)(180,891)(181,890)(182,889)(183,888)(184,887)(185,886)(186,885)(187,884)(188,883)(189,882)(190,881)(191,880)(192,879)(193,878)(194,877)(195,876)(196,875)(197,874)(198,873)(199,872)(200,871)(201,870)(202,869)(203,868)(204,867)(205,866)(206,865)(207,864)(208,863)(209,862)(210,861)(211,860)(212,859)(213,858)(214,857)(215,856)(216,855)(217,854)(218,853)(219,852)(220,851)(221,850)(222,849)(223,848)(224,847)(225,846)(226,845)(227,844)(228,843)(229,842)(230,841)(231,840)(232,839)(233,838)(234,837)(235,836)(236,835)(237,834)(238,833)(239,832)(240,831)(241,830)(242,829)(243,828)(244,827)(245,826)(246,825)(247,824)(248,823)(249,822)(250,821)(251,820)(252,819)(253,818)(254,817)(255,816)(256,815)(257,814)(258,813)(259,812)(260,811)(261,810)(262,809)(263,808)(264,807)(265,806)(266,805)(267,804)(268,803)(269,802)(270,801)(271,800)(272,799)(273,798)(274,797)(275,796)(276,795)(277,794)(278,793)(279,792)(280,791)(281,790)(282,789)(283,788)(284,787)(285,786)(286,785)(287,784)(288,783)(289,782)(290,781)(291,780)(292,779)(293,778)(294,777)(295,776)(296,775)(297,774)(298,773)(299,772)(300,771)(301,770)(302,769)(303,768)(304,767)(305,766)(306,765)(307,764)(308,763)(309,762)(310,761)(311,760)(312,759)(313,758)(314,757)(315,756)(316,755)(317,754)(318,753)(319,752)(320,751)(321,750)(322,749)(323,748)(324,747)(325,746)(326,745)(327,744)(328,743)(329,742)(330,741)(331,740)(332,739)(333,738)(334,737)(335,736)(336,735)(337,734)(338,733)(339,732)(340,731)(341,730)(342,729)(343,728)(344,727)(345,726)(346,725)(347,724)(348,723)(349,722)(350,721)(351,720)(352,719)(353,718)(354,717)(355,716)(356,715)(357,714)(358,713)(359,712)(360,711)(361,710)(362,709)(363,708)(364,707)(365,706)(366,705)(367,704)(368,703)(369,702)(370,701)(371,700)(372,699)(373,698)(374,697)(375,696)(376,695)(377,694)(378,693)(379,692)(380,691)(381,690)(382,689)(383,688)(384,687)(385,686)(386,685)(387,684)(388,683)(389,682)(390,681)(391,680)(392,679)(393,678)(394,677)(395,676)(396,675)(397,674)(398,673)(399,672)(400,671)(401,670)(402,669)(403,668)(404,667)(405,666)(406,665)(407,664)(408,663)(409,662)(410,661)(411,660)(412,659)(413,658)(414,657)(415,656)(416,655)(417,654)(418,653)(419,652)(420,651)(421,650)(422,649)(423,648)(424,647)(425,646)(426,645)(427,644)(428,643)(429,642)(430,641)(431,640)(432,639)(433,638)(434,637)(435,636)(436,635)(437,634)(438,633)(439,632)(440,631)(441,630)(442,629)(443,628)(444,627)(445,626)(446,625)(447,624)(448,623)(449,622)(450,621)(451,620)(452,619)(453,618)(454,617)(455,616)(456,615)(457,614)(458,613)(459,612)(460,611)(461,610)(462,609)(463,608)(464,607)(465,606)(466,605)(467,604)(468,603)(469,602)(470,601)(471,600)(472,599)(473,598)(474,597)(475,596)(476,595)(477,594)(478,593)(479,592)(480,591)(481,590)(482,589)(483,588)(484,587)(485,586)(486,585)(487,584)(488,583)(489,582)(490,581)(491,580)(492,579)(493,578)(494,577)(495,576)(496,575)(497,574)(498,573)(499,572)(500,571)(501,570)(502,569)(503,568)(504,567)(505,566)(506,565)(507,564)(508,563)(509,562)(510,561)(511,560)(512,559)(513,558)(514,557)(515,556)(516,555)(517,554)(518,553)(519,552)(520,551)(521,550)(522,549)(523,548)(524,547)(525,546)(526,545)(527,544)(528,543)(529,542)(530,541)(531,540)(532,539)(533,538)(534,537)(535,536)Polycyclic presentation:
2^3,7,1039|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^103814549 conjugacy classes. The character table is too large to display.
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, 1039]);;
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.5^1038);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);