C₂²×C₆₆₁⋊C₅ #27210247

order	13220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²×C₆₆₁⋊C₅
unambiguous semidirect product	(C₂²×C₆₆₁)⋊C₅

Splittings

Fitting, socle, Hall	C₂²×C₆₆₁ ⋊ C₅
derived, Sylow	C₆₆₁ ⋊ C₂²×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	C₆₆₁⋊C₅
commutator, cosocle, hypocenter	C₆₆₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₆₆₁	C₅
LCS	G ▷ C₆₆₁
UCS	C₁ ◁ C₂²

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 665
transitive degree	μ_t(G) = 2644

A minimal faithful permutation representation — 2 generators on 665 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)(16,31,60,113,199)(20,38,71,49,91)(22,42,78,75,65)(24,46,86,153,185)(26,50,93,35,66)(28,54,101,178,161)(30,58,109,55,103)(32,62,117,206,332)(37,69,41,76,68)(39,73,131,106,186)(43,80,142,146,162)(45,84,149,171,120)(47,88,157,262,322)(51,95,168,92,163)(53,99,174,135,121)(57,107,188,67,122)(59,111,195,223,269)(61,115,184,301,182)(63,119,210,340,290)(70,126,83,147,165)(72,129,98,172,124)(74,133,227,200,323)(77,137,128,94,166)(79,140,139,136,125)(81,144,242,256,303)(85,151,253,292,270)(87,155,258,104,183)(89,159,266,404,478)(96,170,280,187,304)(100,176,289,246,271)(102,180,295,282,212)(108,190,309,164,272)(110,193,152,231,213)(112,197,132,225,192)(114,201,324,123,214)(116,204,329,475,409)(118,208,336,482,299)(127,218,154,257,306)(130,222,179,293,274)(134,229,361,333,479)(138,234,224,189,307)(141,237,239,247,275)(143,240,250,283,216)(145,244,379,398,467)(148,248,277,167,276)(150,251,286,232,217)(156,260,396,353,410)(158,264,400,444,300)(160,268,407,546,430)(169,278,173,284,220)(175,287,236,233,221)(177,291,431,392,446)(181,297,440,433,411)(191,311,453,305,447)(194,314,298,383,412)(196,317,375,423,344)(198,320,243,377,394)(202,326,470,273,369)(203,321,465,364,345)(205,330,281,421,312)(207,334,480,215,346)(209,338,486,490,549)(211,342,491,457,442)(219,350,263,399,468)(226,356,316,357,414)(228,359,302,445,319)(230,363,331,477,432)(235,368,358,325,469)(238,371,376,393,449)(241,374,389,434,415)(245,381,525,542,599)(249,386,420,308,450)(252,387,429,384,416)(254,390,437,424,348)(255,365,349,259,395)(261,370,367,355,313)(265,402,541,597,550)(267,406,544,602,443)(279,419,294,435,417)(285,425,354,310,451)(288,428,373,385,418)(296,438,427,366,352)(315,458,462,534,574)(318,461,533,566,551)(327,472,592,448,575)(328,473,594,528,514)(335,481,601,413,517)(337,484,604,510,495)(339,487,454,578,455)(341,489,441,347,496)(343,493,610,595,637)(351,499,405,543,600)(360,505,476,598,552)(362,507,483,603,463)(372,516,522,538,590)(378,520,536,502,553)(380,523,539,573,464)(382,527,616,633,565)(388,531,564,535,576)(391,532,570,567,554)(397,537,519,560,497)(401,540,588,511,498)(403,518,530,558,456)(408,548,635,581,572)(422,559,459,506,555)(426,561,503,471,591)(436,568,557,452,577)(439,569,563,529,556)(460,521,562,512,501)(466,589,515,513,504)(474,596,649,629,643)(485,605,654,627,622)(488,607,582,586,509)(492,608,585,620,613)(494,611,593,648,580)(500,615,547,634,571)(508,617,606,609,638)(524,623,630,651,639)(526,625,631,652,587)(545,632,653,621,614)(579,644,583,618,640)(584,624,642,628,641)(612,655,650,661,619)(626,657,659,645,647)(636,660,646,658,656)(664,665);(1,2,4,8,16,32,63,36,68,124,197,319,463,586,168,277,417,555,293,434,567,453,144,243,378,521,97,171,282,423,23,45,85,152,255,329,476,222,352,489,594,386,530,628,559,641,660,206,333,137,233,366,475,597,544,631,428,34,65,120,183,300,443,572,166,275,415,553,639,266,405,451,133,228,360,506,616,605,648,516,15,30,59,112,198,321,466,218,349,497,589,105,185,109,192,312,455,580,324,468,396,536,527,626,543,601,425,342,492,609,159,267,262,398,163,271,411,482,602,264,401,450,576,461,585,646,625,60,114,202,327,180,296,439,361,326,471,214,345,44,83,148,249,107,189,308,242,376,323,467,395,25,49,92,164,273,413,338,149,250,119,211,343,494,612,440,570,615,656,260,397,447,574,458,582,645,623,58,110,194,315,176,290,430,359,504,610,655,532,42,79,141,238,372,517,622,240,5,10,20,39,74,134,230,71,128,220,330,147,247,384,280,420,557,640,438,40,75,135,231,364,510,620,174,286,427,562,54,102,181,298,170,281,422,356,501,608,649,531,38,72,130,223,353,336,483,237,3,6,12,24,47,89,160,69,125,216,320,464,587,647,278,418,546,633,435,407,547,578,244,380,524,624,172,283,424,560,46,87,156,261,48,90,161,269,409,549,637,188,306,195,316,459,583,334,470,234,367,512,598,651,632,113,200,66,121,212,301,444,400,539,276,416,554,604,653,406,545,577,229,362,508,618,657,117,207,335,31,61,116,205,331,473,595,350,498,86,154,186,303,193,313,456,379,522,469,590,537,52,98,173,285,426,561,493,253,389,268,408,399,538,272,412,551,603,652,402,153,256,392,534,629,332,478,115,203,328,474,297,441,571,507,472,593,346,495,84,150,252,388,190,310,452,377,13,26,51,96,50,94,167,131,224,354,487,251,9,18,35,67,123,215,347,78,139,236,370,515,575,643,289,429,563,642,111,196,318,462,291,432,565,505,178,292,433,566,80,143,241,375,519,486,606,374,11,22,43,81,145,245,382,129,221,340,477,248,385,529,421,558,635,659,569,76,136,232,365,511,621,658,287,17,33,64,103,182,299,442,93,165,274,414,552,638,661,309,73,132,226,357,502,484,592,371,7,14,28,55,104,184,302,126,217,348,465,588,541,630,419,556,634,654,568,548,636,644,381,526,404,542,284,210,341,490,88,158,265,403,91,162,270,410,550,157,263,307,449,317,460,584,525,481,591,368,513,101,179,294,436,201,325,122,213,344,445,573,540,56,106,187,305,448,199,322,258,394,363,509,619,480,600,208,337,485,62,118,209,339,488,596,650,499,614,155,259,304,446,314,457,581,523,29,57,108,191,99,175,288,227,358,503,611,390,21,41,77,138,235,369,514,142,239,373,518,257,393,535,431,564,479,599,204,82,146,246,383,528,627,617,295,437,496,613,151,254,391,533,311,454,579,520,27,53,100,177,95,169,279,225,355,491,607,387,19,37,70,127,219,351,500,140)(662,663)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13220, 10);

C22×C661⋊C5 #27210247