C₅×C₈₅₉⋊C₃ #27166275

order	12885
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₈₅₉⋊C₃
unambiguous semidirect product	C₈₅₉⋊(C₃×C₅)
unambiguous semidirect product	(C₅×C₈₅₉)⋊C₃

Splittings

Fitting, socle, Hall	C₅×C₈₅₉ ⋊ C₃
derived, Sylow	C₈₅₉ ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₈₅₉⋊C₃
commutator, cosocle, hypocenter	C₈₅₉	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₈₅₉	C₃
LCS	G ▷ C₈₅₉
UCS	C₁ ◁ C₅

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 864
transitive degree	μ_t(G) = 4295

A minimal faithful permutation representation — 2 generators on 864 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,330)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(290,332,376)(293,335,373)(296,338,370)(299,341,367)(302,344,364)(305,347,361)(308,350,358)(311,353,355)(314,356,352)(317,359,349)(320,362,346)(323,365,343)(326,368,340)(329,371,337)(331,374,334)(333,378,425)(336,381,422)(339,384,419)(342,387,416)(345,390,413)(348,393,410)(351,396,407)(354,399,404)(357,402,401)(360,405,398)(363,408,395)(366,411,392)(369,414,389)(372,417,386)(375,420,383)(377,423,380)(379,427,477)(382,430,474)(385,433,471)(388,436,468)(391,439,465)(394,442,462)(397,445,459)(400,448,456)(403,451,453)(406,454,450)(409,457,447)(412,460,444)(415,463,441)(418,466,438)(421,469,435)(424,472,432)(426,475,429)(428,479,531)(431,482,528)(434,485,525)(437,488,522)(440,480,519)(443,493,516)(446,496,513)(449,499,510)(452,502,507)(455,505,504)(458,508,501)(461,511,498)(464,514,495)(467,517,492)(470,520,490)(473,523,487)(476,526,484)(478,529,481)(483,534,582)(486,537,579)(489,540,576)(491,535,573)(494,545,570)(497,547,567)(500,550,564)(503,553,561)(506,556,558)(509,559,555)(512,562,552)(515,565,549)(518,568,546)(521,571,544)(524,574,542)(527,577,539)(530,580,536)(532,583,533)(538,586,628)(541,589,625)(543,587,622)(548,595,617)(551,597,614)(554,600,611)(557,603,608)(560,606,605)(563,609,602)(566,612,599)(569,615,596)(572,618,594)(575,620,593)(578,623,591)(581,626,588)(584,629,585)(590,632,668)(592,633,665)(598,639,658)(601,641,655)(604,644,652)(607,647,649)(610,650,646)(613,653,643)(616,656,640)(619,659,638)(621,661,637)(624,663,636)(627,666,634)(630,669,631)(635,672,702)(642,677,693)(645,679,690)(648,682,687)(651,685,684)(654,688,681)(657,691,678)(660,694,676)(662,696,675)(664,698,674)(667,700,673)(670,703,671)(680,710,723)(683,712,720)(686,715,717)(689,718,714)(692,721,711)(695,724,709)(697,726,708)(699,728,707)(701,730,706)(704,732,705)(713,739,749)(716,741,746)(719,744,743)(722,747,740)(725,750,738)(727,752,737)(729,754,736)(731,756,735)(733,758,734)(742,765,772)(745,767,769)(748,770,766)(751,773,764)(753,775,763)(755,777,762)(757,779,761)(759,781,760)(768,788,792)(771,790,789)(774,793,787)(776,795,786)(778,797,785)(780,799,784)(782,801,783)(791,808,809)(794,810,807)(796,812,806)(798,814,805)(800,816,804)(802,818,803)(811,825,824)(813,826,823)(815,828,822)(817,830,821)(819,832,820)(827,838,837)(829,839,836)(831,841,835)(833,843,834)(840,848,847)(842,849,846)(844,851,845)(850,855,854)(852,856,853)(857,859,858)(860,861,862,863,864);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,428,480,387,295,215,147,177,209,243,280,320,363,409,458,509,560,607,648,683,713,732,666,577,472,374,287,327,370,416,465,516,567,614,655,690,720,746,769,789,807,823,836,846,853,855,839,812,773,721,653,562,457,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,435,487,539,588,631,639,547,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,500,551,598,629,529,425,474,525,576,622,534,427,478,530,581,627,667,701,731,757,780,800,817,831,842,850,851,830,797,752,694,615,514,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,444,495,546,594,637,674,706,734,739,679,600,499,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,440,491,543,592,635,672,589,485,381,289,331,375,421,470,521,572,619,660,695,725,751,774,794,811,818,779,728,661,571,466,368,282,208,145,173,204,238,275,315,358,404,453,504,555,602,643,678,709,737,762,784,803,808,767,715,647,556,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,535,436,338,252,178,211,246,283,323,366,412,461,512,563,610,651,686,716,742,758,700,623,523,420,328,247,284,324,367,413,462,513,564,611,652,687,717,743,766,787,806,822,835,845,848,826,793,747,688,609,508,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,536,585,595,493,390,298,218,150,94,118,144,172,203,237,274,314,357,403,452,503,554,601,642,669,580,475,376,422,471,522,573,479,532,584,630,670,704,733,759,782,802,819,833,844,852,857,856,841,814,775,724,656,565,460,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,441,492,544,593,636,673,705,710,641,550,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,494,531,582,628,668,702,632,537,430,332,377,424,473,524,575,621,662,697,727,753,776,796,813,827,832,799,754,696,618,517,414,322,242,174,117,142,170,201,235,272,312,355,401,450,501,552,599,640,676,708,736,761,783,788,741,682,603,502,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,538,587,488,384,292,212,248,286,326,369,415,464,515,566,613,654,689,719,745,768,781,730,663,574,469,371,285,210,244,281,321,364,410,459,510,561,608,649,684,714,740,764,786,805,821,834,838,810,770,718,650,559,454,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,429,481,533,545,439,341,255,181,119,146,175,206,240,277,317,360,406,455,506,557,604,645,680,703,626,526,423,330,373,419,468,519,570,617,658,693,723,749,772,792,809,824,837,847,854,858,859,849,828,795,750,691,612,511,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,542,591,634,671,677,597,496,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,446,497,548,583,477,528,579,625,665,586,482,378,426,476,527,578,624,664,699,729,755,778,798,815,829,840,843,816,777,726,659,568,463,365,279,205,143,92,114,139,167,198,232,269,309,352,398,447,498,549,596,638,675,707,735,760,765,712,644,553,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,541,590,633,540,433,335,249,288,329,372,418,467,518,569,616,657,692,722,748,771,791,801,756,698,620,520,417,325,245,176,207,241,278,318,361,407,456,507,558,605,646,681,711,738,763,785,804,820,825,790,744,685,606,505,402,310,230,162,106,62,30,10)

Conjugacy classes & character table

1445 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₅×C₈₅₉⋊C₃, C₅×D₈₅₉⋊C₃, D₅×C₈₅₉⋊C₃, …
maximal normal subgroup of	C₂×C₅×C₈₅₉⋊C₃, C₅×D₈₅₉⋊C₃, D₅×C₈₅₉⋊C₃, …
maximal quotient of	C₂×C₅×C₈₅₉⋊C₃, C₅×C₈₅₉⋊C₉, C₃×C₅×C₈₅₉⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12885, 1);

C5×C859⋊C3 #27166275