Finite Group DatabasebetaFinite Group Databasebeta| order | 59864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic14966 |
| Hall | C7×C1069 ⋊ Q8 |
| Sylow | C1069 ⋊ Dic14 |
| Sylow | C7 ⋊ Dic2138 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7483 |
| commutator, socle, prime radical, cosocle | C2×C7×C1069 | C22 |
| Fitting | C4×C7×C1069 | C2 |
| hypercenter | C4 | D7483 |
| hypocenter | C7×C1069 | Q8 |
| LCS | G ▷ C2×C7×C1069 ▷ C7×C1069 | |
| UCS | C1 ◁ C2 ◁ C4 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1084 |
| transitive degree | μt(G) = 59864 |
A minimal faithful permutation representation — 2 generators on 1084 points:
(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,1084)(18,1083)(19,1082)(20,1081)(21,1080)(22,1079)(23,1078)(24,1077)(25,1076)(26,1075)(27,1074)(28,1073)(29,1072)(30,1071)(31,1070)(32,1069)(33,1068)(34,1067)(35,1066)(36,1065)(37,1064)(38,1063)(39,1062)(40,1061)(41,1060)(42,1059)(43,1058)(44,1057)(45,1056)(46,1055)(47,1054)(48,1053)(49,1052)(50,1051)(51,1050)(52,1049)(53,1048)(54,1047)(55,1046)(56,1045)(57,1044)(58,1043)(59,1042)(60,1041)(61,1040)(62,1039)(63,1038)(64,1037)(65,1036)(66,1035)(67,1034)(68,1033)(69,1032)(70,1031)(71,1030)(72,1029)(73,1028)(74,1027)(75,1026)(76,1025)(77,1024)(78,1023)(79,1022)(80,1021)(81,1020)(82,1019)(83,1018)(84,1017)(85,1016)(86,1015)(87,1014)(88,1013)(89,1012)(90,1011)(91,1010)(92,1009)(93,1008)(94,1007)(95,1006)(96,1005)(97,1004)(98,1003)(99,1002)(100,1001)(101,1000)(102,999)(103,998)(104,997)(105,996)(106,995)(107,994)(108,993)(109,992)(110,991)(111,990)(112,989)(113,988)(114,987)(115,986)(116,985)(117,984)(118,983)(119,982)(120,981)(121,980)(122,979)(123,978)(124,977)(125,976)(126,975)(127,974)(128,973)(129,972)(130,971)(131,970)(132,969)(133,968)(134,967)(135,966)(136,965)(137,964)(138,963)(139,962)(140,961)(141,960)(142,959)(143,958)(144,957)(145,956)(146,955)(147,954)(148,953)(149,952)(150,951)(151,950)(152,949)(153,948)(154,947)(155,946)(156,945)(157,944)(158,943)(159,942)(160,941)(161,940)(162,939)(163,938)(164,937)(165,936)(166,935)(167,934)(168,933)(169,932)(170,931)(171,930)(172,929)(173,928)(174,927)(175,926)(176,925)(177,924)(178,923)(179,922)(180,921)(181,920)(182,919)(183,918)(184,917)(185,916)(186,915)(187,914)(188,913)(189,912)(190,911)(191,910)(192,909)(193,908)(194,907)(195,906)(196,905)(197,904)(198,903)(199,902)(200,901)(201,900)(202,899)(203,898)(204,897)(205,896)(206,895)(207,894)(208,893)(209,892)(210,891)(211,890)(212,889)(213,888)(214,887)(215,886)(216,885)(217,884)(218,883)(219,882)(220,881)(221,880)(222,879)(223,878)(224,877)(225,876)(226,875)(227,874)(228,873)(229,872)(230,871)(231,870)(232,869)(233,868)(234,867)(235,866)(236,865)(237,864)(238,863)(239,862)(240,861)(241,860)(242,859)(243,858)(244,857)(245,856)(246,855)(247,854)(248,853)(249,852)(250,851)(251,850)(252,849)(253,848)(254,847)(255,846)(256,845)(257,844)(258,843)(259,842)(260,841)(261,840)(262,839)(263,838)(264,837)(265,836)(266,835)(267,834)(268,833)(269,832)(270,831)(271,830)(272,829)(273,828)(274,827)(275,826)(276,825)(277,824)(278,823)(279,822)(280,821)(281,820)(282,819)(283,818)(284,817)(285,816)(286,815)(287,814)(288,813)(289,812)(290,811)(291,810)(292,809)(293,808)(294,807)(295,806)(296,805)(297,804)(298,803)(299,802)(300,801)(301,800)(302,799)(303,798)(304,797)(305,796)(306,795)(307,794)(308,793)(309,792)(310,791)(311,790)(312,789)(313,788)(314,787)(315,786)(316,785)(317,784)(318,783)(319,782)(320,781)(321,780)(322,779)(323,778)(324,777)(325,776)(326,775)(327,774)(328,773)(329,772)(330,771)(331,770)(332,769)(333,768)(334,767)(335,766)(336,765)(337,764)(338,763)(339,762)(340,761)(341,760)(342,759)(343,758)(344,757)(345,756)(346,755)(347,754)(348,753)(349,752)(350,751)(351,750)(352,749)(353,748)(354,747)(355,746)(356,745)(357,744)(358,743)(359,742)(360,741)(361,740)(362,739)(363,738)(364,737)(365,736)(366,735)(367,734)(368,733)(369,732)(370,731)(371,730)(372,729)(373,728)(374,727)(375,726)(376,725)(377,724)(378,723)(379,722)(380,721)(381,720)(382,719)(383,718)(384,717)(385,716)(386,715)(387,714)(388,713)(389,712)(390,711)(391,710)(392,709)(393,708)(394,707)(395,706)(396,705)(397,704)(398,703)(399,702)(400,701)(401,700)(402,699)(403,698)(404,697)(405,696)(406,695)(407,694)(408,693)(409,692)(410,691)(411,690)(412,689)(413,688)(414,687)(415,686)(416,685)(417,684)(418,683)(419,682)(420,681)(421,680)(422,679)(423,678)(424,677)(425,676)(426,675)(427,674)(428,673)(429,672)(430,671)(431,670)(432,669)(433,668)(434,667)(435,666)(436,665)(437,664)(438,663)(439,662)(440,661)(441,660)(442,659)(443,658)(444,657)(445,656)(446,655)(447,654)(448,653)(449,652)(450,651)(451,650)(452,649)(453,648)(454,647)(455,646)(456,645)(457,644)(458,643)(459,642)(460,641)(461,640)(462,639)(463,638)(464,637)(465,636)(466,635)(467,634)(468,633)(469,632)(470,631)(471,630)(472,629)(473,628)(474,627)(475,626)(476,625)(477,624)(478,623)(479,622)(480,621)(481,620)(482,619)(483,618)(484,617)(485,616)(486,615)(487,614)(488,613)(489,612)(490,611)(491,610)(492,609)(493,608)(494,607)(495,606)(496,605)(497,604)(498,603)(499,602)(500,601)(501,600)(502,599)(503,598)(504,597)(505,596)(506,595)(507,594)(508,593)(509,592)(510,591)(511,590)(512,589)(513,588)(514,587)(515,586)(516,585)(517,584)(518,583)(519,582)(520,581)(521,580)(522,579)(523,578)(524,577)(525,576)(526,575)(527,574)(528,573)(529,572)(530,571)(531,570)(532,569)(533,568)(534,567)(535,566)(536,565)(537,564)(538,563)(539,562)(540,561)(541,560)(542,559)(543,558)(544,557)(545,556)(546,555)(547,554)(548,553)(549,552)(550,551)Polycyclic presentation:
2^3,7,1069|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^106814969 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, 1069]);;
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.5^1068);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);