#27212465

order	13473
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₉×C₄₉₉ ⋊ C₃
Sylow	C₄₉₉ ⋊ 3₋¹⁺²

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃×C₄₉₉⋊C₃
commutator, socle, cosocle	C₃×C₄₉₉	C₃²
Fitting	C₉×C₄₉₉	C₃
prime radical	C₃×C₄₉₉⋊C₃	C₃
hypercenter	C₉	C₄₉₉⋊C₃
hypocenter	C₄₉₉	3₋¹⁺²
LCS	G ▷ C₃×C₄₉₉ ▷ C₄₉₉
UCS	C₁ ◁ C₃ ◁ C₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 508
transitive degree	μ_t(G) = 4491

A minimal faithful permutation representation — 2 generators on 508 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)(250,289,322)(253,291,326)(256,294,323)(259,297,320)(262,300,317)(265,303,314)(268,306,311)(271,309,308)(274,312,305)(277,315,302)(280,318,299)(283,321,296)(286,324,293)(288,327,290)(292,331,359)(295,333,363)(298,336,360)(301,338,358)(304,341,355)(307,344,352)(310,347,349)(313,350,346)(316,353,343)(319,356,340)(325,361,335)(328,364,332)(329,362,330)(334,368,392)(337,370,394)(339,372,393)(342,374,391)(345,377,388)(348,380,385)(351,383,382)(354,386,379)(357,389,376)(365,395,369)(366,396,367)(371,399,419)(373,400,421)(375,402,420)(378,404,418)(381,407,415)(384,410,412)(387,413,409)(390,416,406)(397,422,398)(401,423,440)(403,424,442)(405,426,441)(408,428,439)(411,431,436)(414,434,433)(417,437,430)(425,443,457)(427,444,459)(429,446,458)(432,448,456)(435,451,453)(438,454,450)(445,460,471)(447,461,473)(449,463,472)(452,465,470)(455,468,467)(462,474,482)(464,475,484)(466,477,483)(469,479,481)(476,485,490)(478,486,492)(480,488,491)(487,493,496)(489,494,497)(495,498,499)(501,504,505)(503,507,508);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,279,205,143,92,114,139,167,198,232,269,308,346,379,406,399,333,252,178,211,246,283,322,359,392,419,440,457,471,482,490,496,499,498,486,463,428,377,303,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,301,339,373,401,416,353,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,296,335,369,398,395,324,245,176,207,241,278,317,355,388,415,436,453,467,460,424,372,297,218,150,94,118,144,172,203,237,274,313,351,384,411,432,449,464,476,479,451,410,347,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,292,318,239,171,115,70,89,111,136,164,195,229,266,305,343,376,368,291,212,248,286,325,362,287,326,363,394,421,442,459,473,484,492,497,493,475,446,404,341,261,187,125,75,37,52,69,88,110,135,163,194,228,265,304,342,375,403,425,437,386,312,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,293,332,367,361,282,208,145,173,204,238,275,314,352,385,412,433,450,443,400,336,255,181,119,146,175,206,240,277,316,354,387,414,435,452,466,478,487,488,465,431,380,306,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,295,334,356,276,202,140,90,51,67,86,108,133,161,192,226,263,302,340,331,249,288,328,365,396,327,247,284,323,360,393,420,441,458,472,483,491,485,461,426,374,300,221,153,97,53,71,91,113,138,166,197,231,268,307,345,378,405,427,445,454,413,350,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,290,330,321,242,174,117,142,170,201,235,272,311,349,382,409,430,423,370,294,215,147,177,209,243,280,319,357,390,417,438,455,469,480,489,495,494,477,448,407,344,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,298,337,371,389,315,236,168,112,68,35,48,64,83,105,130,158,189,223,260,299,289,329,366,397,422,364,285,210,244,281,320,358,391,418,439,456,470,481,474,444,402,338,258,184,122,72,93,116,141,169,200,234,271,310,348,381,408,429,447,462,468,434,383,309,230,162,106,62,30,10)(500,501,503,502,505,507,506,504,508)

Polycyclic presentation:

3^3,499|a^3=c,b^a=bc,d^b=d^139

Conjugacy classes & character table

1505 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13473, 10);