C₃×D₅₀₆₅ #44477237

order	30390
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅₀₆₅
unambiguous semidirect product	C₅⋊(C₃×D₁₀₁₃)
unambiguous semidirect product	C₁₀₁₃⋊(C₃×D₅)

Splittings

Fitting, socle, Hall	C₃×C₅×C₁₀₁₃ ⋊ C₂
derived, Hall	C₅×C₁₀₁₃ ⋊ C₂×C₃
Hall	C₃×C₁₀₁₃ ⋊ D₅
Sylow	C₁₀₁₃ ⋊ C₃×D₅
Hall	C₃×C₅ ⋊ D₁₀₁₃
Sylow	C₅ ⋊ C₃×D₁₀₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₅₀₆₅
commutator, cosocle, hypocenter	C₅×C₁₀₁₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₀₁₃	C₂
LCS	G ▷ C₅×C₁₀₁₃
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1021
transitive degree	μ_t(G) = 15195

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

Conjugacy classes & character table

7602 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30390, 4);

C3×D5065 #44477237