Finite Group DatabasebetaFinite Group Databasebeta| order | 57736 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic14434 |
| Hall | C7×C1031 ⋊ Q8 |
| Sylow | C1031 ⋊ Dic14 |
| Sylow | C7 ⋊ Dic2062 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7217 |
| commutator, socle, prime radical, cosocle | C2×C7×C1031 | C22 |
| Fitting | C4×C7×C1031 | C2 |
| hypercenter | C4 | D7217 |
| hypocenter | C7×C1031 | Q8 |
| LCS | G ▷ C2×C7×C1031 ▷ C7×C1031 | |
| UCS | C1 ◁ C2 ◁ C4 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1046 |
| transitive degree | μt(G) = 57736 |
A minimal faithful permutation representation — 2 generators on 1046 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);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,1046)(18,1045)(19,1044)(20,1043)(21,1042)(22,1041)(23,1040)(24,1039)(25,1038)(26,1037)(27,1036)(28,1035)(29,1034)(30,1033)(31,1032)(32,1031)(33,1030)(34,1029)(35,1028)(36,1027)(37,1026)(38,1025)(39,1024)(40,1023)(41,1022)(42,1021)(43,1020)(44,1019)(45,1018)(46,1017)(47,1016)(48,1015)(49,1014)(50,1013)(51,1012)(52,1011)(53,1010)(54,1009)(55,1008)(56,1007)(57,1006)(58,1005)(59,1004)(60,1003)(61,1002)(62,1001)(63,1000)(64,999)(65,998)(66,997)(67,996)(68,995)(69,994)(70,993)(71,992)(72,991)(73,990)(74,989)(75,988)(76,987)(77,986)(78,985)(79,984)(80,983)(81,982)(82,981)(83,980)(84,979)(85,978)(86,977)(87,976)(88,975)(89,974)(90,973)(91,972)(92,971)(93,970)(94,969)(95,968)(96,967)(97,966)(98,965)(99,964)(100,963)(101,962)(102,961)(103,960)(104,959)(105,958)(106,957)(107,956)(108,955)(109,954)(110,953)(111,952)(112,951)(113,950)(114,949)(115,948)(116,947)(117,946)(118,945)(119,944)(120,943)(121,942)(122,941)(123,940)(124,939)(125,938)(126,937)(127,936)(128,935)(129,934)(130,933)(131,932)(132,931)(133,930)(134,929)(135,928)(136,927)(137,926)(138,925)(139,924)(140,923)(141,922)(142,921)(143,920)(144,919)(145,918)(146,917)(147,916)(148,915)(149,914)(150,913)(151,912)(152,911)(153,910)(154,909)(155,908)(156,907)(157,906)(158,905)(159,904)(160,903)(161,902)(162,901)(163,900)(164,899)(165,898)(166,897)(167,896)(168,895)(169,894)(170,893)(171,892)(172,891)(173,890)(174,889)(175,888)(176,887)(177,886)(178,885)(179,884)(180,883)(181,882)(182,881)(183,880)(184,879)(185,878)(186,877)(187,876)(188,875)(189,874)(190,873)(191,872)(192,871)(193,870)(194,869)(195,868)(196,867)(197,866)(198,865)(199,864)(200,863)(201,862)(202,861)(203,860)(204,859)(205,858)(206,857)(207,856)(208,855)(209,854)(210,853)(211,852)(212,851)(213,850)(214,849)(215,848)(216,847)(217,846)(218,845)(219,844)(220,843)(221,842)(222,841)(223,840)(224,839)(225,838)(226,837)(227,836)(228,835)(229,834)(230,833)(231,832)(232,831)(233,830)(234,829)(235,828)(236,827)(237,826)(238,825)(239,824)(240,823)(241,822)(242,821)(243,820)(244,819)(245,818)(246,817)(247,816)(248,815)(249,814)(250,813)(251,812)(252,811)(253,810)(254,809)(255,808)(256,807)(257,806)(258,805)(259,804)(260,803)(261,802)(262,801)(263,800)(264,799)(265,798)(266,797)(267,796)(268,795)(269,794)(270,793)(271,792)(272,791)(273,790)(274,789)(275,788)(276,787)(277,786)(278,785)(279,784)(280,783)(281,782)(282,781)(283,780)(284,779)(285,778)(286,777)(287,776)(288,775)(289,774)(290,773)(291,772)(292,771)(293,770)(294,769)(295,768)(296,767)(297,766)(298,765)(299,764)(300,763)(301,762)(302,761)(303,760)(304,759)(305,758)(306,757)(307,756)(308,755)(309,754)(310,753)(311,752)(312,751)(313,750)(314,749)(315,748)(316,747)(317,746)(318,745)(319,744)(320,743)(321,742)(322,741)(323,740)(324,739)(325,738)(326,737)(327,736)(328,735)(329,734)(330,733)(331,732)(332,731)(333,730)(334,729)(335,728)(336,727)(337,726)(338,725)(339,724)(340,723)(341,722)(342,721)(343,720)(344,719)(345,718)(346,717)(347,716)(348,715)(349,714)(350,713)(351,712)(352,711)(353,710)(354,709)(355,708)(356,707)(357,706)(358,705)(359,704)(360,703)(361,702)(362,701)(363,700)(364,699)(365,698)(366,697)(367,696)(368,695)(369,694)(370,693)(371,692)(372,691)(373,690)(374,689)(375,688)(376,687)(377,686)(378,685)(379,684)(380,683)(381,682)(382,681)(383,680)(384,679)(385,678)(386,677)(387,676)(388,675)(389,674)(390,673)(391,672)(392,671)(393,670)(394,669)(395,668)(396,667)(397,666)(398,665)(399,664)(400,663)(401,662)(402,661)(403,660)(404,659)(405,658)(406,657)(407,656)(408,655)(409,654)(410,653)(411,652)(412,651)(413,650)(414,649)(415,648)(416,647)(417,646)(418,645)(419,644)(420,643)(421,642)(422,641)(423,640)(424,639)(425,638)(426,637)(427,636)(428,635)(429,634)(430,633)(431,632)(432,631)(433,630)(434,629)(435,628)(436,627)(437,626)(438,625)(439,624)(440,623)(441,622)(442,621)(443,620)(444,619)(445,618)(446,617)(447,616)(448,615)(449,614)(450,613)(451,612)(452,611)(453,610)(454,609)(455,608)(456,607)(457,606)(458,605)(459,604)(460,603)(461,602)(462,601)(463,600)(464,599)(465,598)(466,597)(467,596)(468,595)(469,594)(470,593)(471,592)(472,591)(473,590)(474,589)(475,588)(476,587)(477,586)(478,585)(479,584)(480,583)(481,582)(482,581)(483,580)(484,579)(485,578)(486,577)(487,576)(488,575)(489,574)(490,573)(491,572)(492,571)(493,570)(494,569)(495,568)(496,567)(497,566)(498,565)(499,564)(500,563)(501,562)(502,561)(503,560)(504,559)(505,558)(506,557)(507,556)(508,555)(509,554)(510,553)(511,552)(512,551)(513,550)(514,549)(515,548)(516,547)(517,546)(518,545)(519,544)(520,543)(521,542)(522,541)(523,540)(524,539)(525,538)(526,537)(527,536)(528,535)(529,534)(530,533)(531,532)Polycyclic presentation:
2^3,7,1031|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^103014437 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, 1031]);;
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.5^1030);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);