D₁₀₃₄₄ #72215331

order	20688
automorphism	Aut(G) order 35583360
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₃₄₄

Splittings

Fitting	C₃×C₈×C₄₃₁ ⋊ C₂
Hall	C₃×C₄₃₁ ⋊ D₈
Sylow	C₄₃₁ ⋊ D₂₄
Sylow	C₃ ⋊ D₃₄₄₈

Subgroups

	subgroup	G/H
center	C₂	D₅₁₇₂
commutator, cosocle	C₃×C₄×C₄₃₁	C₂²
Frattini	C₄	C₂×D₁₂₉₃
Fitting	C₃×C₈×C₄₃₁	C₂
socle	C₂×C₃×C₄₃₁	D₄
hypercenter	C₈	D₁₂₉₃
hypocenter	C₃×C₄₃₁	D₈
LCS	G ▷ C₃×C₄×C₄₃₁ ▷ C₂×C₃×C₄₃₁ ▷ C₃×C₄₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 442
transitive degree	μ_t(G) = 10344

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

Polycyclic presentation:

2^4,3,431|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^430

Conjugacy classes & character table

5175 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₀₆₈₈, D₃₁₀₃₂
maximal normal subgroup of	D₂₀₆₈₈
maximal quotient of	Dic₁₀₃₄₄, D₂₀₆₈₈, D₃₁₀₃₂

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, 431]);;
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.6^430);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10344 #72215331