D₁₀₈₀ #72210112

order	2160
automorphism	Aut(G) order 311040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₈₀

Splittings

Fitting	C₅×C₈×C₂₇ ⋊ C₂
Hall	C₅×C₂₇ ⋊ D₈
Sylow	C₂₇ ⋊ D₄₀
Sylow	C₅ ⋊ D₂₁₆

Subgroups

	subgroup	G/H
center	C₂	D₅₄₀
commutator, cosocle	C₄×C₅×C₂₇	C₂²
Frattini	C₄×C₉	C₂×D₁₅
Fitting	C₅×C₈×C₂₇	C₂
socle	C₂×C₃×C₅	D₃₆
hypercenter	C₈	D₁₃₅
hypocenter	C₅×C₂₇	D₈
LCS	G ▷ C₄×C₅×C₂₇ ▷ C₂×C₅×C₂₇ ▷ C₅×C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1080

A minimal faithful permutation representation — 2 generators on 40 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);(2,8)(3,7)(4,6)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

543 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₁₆₀, D₃₂₄₀, D₅₄₀₀, …
maximal normal subgroup of	D₂₁₆₀
maximal quotient of	D₂₁₆₀, D₃₂₄₀, D₅₄₀₀, …

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

D1080 #72210112