D₃₂₂₈ #72211067

order	6456
automorphism	Aut(G) order 3460416
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₂₈

Splittings

Fitting	C₃×C₄×C₂₆₉ ⋊ C₂
Hall	C₃×C₂₆₉ ⋊ D₄
Sylow	C₂₆₉ ⋊ D₁₂
Sylow	C₃ ⋊ D₁₀₇₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₈₀₇
commutator, socle, cosocle	C₂×C₃×C₂₆₉	C₂²
Fitting	C₃×C₄×C₂₆₉	C₂
hypercenter	C₄	D₈₀₇
hypocenter	C₃×C₂₆₉	D₄
LCS	G ▷ C₂×C₃×C₂₆₉ ▷ C₃×C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 276
transitive degree	μ_t(G) = 3228

A minimal faithful permutation representation — 2 generators on 276 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);(2,4)(6,7)(9,276)(10,275)(11,274)(12,273)(13,272)(14,271)(15,270)(16,269)(17,268)(18,267)(19,266)(20,265)(21,264)(22,263)(23,262)(24,261)(25,260)(26,259)(27,258)(28,257)(29,256)(30,255)(31,254)(32,253)(33,252)(34,251)(35,250)(36,249)(37,248)(38,247)(39,246)(40,245)(41,244)(42,243)(43,242)(44,241)(45,240)(46,239)(47,238)(48,237)(49,236)(50,235)(51,234)(52,233)(53,232)(54,231)(55,230)(56,229)(57,228)(58,227)(59,226)(60,225)(61,224)(62,223)(63,222)(64,221)(65,220)(66,219)(67,218)(68,217)(69,216)(70,215)(71,214)(72,213)(73,212)(74,211)(75,210)(76,209)(77,208)(78,207)(79,206)(80,205)(81,204)(82,203)(83,202)(84,201)(85,200)(86,199)(87,198)(88,197)(89,196)(90,195)(91,194)(92,193)(93,192)(94,191)(95,190)(96,189)(97,188)(98,187)(99,186)(100,185)(101,184)(102,183)(103,182)(104,181)(105,180)(106,179)(107,178)(108,177)(109,176)(110,175)(111,174)(112,173)(113,172)(114,171)(115,170)(116,169)(117,168)(118,167)(119,166)(120,165)(121,164)(122,163)(123,162)(124,161)(125,160)(126,159)(127,158)(128,157)(129,156)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)

Polycyclic presentation:

2^3,3,269|b^2=c,b^a=bc,d^a=d^2,e^a=e^268

Conjugacy classes & character table

1617 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₆₄₅₆, D₉₆₈₄, D₁₆₁₄₀, …
maximal normal subgroup of	D₆₄₅₆
maximal quotient of	D₆₄₅₆, Dic₃₂₂₈, D₉₆₈₄, …

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 269]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^268);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3228 #72211067