D₂₀₂₀ #72211073

order	4040
automorphism	Aut(G) order 1616000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₂₀

Splittings

Fitting	C₄×C₅×C₁₀₁ ⋊ C₂
Hall	C₅×C₁₀₁ ⋊ D₄
Sylow	C₁₀₁ ⋊ D₂₀
Sylow	C₅ ⋊ D₄₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₀₅
commutator, socle, cosocle	C₂×C₅×C₁₀₁	C₂²
Fitting	C₄×C₅×C₁₀₁	C₂
hypercenter	C₄	D₅₀₅
hypocenter	C₅×C₁₀₁	D₄
LCS	G ▷ C₂×C₅×C₁₀₁ ▷ C₅×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 110
transitive degree	μ_t(G) = 2020

A minimal faithful permutation representation — 2 generators on 110 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);(2,4)(6,9)(7,8)(11,110)(12,109)(13,108)(14,107)(15,106)(16,105)(17,104)(18,103)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2^3,5,101|b^2=c,b^a=bc,d^a=d^4,e^a=e^100

Conjugacy classes & character table

1013 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₀₄₀, D₆₀₆₀, D₁₀₁₀₀, …
maximal normal subgroup of	D₄₀₄₀
maximal quotient of	D₄₀₄₀, D₆₀₆₀, Dic₂₀₂₀, …

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, 5, 101]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2020 #72211073