D₁₂₂₁₆ #72224060

order	24432
automorphism	Aut(G) order 49645824
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₂₁₆

Splittings

Fitting	C₃×C₈×C₅₀₉ ⋊ C₂
Hall	C₃×C₅₀₉ ⋊ D₈
Sylow	C₅₀₉ ⋊ D₂₄
Sylow	C₃ ⋊ D₄₀₇₂

Subgroups

	subgroup	G/H
center	C₂	D₆₁₀₈
commutator, cosocle	C₃×C₄×C₅₀₉	C₂²
Frattini	C₄	C₂×D₁₅₂₇
Fitting	C₃×C₈×C₅₀₉	C₂
socle	C₂×C₃×C₅₀₉	D₄
hypercenter	C₈	D₁₅₂₇
hypocenter	C₃×C₅₀₉	D₈
LCS	G ▷ C₃×C₄×C₅₀₉ ▷ C₂×C₃×C₅₀₉ ▷ C₃×C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 520
transitive degree	μ_t(G) = 12216

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

Polycyclic presentation:

2^4,3,509|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^508

Conjugacy classes & character table

6111 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₄₄₃₂
maximal normal subgroup of	D₂₄₄₃₂
maximal quotient of	Dic₁₂₂₁₆, D₂₄₄₃₂

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 509]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^508);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D12216 #72224060