Dic₂₀₂₀ #72212374

order	8080
automorphism	Aut(G) order 6464000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₀₂₀

Splittings

Hall	C₅×C₁₀₁ ⋊ Q₁₆
Sylow	C₁₀₁ ⋊ Dic₂₀
Sylow	C₅ ⋊ Dic₄₀₄

Subgroups

	subgroup	G/H
center	C₂	D₂₀₂₀
commutator, cosocle	C₄×C₅×C₁₀₁	C₂²
Frattini	C₄	C₂×D₅₀₅
Fitting	C₅×C₈×C₁₀₁	C₂
socle, prime radical	C₂×C₅×C₁₀₁	D₄
hypercenter	C₈	D₅₀₅
hypocenter	C₅×C₁₀₁	Q₁₆
LCS	G ▷ C₄×C₅×C₁₀₁ ▷ C₂×C₅×C₁₀₁ ▷ C₅×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 8080

A minimal faithful permutation representation — 2 generators on 122 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,21)(19,20)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

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

Conjugacy classes & character table

2023 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₀₄₀, Dic₆₀₆₀, Dic₁₀₁₀₀, …
maximal normal subgroup of	Dic₄₀₄₀
maximal quotient of	Dic₆₀₆₀, Dic₁₀₁₀₀, Dic₁₄₁₄₀

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, 101]);;
gap> SetPower(coll, 1, F.4);;
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.6^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2020 #72212374