D₂₄₄₃₂ #72287897

order	48864
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₄₄₃₂

Splittings

Fitting	C₃×C₁₆×C₅₀₉ ⋊ C₂
Hall	C₃×C₅₀₉ ⋊ D₁₆
Sylow	C₅₀₉ ⋊ D₄₈
Sylow	C₃ ⋊ D₈₁₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₂₂₁₆
commutator, cosocle	C₃×C₈×C₅₀₉	C₂²
Frattini	C₈	C₂×D₁₅₂₇
Fitting	C₃×C₁₆×C₅₀₉	C₂
socle	C₂×C₃×C₅₀₉	D₈
hypercenter	C₁₆	D₁₅₂₇
hypocenter	C₃×C₅₀₉	D₁₆
LCS	G ▷ C₃×C₈×C₅₀₉ ▷ C₃×C₄×C₅₀₉ ▷ C₂×C₃×C₅₀₉ ▷ C₃×C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 528
transitive degree	μ_t(G) = 24432

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

Polycyclic presentation:

2^5,3,509|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^508

Conjugacy classes & character table

12219 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", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 509]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^508);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D24432 #72287897