D₄₀₀₀ #72211274

order	8000
automorphism	Aut(G) order 6400000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₀₀₀

Splittings

Fitting	C₃₂×C₁₂₅ ⋊ C₂
Sylow	C₁₂₅ ⋊ D₃₂

Subgroups

	subgroup	G/H
center	C₂	D₂₀₀₀
commutator, cosocle	C₁₆×C₁₂₅	C₂²
Frattini	C₁₆×C₂₅	C₂×D₅
Fitting	C₃₂×C₁₂₅	C₂
socle	C₂×C₅	D₄₀₀
hypercenter	C₃₂	D₁₂₅
hypocenter	C₁₂₅	D₃₂
LCS	G ▷ C₁₆×C₁₂₅ ▷ C₈×C₁₂₅ ▷ C₄×C₁₂₅ ▷ C₂×C₁₂₅ ▷ C₁₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 4000

A minimal faithful permutation representation — 2 generators on 157 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)

Polycyclic presentation:

2^6,5^3|b^2=cd,c^2=de,d^2=ef,e^2=f,g^5=h,h^5=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4h^4i^4,h^a=h^4i^4,i^a=i^4

Conjugacy classes & character table

2003 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₈₀₀₀, D₁₂₀₀₀, D₂₀₀₀₀, …
maximal normal subgroup of	D₈₀₀₀
maximal quotient of	D₈₀₀₀, Dic₄₀₀₀, D₁₂₀₀₀, …

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", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 5, 5, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.8^4*F.9^4);;
gap> SetConjugate(coll, 8, 1, F.8^4*F.9^4);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4000 #72211274