D₂₂₁₃₂ #72275604

order	44264
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₂₁₃₂

Splittings

Fitting	C₄×C₁₁×C₅₀₃ ⋊ C₂
Hall	C₁₁×C₅₀₃ ⋊ D₄
Sylow	C₅₀₃ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₂₀₁₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₅₃₃
commutator, socle, cosocle	C₂×C₁₁×C₅₀₃	C₂²
Fitting	C₄×C₁₁×C₅₀₃	C₂
hypercenter	C₄	D₅₅₃₃
hypocenter	C₁₁×C₅₀₃	D₄
LCS	G ▷ C₂×C₁₁×C₅₀₃ ▷ C₁₁×C₅₀₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 518
transitive degree	μ_t(G) = 22132

A minimal faithful permutation representation — 2 generators on 518 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518);(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,518)(18,517)(19,516)(20,515)(21,514)(22,513)(23,512)(24,511)(25,510)(26,509)(27,508)(28,507)(29,506)(30,505)(31,504)(32,503)(33,502)(34,501)(35,500)(36,499)(37,498)(38,497)(39,496)(40,495)(41,494)(42,493)(43,492)(44,491)(45,490)(46,489)(47,488)(48,487)(49,486)(50,485)(51,484)(52,483)(53,482)(54,481)(55,480)(56,479)(57,478)(58,477)(59,476)(60,475)(61,474)(62,473)(63,472)(64,471)(65,470)(66,469)(67,468)(68,467)(69,466)(70,465)(71,464)(72,463)(73,462)(74,461)(75,460)(76,459)(77,458)(78,457)(79,456)(80,455)(81,454)(82,453)(83,452)(84,451)(85,450)(86,449)(87,448)(88,447)(89,446)(90,445)(91,444)(92,443)(93,442)(94,441)(95,440)(96,439)(97,438)(98,437)(99,436)(100,435)(101,434)(102,433)(103,432)(104,431)(105,430)(106,429)(107,428)(108,427)(109,426)(110,425)(111,424)(112,423)(113,422)(114,421)(115,420)(116,419)(117,418)(118,417)(119,416)(120,415)(121,414)(122,413)(123,412)(124,411)(125,410)(126,409)(127,408)(128,407)(129,406)(130,405)(131,404)(132,403)(133,402)(134,401)(135,400)(136,399)(137,398)(138,397)(139,396)(140,395)(141,394)(142,393)(143,392)(144,391)(145,390)(146,389)(147,388)(148,387)(149,386)(150,385)(151,384)(152,383)(153,382)(154,381)(155,380)(156,379)(157,378)(158,377)(159,376)(160,375)(161,374)(162,373)(163,372)(164,371)(165,370)(166,369)(167,368)(168,367)(169,366)(170,365)(171,364)(172,363)(173,362)(174,361)(175,360)(176,359)(177,358)(178,357)(179,356)(180,355)(181,354)(182,353)(183,352)(184,351)(185,350)(186,349)(187,348)(188,347)(189,346)(190,345)(191,344)(192,343)(193,342)(194,341)(195,340)(196,339)(197,338)(198,337)(199,336)(200,335)(201,334)(202,333)(203,332)(204,331)(205,330)(206,329)(207,328)(208,327)(209,326)(210,325)(211,324)(212,323)(213,322)(214,321)(215,320)(216,319)(217,318)(218,317)(219,316)(220,315)(221,314)(222,313)(223,312)(224,311)(225,310)(226,309)(227,308)(228,307)(229,306)(230,305)(231,304)(232,303)(233,302)(234,301)(235,300)(236,299)(237,298)(238,297)(239,296)(240,295)(241,294)(242,293)(243,292)(244,291)(245,290)(246,289)(247,288)(248,287)(249,286)(250,285)(251,284)(252,283)(253,282)(254,281)(255,280)(256,279)(257,278)(258,277)(259,276)(260,275)(261,274)(262,273)(263,272)(264,271)(265,270)(266,269)(267,268)

Polycyclic presentation:

2^3,11,503|b^2=c,b^a=bc,d^a=d^10,e^a=e^502

Conjugacy classes & character table

11069 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 11, 503]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^502);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D22132 #72275604