D₁₁₂₄₀ #72215878

order	22480
automorphism	Aut(G) order 50355200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₂₄₀

Splittings

Fitting	C₅×C₈×C₂₈₁ ⋊ C₂
Hall	C₅×C₂₈₁ ⋊ D₈
Sylow	C₂₈₁ ⋊ D₄₀
Sylow	C₅ ⋊ D₂₂₄₈

Subgroups

	subgroup	G/H
center	C₂	D₅₆₂₀
commutator, cosocle	C₄×C₅×C₂₈₁	C₂²
Frattini	C₄	C₂×D₁₄₀₅
Fitting	C₅×C₈×C₂₈₁	C₂
socle	C₂×C₅×C₂₈₁	D₄
hypercenter	C₈	D₁₄₀₅
hypocenter	C₅×C₂₈₁	D₈
LCS	G ▷ C₄×C₅×C₂₈₁ ▷ C₂×C₅×C₂₈₁ ▷ C₅×C₂₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 294
transitive degree	μ_t(G) = 11240

A minimal faithful permutation representation — 2 generators on 294 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);(2,8)(3,7)(4,6)(10,13)(11,12)(15,294)(16,293)(17,292)(18,291)(19,290)(20,289)(21,288)(22,287)(23,286)(24,285)(25,284)(26,283)(27,282)(28,281)(29,280)(30,279)(31,278)(32,277)(33,276)(34,275)(35,274)(36,273)(37,272)(38,271)(39,270)(40,269)(41,268)(42,267)(43,266)(44,265)(45,264)(46,263)(47,262)(48,261)(49,260)(50,259)(51,258)(52,257)(53,256)(54,255)(55,254)(56,253)(57,252)(58,251)(59,250)(60,249)(61,248)(62,247)(63,246)(64,245)(65,244)(66,243)(67,242)(68,241)(69,240)(70,239)(71,238)(72,237)(73,236)(74,235)(75,234)(76,233)(77,232)(78,231)(79,230)(80,229)(81,228)(82,227)(83,226)(84,225)(85,224)(86,223)(87,222)(88,221)(89,220)(90,219)(91,218)(92,217)(93,216)(94,215)(95,214)(96,213)(97,212)(98,211)(99,210)(100,209)(101,208)(102,207)(103,206)(104,205)(105,204)(106,203)(107,202)(108,201)(109,200)(110,199)(111,198)(112,197)(113,196)(114,195)(115,194)(116,193)(117,192)(118,191)(119,190)(120,189)(121,188)(122,187)(123,186)(124,185)(125,184)(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)(135,174)(136,173)(137,172)(138,171)(139,170)(140,169)(141,168)(142,167)(143,166)(144,165)(145,164)(146,163)(147,162)(148,161)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)

Polycyclic presentation:

2^4,5,281|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^280

Conjugacy classes & character table

5623 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₄₈₀
maximal normal subgroup of	D₂₂₄₈₀
maximal quotient of	Dic₁₁₂₄₀, D₂₂₄₈₀

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, 5, 281]);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^280);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11240 #72215878