C₃×C₈₈₁⋊C₅ #27210228

order	13215
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₈₈₁⋊C₅
unambiguous semidirect product	C₈₈₁⋊(C₃×C₅)
unambiguous semidirect product	(C₃×C₈₈₁)⋊C₅

Splittings

Fitting, socle, Hall	C₃×C₈₈₁ ⋊ C₅
derived, Sylow	C₈₈₁ ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₈₈₁⋊C₅
commutator, cosocle, hypocenter	C₈₈₁	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₈₈₁	C₅
LCS	G ▷ C₈₈₁
UCS	C₁ ◁ C₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 884
transitive degree	μ_t(G) = 2643

A minimal faithful permutation representation — 2 generators on 884 points:

(2,3,5,9,17)(4,7,13,25,48)(6,11,21,40,33)(8,15,29,56,105)(10,19,36,18,34)(12,23,44,82,90)(14,27,52,97,64)(16,31,60,113,149)(20,38,71,49,91)(22,42,78,75,65)(24,46,86,154,187)(26,50,93,35,66)(28,54,101,179,162)(30,58,109,194,121)(32,62,117,208,342)(37,69,41,76,68)(39,73,133,106,188)(43,80,143,147,163)(45,84,150,172,122)(47,88,158,267,252)(51,95,168,92,164)(53,99,175,136,123)(55,103,183,303,311)(57,107,190,67,124)(59,111,198,328,275)(61,115,204,298,216)(63,119,212,349,235)(70,128,83,148,166)(72,131,98,173,126)(74,135,234,201,255)(77,138,130,94,167)(79,141,140,137,127)(81,145,247,260,312)(85,152,256,296,276)(87,156,263,321,217)(89,160,271,421,511)(96,170,283,189,313)(100,177,292,250,277)(102,181,299,286,218)(104,185,307,466,396)(108,192,317,165,278)(110,196,324,236,219)(112,200,331,499,473)(114,202,262,125,220)(116,206,339,496,429)(118,210,345,514,357)(120,214,353,526,560)(129,225,155,261,315)(132,229,180,297,280)(134,232,195,322,222)(139,239,231,191,316)(142,242,244,251,281)(144,245,253,287,223)(146,249,393,414,398)(151,254,290,237,224)(153,258,402,459,474)(157,265,411,494,430)(159,269,417,452,358)(161,273,425,598,379)(169,282,174,288,227)(171,285,438,333,401)(176,291,241,238,228)(178,294,447,405,475)(182,301,455,450,431)(184,305,462,484,359)(186,309,354,527,680)(193,319,481,314,476)(197,326,490,395,432)(199,330,497,440,360)(203,335,416,279,433)(205,337,461,380,361)(207,341,509,426,600)(209,343,512,221,362)(211,347,517,686,606)(213,351,522,439,533)(215,355,529,698,766)(226,367,268,415,407)(230,372,304,460,478)(233,376,329,495,435)(240,384,378,334,410)(243,386,390,406,479)(246,388,399,451,436)(248,391,408,485,364)(257,400,453,441,365)(259,404,575,636,570)(264,409,488,381,366)(266,413,583,670,647)(270,419,589,667,607)(272,423,594,683,534)(274,427,602,752,724)(284,437,323,486,369)(289,443,374,318,480)(293,446,397,442,370)(295,449,620,585,571)(300,454,445,382,371)(302,457,625,628,648)(306,464,633,665,608)(308,468,639,622,535)(310,471,603,753,555)(320,483,654,502,574)(325,489,385,383,375)(327,492,662,577,649)(332,501,428,604,536)(336,504,593,477,650)(338,506,638,568,609)(340,508,672,613,537)(344,513,682,434,610)(346,515,684,556,538)(348,519,688,807,749)(350,520,394,363,539)(352,524,693,655,758)(356,531,701,814,855)(368,543,422,592,681)(373,548,467,637,578)(377,553,500,671,651)(387,561,566,586,580)(389,562,572,629,652)(392,564,581,666,612)(403,573,631,656,540)(412,582,668,614,541)(418,587,630,557,542)(420,591,740,601,750)(424,596,744,805,759)(444,617,554,503,588)(448,619,579,657,545)(456,624,623,615,546)(458,627,768,775,729)(463,632,660,558,547)(465,635,773,796,785)(469,641,778,794,760)(470,530,699,802,704)(472,645,783,854,828)(482,653,487,658,550)(491,661,569,616,551)(493,664,792,738,730)(498,669,618,559,552)(505,676,748,674,732)(507,677,697,733,786)(510,679,644,755,705)(516,685,804,728,761)(518,687,806,763,706)(521,690,576,611,762)(523,691,621,720,707)(525,695,811,799,836)(528,696,801,549,715)(532,703,817,863,812)(544,711,599,747,567)(563,725,731,776,735)(565,726,736,797,787)(584,737,605,756,708)(590,739,798,764,709)(595,742,803,721,710)(597,746,834,860,837)(626,767,771,788,712)(634,772,795,765,713)(640,777,770,722,714)(642,780,851,751,838)(643,702,815,858,844)(646,784,723,718,673)(659,790,719,675,743)(663,791,734,789,717)(678,800,840,831,830)(689,808,861,841,819)(692,809,769,829,845)(694,810,793,846,820)(700,813,857,827,825)(716,826,754,839,727)(741,832,782,842,821)(745,833,859,847,822)(757,843,823,774,849)(779,850,856,848,824)(781,818,864,876,870)(816,862,875,873,868)(835,869,877,871,866)(852,874,853,872,867)(865,878,881,880,879)(882,883,884);(1,2,4,8,16,32,63,120,215,356,532,564,27,53,100,178,295,80,144,246,389,563,170,284,206,340,462,631,771,413,584,637,776,768,627,769,695,812,849,256,399,514,683,802,409,25,49,92,165,279,434,611,86,155,262,407,578,181,300,212,350,521,326,491,423,595,743,504,675,641,779,699,316,479,652,787,839,486,657,788,843,872,880,878,589,711,807,860,876,742,113,201,333,502,674,443,616,331,500,672,798,856,508,427,603,754,840,685,225,366,541,708,490,384,560,724,828,669,97,172,286,440,613,763,846,283,88,159,270,420,446,36,68,126,222,349,76,137,237,381,557,721,688,232,375,551,717,810,388,11,22,43,81,146,38,72,132,230,373,549,716,111,199,303,459,628,258,403,574,730,625,449,621,519,689,772,150,253,397,569,698,261,406,393,566,682,650,786,573,44,83,149,252,396,99,176,117,209,344,192,318,269,418,588,610,761,464,634,527,188,312,474,647,750,837,851,800,703,818,865,862,417,539,707,820,863,592,164,277,431,608,760,844,832,198,329,496,667,794,337,214,354,528,697,513,124,219,360,536,324,488,660,784,854,495,666,654,548,714,824,867,791,168,31,61,116,207,291,17,33,64,121,216,357,533,309,470,643,781,517,131,228,370,545,691,242,3,6,12,24,47,89,161,274,428,605,757,726,54,102,182,302,458,145,248,392,565,455,285,439,341,510,632,82,147,250,395,568,728,829,247,390,512,681,801,400,21,41,77,139,240,50,94,60,114,203,336,505,156,264,410,580,735,301,456,351,523,692,492,663,596,745,778,676,799,780,852,813,267,414,585,738,831,658,789,583,736,806,859,875,739,109,195,323,487,659,202,334,307,467,638,617,766,501,673,602,194,321,484,604,755,841,817,481,367,542,709,821,661,93,166,280,435,598,173,287,441,614,764,847,834,437,160,272,424,597,345,69,127,223,364,520,138,238,382,558,722,827,808,376,552,499,670,796,562,23,45,85,153,259,73,134,233,377,299,453,623,200,332,460,629,620,404,576,732,830,767,143,244,342,511,680,254,9,18,35,67,125,221,363,29,57,108,193,320,84,151,255,398,570,177,293,210,346,516,319,482,419,590,639,762,845,635,774,696,313,475,648,785,838,870,874,411,581,686,805,858,587,56,106,189,314,477,278,432,183,304,461,630,770,330,498,425,599,748,506,355,530,700,701,317,220,361,537,705,489,48,90,162,275,429,606,758,715,825,868,879,744,282,62,118,211,348,208,34,65,122,217,358,534,509,471,644,782,853,687,229,371,546,712,809,386,7,14,28,55,104,186,310,472,646,179,296,450,103,184,306,465,447,249,394,567,727,624,78,140,241,385,526,148,251,234,378,554,719,811,391,13,26,51,96,171,42,79,142,243,387,95,169,115,205,338,507,678,265,412,466,636,775,457,626,524,694,773,664,793,746,835,850,263,408,579,734,814,415,586,575,731,804,790,855,737,101,180,298,452,622,196,325,271,422,593,335,503,468,640,529,190,315,478,651,753,322,485,656,756,842,871,864,653,543,710,822,866,594,167,281,436,612,747,288,442,615,765,848,873,869,339,273,426,601,751,515,128,224,365,540,690,239,383,559,723,497,668,795,553,718,671,797,792,725,46,87,157,266,292,135,235,379,555,454,40,75,136,236,380,556,720,133,231,374,550,693,245,5,10,20,39,74,19,37,70,129,226,368,544,58,110,197,327,493,152,257,401,571,729,294,448,347,518,633,483,655,591,741,777,154,260,405,577,733,476,649,402,572,684,803,857,582,52,98,174,289,444,107,191,158,268,416,433,609,305,463,353,105,187,311,473,600,749,836,677,531,702,816,815,480,362,538,706,819,421,91,163,276,430,607,759,740,826,861,877,881,833,204,119,213,352,525,343,66,123,218,359,535,704,679,645,783,328,494,665,372,547,713,823,662,561,15,30,59,112,175,290,445,618,752,297,451,438,185,308,469,642,619,71,130,227,369,522,141)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃×C₈₈₁⋊C₅, C₃×D₈₈₁⋊C₅, D₃×C₈₈₁⋊C₅, …
maximal normal subgroup of	C₂×C₃×C₈₈₁⋊C₅, C₃×D₈₈₁⋊C₅, D₃×C₈₈₁⋊C₅, …
maximal quotient of	C₂×C₃×C₈₈₁⋊C₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13215, 1);

C3×C881⋊C5 #27210228