Finite Group DatabasebetaFinite Group Databasebeta| order | 60540 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic15135 |
| derived, Hall | C3×C5×C1009 ⋊ C4 |
| Hall | C5×C1009 ⋊ Dic3 |
| Hall | C3×C1009 ⋊ Dic5 |
| Sylow | C1009 ⋊ Dic15 |
| Hall | C3×C5 ⋊ Dic1009 |
| Sylow | C5 ⋊ Dic3027 |
| Sylow | C3 ⋊ Dic5045 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D15135 |
| commutator, hypocenter | C3×C5×C1009 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C3×C5×C1009 | C2 |
| LCS | G ▷ C3×C5×C1009 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1021 |
| transitive degree | μt(G) = 60540 |
A minimal faithful permutation representation — 2 generators on 1021 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,1021)(15,1020)(16,1019)(17,1018)(18,1017)(19,1016)(20,1015)(21,1014)(22,1013)(23,1012)(24,1011)(25,1010)(26,1009)(27,1008)(28,1007)(29,1006)(30,1005)(31,1004)(32,1003)(33,1002)(34,1001)(35,1000)(36,999)(37,998)(38,997)(39,996)(40,995)(41,994)(42,993)(43,992)(44,991)(45,990)(46,989)(47,988)(48,987)(49,986)(50,985)(51,984)(52,983)(53,982)(54,981)(55,980)(56,979)(57,978)(58,977)(59,976)(60,975)(61,974)(62,973)(63,972)(64,971)(65,970)(66,969)(67,968)(68,967)(69,966)(70,965)(71,964)(72,963)(73,962)(74,961)(75,960)(76,959)(77,958)(78,957)(79,956)(80,955)(81,954)(82,953)(83,952)(84,951)(85,950)(86,949)(87,948)(88,947)(89,946)(90,945)(91,944)(92,943)(93,942)(94,941)(95,940)(96,939)(97,938)(98,937)(99,936)(100,935)(101,934)(102,933)(103,932)(104,931)(105,930)(106,929)(107,928)(108,927)(109,926)(110,925)(111,924)(112,923)(113,922)(114,921)(115,920)(116,919)(117,918)(118,917)(119,916)(120,915)(121,914)(122,913)(123,912)(124,911)(125,910)(126,909)(127,908)(128,907)(129,906)(130,905)(131,904)(132,903)(133,902)(134,901)(135,900)(136,899)(137,898)(138,897)(139,896)(140,895)(141,894)(142,893)(143,892)(144,891)(145,890)(146,889)(147,888)(148,887)(149,886)(150,885)(151,884)(152,883)(153,882)(154,881)(155,880)(156,879)(157,878)(158,877)(159,876)(160,875)(161,874)(162,873)(163,872)(164,871)(165,870)(166,869)(167,868)(168,867)(169,866)(170,865)(171,864)(172,863)(173,862)(174,861)(175,860)(176,859)(177,858)(178,857)(179,856)(180,855)(181,854)(182,853)(183,852)(184,851)(185,850)(186,849)(187,848)(188,847)(189,846)(190,845)(191,844)(192,843)(193,842)(194,841)(195,840)(196,839)(197,838)(198,837)(199,836)(200,835)(201,834)(202,833)(203,832)(204,831)(205,830)(206,829)(207,828)(208,827)(209,826)(210,825)(211,824)(212,823)(213,822)(214,821)(215,820)(216,819)(217,818)(218,817)(219,816)(220,815)(221,814)(222,813)(223,812)(224,811)(225,810)(226,809)(227,808)(228,807)(229,806)(230,805)(231,804)(232,803)(233,802)(234,801)(235,800)(236,799)(237,798)(238,797)(239,796)(240,795)(241,794)(242,793)(243,792)(244,791)(245,790)(246,789)(247,788)(248,787)(249,786)(250,785)(251,784)(252,783)(253,782)(254,781)(255,780)(256,779)(257,778)(258,777)(259,776)(260,775)(261,774)(262,773)(263,772)(264,771)(265,770)(266,769)(267,768)(268,767)(269,766)(270,765)(271,764)(272,763)(273,762)(274,761)(275,760)(276,759)(277,758)(278,757)(279,756)(280,755)(281,754)(282,753)(283,752)(284,751)(285,750)(286,749)(287,748)(288,747)(289,746)(290,745)(291,744)(292,743)(293,742)(294,741)(295,740)(296,739)(297,738)(298,737)(299,736)(300,735)(301,734)(302,733)(303,732)(304,731)(305,730)(306,729)(307,728)(308,727)(309,726)(310,725)(311,724)(312,723)(313,722)(314,721)(315,720)(316,719)(317,718)(318,717)(319,716)(320,715)(321,714)(322,713)(323,712)(324,711)(325,710)(326,709)(327,708)(328,707)(329,706)(330,705)(331,704)(332,703)(333,702)(334,701)(335,700)(336,699)(337,698)(338,697)(339,696)(340,695)(341,694)(342,693)(343,692)(344,691)(345,690)(346,689)(347,688)(348,687)(349,686)(350,685)(351,684)(352,683)(353,682)(354,681)(355,680)(356,679)(357,678)(358,677)(359,676)(360,675)(361,674)(362,673)(363,672)(364,671)(365,670)(366,669)(367,668)(368,667)(369,666)(370,665)(371,664)(372,663)(373,662)(374,661)(375,660)(376,659)(377,658)(378,657)(379,656)(380,655)(381,654)(382,653)(383,652)(384,651)(385,650)(386,649)(387,648)(388,647)(389,646)(390,645)(391,644)(392,643)(393,642)(394,641)(395,640)(396,639)(397,638)(398,637)(399,636)(400,635)(401,634)(402,633)(403,632)(404,631)(405,630)(406,629)(407,628)(408,627)(409,626)(410,625)(411,624)(412,623)(413,622)(414,621)(415,620)(416,619)(417,618)(418,617)(419,616)(420,615)(421,614)(422,613)(423,612)(424,611)(425,610)(426,609)(427,608)(428,607)(429,606)(430,605)(431,604)(432,603)(433,602)(434,601)(435,600)(436,599)(437,598)(438,597)(439,596)(440,595)(441,594)(442,593)(443,592)(444,591)(445,590)(446,589)(447,588)(448,587)(449,586)(450,585)(451,584)(452,583)(453,582)(454,581)(455,580)(456,579)(457,578)(458,577)(459,576)(460,575)(461,574)(462,573)(463,572)(464,571)(465,570)(466,569)(467,568)(468,567)(469,566)(470,565)(471,564)(472,563)(473,562)(474,561)(475,560)(476,559)(477,558)(478,557)(479,556)(480,555)(481,554)(482,553)(483,552)(484,551)(485,550)(486,549)(487,548)(488,547)(489,546)(490,545)(491,544)(492,543)(493,542)(494,541)(495,540)(496,539)(497,538)(498,537)(499,536)(500,535)(501,534)(502,533)(503,532)(504,531)(505,530)(506,529)(507,528)(508,527)(509,526)(510,525)(511,524)(512,523)(513,522)(514,521)(515,520)(516,519)(517,518)Polycyclic presentation:
2^2,3,5,1009|a^2=b,c^a=c^2,d^a=d^4,e^a=e^100815138 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, 3, 5, 1009]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^1008);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);