D₁₁₁₁₂ #72215817

order	22224
automorphism	Aut(G) order 41069952
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₁₂

Splittings

Fitting	C₃×C₈×C₄₆₃ ⋊ C₂
Hall	C₃×C₄₆₃ ⋊ D₈
Sylow	C₄₆₃ ⋊ D₂₄
Sylow	C₃ ⋊ D₃₇₀₄

Subgroups

	subgroup	G/H
center	C₂	D₅₅₅₆
commutator, cosocle	C₃×C₄×C₄₆₃	C₂²
Frattini	C₄	C₂×D₁₃₈₉
Fitting	C₃×C₈×C₄₆₃	C₂
socle	C₂×C₃×C₄₆₃	D₄
hypercenter	C₈	D₁₃₈₉
hypocenter	C₃×C₄₆₃	D₈
LCS	G ▷ C₃×C₄×C₄₆₃ ▷ C₂×C₃×C₄₆₃ ▷ C₃×C₄₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 474
transitive degree	μ_t(G) = 11112

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

Polycyclic presentation:

2^4,3,463|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^462

Conjugacy classes & character table

5559 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₂₂₄
maximal normal subgroup of	D₂₂₂₂₄
maximal quotient of	Dic₁₁₁₁₂, D₂₂₂₂₄

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, 463]);;
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.6^462);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11112 #72215817