D₁₀₄₂₀ #72215359

order	20840
automorphism	Aut(G) order 43347200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₄₂₀

Splittings

Fitting	C₄×C₅×C₅₂₁ ⋊ C₂
Hall	C₅×C₅₂₁ ⋊ D₄
Sylow	C₅₂₁ ⋊ D₂₀
Sylow	C₅ ⋊ D₂₀₈₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₆₀₅
commutator, socle, cosocle	C₂×C₅×C₅₂₁	C₂²
Fitting	C₄×C₅×C₅₂₁	C₂
hypercenter	C₄	D₂₆₀₅
hypocenter	C₅×C₅₂₁	D₄
LCS	G ▷ C₂×C₅×C₅₂₁ ▷ C₅×C₅₂₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 530
transitive degree	μ_t(G) = 10420

A minimal faithful permutation representation — 2 generators on 530 points:

(1,2,3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530);(2,4)(6,9)(7,8)(11,530)(12,529)(13,528)(14,527)(15,526)(16,525)(17,524)(18,523)(19,522)(20,521)(21,520)(22,519)(23,518)(24,517)(25,516)(26,515)(27,514)(28,513)(29,512)(30,511)(31,510)(32,509)(33,508)(34,507)(35,506)(36,505)(37,504)(38,503)(39,502)(40,501)(41,500)(42,499)(43,498)(44,497)(45,496)(46,495)(47,494)(48,493)(49,492)(50,491)(51,490)(52,489)(53,488)(54,487)(55,486)(56,485)(57,484)(58,483)(59,482)(60,481)(61,480)(62,479)(63,478)(64,477)(65,476)(66,475)(67,474)(68,473)(69,472)(70,471)(71,470)(72,469)(73,468)(74,467)(75,466)(76,465)(77,464)(78,463)(79,462)(80,461)(81,460)(82,459)(83,458)(84,457)(85,456)(86,455)(87,454)(88,453)(89,452)(90,451)(91,450)(92,449)(93,448)(94,447)(95,446)(96,445)(97,444)(98,443)(99,442)(100,441)(101,440)(102,439)(103,438)(104,437)(105,436)(106,435)(107,434)(108,433)(109,432)(110,431)(111,430)(112,429)(113,428)(114,427)(115,426)(116,425)(117,424)(118,423)(119,422)(120,421)(121,420)(122,419)(123,418)(124,417)(125,416)(126,415)(127,414)(128,413)(129,412)(130,411)(131,410)(132,409)(133,408)(134,407)(135,406)(136,405)(137,404)(138,403)(139,402)(140,401)(141,400)(142,399)(143,398)(144,397)(145,396)(146,395)(147,394)(148,393)(149,392)(150,391)(151,390)(152,389)(153,388)(154,387)(155,386)(156,385)(157,384)(158,383)(159,382)(160,381)(161,380)(162,379)(163,378)(164,377)(165,376)(166,375)(167,374)(168,373)(169,372)(170,371)(171,370)(172,369)(173,368)(174,367)(175,366)(176,365)(177,364)(178,363)(179,362)(180,361)(181,360)(182,359)(183,358)(184,357)(185,356)(186,355)(187,354)(188,353)(189,352)(190,351)(191,350)(192,349)(193,348)(194,347)(195,346)(196,345)(197,344)(198,343)(199,342)(200,341)(201,340)(202,339)(203,338)(204,337)(205,336)(206,335)(207,334)(208,333)(209,332)(210,331)(211,330)(212,329)(213,328)(214,327)(215,326)(216,325)(217,324)(218,323)(219,322)(220,321)(221,320)(222,319)(223,318)(224,317)(225,316)(226,315)(227,314)(228,313)(229,312)(230,311)(231,310)(232,309)(233,308)(234,307)(235,306)(236,305)(237,304)(238,303)(239,302)(240,301)(241,300)(242,299)(243,298)(244,297)(245,296)(246,295)(247,294)(248,293)(249,292)(250,291)(251,290)(252,289)(253,288)(254,287)(255,286)(256,285)(257,284)(258,283)(259,282)(260,281)(261,280)(262,279)(263,278)(264,277)(265,276)(266,275)(267,274)(268,273)(269,272)(270,271)

Polycyclic presentation:

2^3,5,521|b^2=c,b^a=bc,d^a=d^4,e^a=e^520

Conjugacy classes & character table

5213 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₀₈₄₀, D₃₁₂₆₀
maximal normal subgroup of	D₂₀₈₄₀
maximal quotient of	Dic₁₀₄₂₀, D₂₀₈₄₀, D₃₁₂₆₀

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, 5, 521]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^520);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10420 #72215359