D₁₁₂₀ #72210121

order	2240
automorphism	Aut(G) order 430080
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₂₀

Splittings

Fitting	C₅×C₇×C₃₂ ⋊ C₂
Hall	C₅×C₇ ⋊ D₃₂
Sylow	C₇ ⋊ D₁₆₀
Sylow	C₅ ⋊ D₂₂₄

Subgroups

	subgroup	G/H
center	C₂	D₅₆₀
commutator, cosocle	C₅×C₇×C₁₆	C₂²
Frattini	C₁₆	C₂×D₃₅
Fitting	C₅×C₇×C₃₂	C₂
socle	C₂×C₅×C₇	D₁₆
hypercenter	C₃₂	D₃₅
hypocenter	C₅×C₇	D₃₂
LCS	G ▷ C₅×C₇×C₁₆ ▷ C₅×C₇×C₈ ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 1120

A minimal faithful permutation representation — 2 generators on 44 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);(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)(34,37)(35,36)(39,44)(40,43)(41,42)

Polycyclic presentation:

2^6,5,7|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4,h^a=h^6

Conjugacy classes & character table

563 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₂₄₀, D₃₃₆₀, D₅₆₀₀, …
maximal normal subgroup of	D₂₂₄₀
maximal quotient of	D₂₂₄₀, D₃₃₆₀, Dic₁₁₂₀, …

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, 2, 2, 5, 7]);;
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> 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);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1120 #72210121