#27210253

order	13220
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅×C₆₆₁ ⋊ C₄
Sylow	C₆₆₁ ⋊ Dic₅
Sylow	C₅ ⋊ #14581558

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₆₆₁	C₄
prime radical, cosocle	C₅×D₆₆₁	C₂
LCS	G ▷ C₅×C₆₆₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 666
transitive degree	μ_t(G) = 3305

A minimal faithful permutation representation — 2 generators on 666 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,153)(99,126,127,149)(103,130,131,145)(107,134,135,141)(111,138,139,137)(115,142,143,133)(119,146,147,129)(121,150,151,125)(124,155,156,190)(128,159,160,186)(132,163,164,182)(136,167,168,178)(140,171,172,174)(144,175,176,170)(148,179,180,166)(152,183,184,162)(154,187,188,158)(157,192,193,231)(161,196,197,227)(165,200,201,223)(169,204,205,219)(173,208,209,215)(177,212,213,211)(181,216,217,207)(185,220,221,203)(189,224,225,199)(191,228,229,195)(194,233,234,276)(198,237,238,272)(202,241,242,268)(206,245,246,264)(210,249,250,260)(214,253,254,256)(218,257,258,252)(222,261,262,248)(226,265,266,244)(230,269,270,240)(232,273,274,236)(235,278,279,325)(239,282,283,321)(243,286,287,317)(247,290,291,313)(251,294,295,309)(255,298,299,305)(259,302,303,301)(263,306,307,297)(267,310,311,293)(271,314,315,289)(275,318,319,285)(277,322,323,281)(280,327,328,378)(284,331,332,374)(288,335,336,370)(292,339,340,366)(296,343,344,362)(300,347,348,358)(304,351,352,354)(308,355,356,350)(312,359,360,346)(316,363,364,342)(320,367,368,338)(324,371,372,334)(326,375,376,330)(329,380,381,435)(333,384,385,431)(337,388,389,427)(341,392,393,423)(345,396,397,419)(349,400,401,415)(353,404,405,411)(357,408,409,407)(361,412,413,403)(365,416,417,399)(369,420,421,395)(373,424,425,391)(377,428,429,387)(379,432,433,383)(382,437,438,478)(386,441,442,489)(390,445,446,487)(394,449,450,483)(398,453,454,479)(402,457,458,475)(406,461,462,472)(410,465,466,468)(414,469,470,464)(418,439,473,460)(422,476,477,456)(426,480,481,452)(430,484,485,448)(434,488,486,444)(436,490,482,440)(443,491,492,521)(447,494,495,525)(451,497,498,524)(455,500,501,523)(459,503,504,522)(463,506,507,520)(467,510,511,517)(471,514,515,513)(474,493,518,509)(496,526,527,552)(499,529,530,556)(502,532,533,555)(505,535,536,554)(508,538,539,553)(512,541,542,551)(516,545,546,548)(519,528,549,544)(531,557,558,579)(534,560,561,583)(537,563,564,582)(540,566,567,581)(543,569,570,580)(547,572,573,578)(550,559,576,575)(562,584,585,602)(565,587,588,606)(568,590,591,605)(571,593,594,604)(574,596,597,603)(577,586,599,601)(589,607,608,621)(592,610,611,625)(595,613,614,624)(598,616,617,623)(600,609,619,622)(612,626,627,637)(615,629,630,640)(618,632,633,639)(620,628,635,638)(631,641,642,649)(634,644,645,651)(636,643,647,650)(646,652,653,657)(648,654,655,658)(656,659,660,661)(663,664)(665,666);(1,2,5,10,19,32,49,70,95,124,157,194,235,280,329,382,439,412,355,302,253,208,167,130,97,68,92,119,148,181,218,259,304,353,406,463,508,540,568,592,612,628,616,593,566,535,500,449,388,331,278,326,377,430,486,433,378,431,487,524,555,582,605,624,639,650,642,630,614,594,570,542,511,466,409,356,307,262,221,184,151,120,149,182,219,260,305,354,407,464,509,492,442,385,332,283,238,197,160,127,98,73,52,35,22,13,6,11,20,33,50,71,96,125,158,195,236,281,330,383,440,476,416,359,306,257,212,171,134,101,72,47,67,90,115,144,177,214,255,300,349,402,459,505,537,565,589,609,596,569,538,503,453,392,335,282,233,277,324,373,426,482,435,489,525,556,583,606,625,640,651,658,653,645,633,617,597,573,546,515,470,413,360,311,266,225,188,153,186,223,264,309,358,411,468,513,544,527,495,446,389,336,287,242,201,164,131,102,77,56,39,26,15,24,37,54,75,100,129,162,199,240,285,334,387,444,480,420,363,310,261,216,175,138,105,76,51,30,46,65,86,111,140,173,210,251,296,345,398,455,502,534,562,586,572,541,506,457,396,339,286,237,192,232,275,320,369,422,478,521,552,579,602,621,637,649,657,661,660,655,647,635,619,599,576,549,518,473,417,364,315,270,229,190,227,268,313,362,415,472,517,548,575,558,530,498,450,393,340,291,246,205,168,135,106,81,60,43,28,41,58,79,104,133,166,203,244,289,338,391,448,484,424,367,314,265,220,179,142,109,80,55,34,17,29,44,61,82,107,136,169,206,247,292,341,394,451,499,531,559,545,510,461,400,343,290,241,196,155,191,230,271,316,365,418,474,519,550,577,600,620,636,648,656,659,652,641,626,607,584,557,526,491,437,477,421,368,319,274,231,272,317,366,419,475,520,551,578,601,585,561,533,501,454,397,344,295,250,209,172,139,110,85,64,45,62,83,108,137,170,207,248,293,342,395,452,488,428,371,318,269,224,183,146,113,84,59,38,21,8,16,27,40,57,78,103,132,165,202,243,288,337,390,447,496,528,514,465,404,347,294,245,200,159,122,154,189,226,267,312,361,414,471,516,547,574,598,618,634,646,654,644,629,610,587,560,529,494,441,380,436,481,425,372,323,276,321,370,423,479,522,553,580,603,622,608,588,564,536,504,458,401,348,299,254,213,176,143,114,89,66,87,112,141,174,211,252,297,346,399,456,490,432,375,322,273,228,187,150,117,88,63,42,25,12,3,7,14,23,36,53,74,99,128,161,198,239,284,333,386,443,493,469,408,351,298,249,204,163,126,93,121,152,185,222,263,308,357,410,467,512,543,571,595,615,631,643,632,613,590,563,532,497,445,384,327,379,434,485,429,376,325,374,427,483,523,554,581,604,623,638,627,611,591,567,539,507,462,405,352,303,258,217,180,147,118,91,116,145,178,215,256,301,350,403,460,438,381,328,279,234,193,156,123,94,69,48,31,18,9,4)(662,663,665,666,664)

Polycyclic presentation:

2^2,5,661|a^2=b,c^a=c^4,d^a=d^555,d^b=d^660

Conjugacy classes & character table

833 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13220, 16);