D₃₂₂₈₀ #72302001

order	64560
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₂₈₀

Splittings

Fitting	C₃×C₅×C₈×C₂₆₉ ⋊ C₂
Hall	C₃×C₅×C₂₆₉ ⋊ D₈
Hall	C₅×C₂₆₉ ⋊ D₂₄
Hall	C₃×C₂₆₉ ⋊ D₄₀
Sylow	C₂₆₉ ⋊ D₁₂₀
Hall	C₃×C₅ ⋊ D₂₁₅₂
Sylow	C₅ ⋊ D₆₄₅₆
Sylow	C₃ ⋊ D₁₀₇₆₀

Subgroups

	subgroup	G/H
center	C₂	D₁₆₁₄₀
commutator, cosocle	C₃×C₄×C₅×C₂₆₉	C₂²
Frattini	C₄	C₂×D₄₀₃₅
Fitting	C₃×C₅×C₈×C₂₆₉	C₂
socle	C₂×C₃×C₅×C₂₆₉	D₄
hypercenter	C₈	D₄₀₃₅
hypocenter	C₃×C₅×C₂₆₉	D₈
LCS	G ▷ C₃×C₄×C₅×C₂₆₉ ▷ C₂×C₃×C₅×C₂₆₉ ▷ C₃×C₅×C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 285
transitive degree	μ_t(G) = 32280

A minimal faithful permutation representation — 2 generators on 285 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);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,285)(19,284)(20,283)(21,282)(22,281)(23,280)(24,279)(25,278)(26,277)(27,276)(28,275)(29,274)(30,273)(31,272)(32,271)(33,270)(34,269)(35,268)(36,267)(37,266)(38,265)(39,264)(40,263)(41,262)(42,261)(43,260)(44,259)(45,258)(46,257)(47,256)(48,255)(49,254)(50,253)(51,252)(52,251)(53,250)(54,249)(55,248)(56,247)(57,246)(58,245)(59,244)(60,243)(61,242)(62,241)(63,240)(64,239)(65,238)(66,237)(67,236)(68,235)(69,234)(70,233)(71,232)(72,231)(73,230)(74,229)(75,228)(76,227)(77,226)(78,225)(79,224)(80,223)(81,222)(82,221)(83,220)(84,219)(85,218)(86,217)(87,216)(88,215)(89,214)(90,213)(91,212)(92,211)(93,210)(94,209)(95,208)(96,207)(97,206)(98,205)(99,204)(100,203)(101,202)(102,201)(103,200)(104,199)(105,198)(106,197)(107,196)(108,195)(109,194)(110,193)(111,192)(112,191)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)

Polycyclic presentation:

2^4,3,5,269|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^268

Conjugacy classes & character table

16143 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, 3, 5, 269]);;
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.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^268);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32280 #72302001