D₁₁₆₈₈ #72223940

order	23376
automorphism	Aut(G) order 45442944
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₆₈₈

Splittings

Fitting	C₃×C₈×C₄₈₇ ⋊ C₂
Hall	C₃×C₄₈₇ ⋊ D₈
Sylow	C₄₈₇ ⋊ D₂₄
Sylow	C₃ ⋊ D₃₈₉₆

Subgroups

	subgroup	G/H
center	C₂	D₅₈₄₄
commutator, cosocle	C₃×C₄×C₄₈₇	C₂²
Frattini	C₄	C₂×D₁₄₆₁
Fitting	C₃×C₈×C₄₈₇	C₂
socle	C₂×C₃×C₄₈₇	D₄
hypercenter	C₈	D₁₄₆₁
hypocenter	C₃×C₄₈₇	D₈
LCS	G ▷ C₃×C₄×C₄₈₇ ▷ C₂×C₃×C₄₈₇ ▷ C₃×C₄₈₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 498
transitive degree	μ_t(G) = 11688

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

Polycyclic presentation:

2^4,3,487|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^486

Conjugacy classes & character table

5847 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₃₇₆
maximal normal subgroup of	D₂₃₃₇₆
maximal quotient of	Dic₁₁₆₈₈, D₂₃₃₇₆

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, 487]);;
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.6^486);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11688 #72223940