D₁₂₂₀ #72210141

order	2440
automorphism	Aut(G) order 585600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₂₀

Splittings

Fitting	C₄×C₅×C₆₁ ⋊ C₂
Hall	C₅×C₆₁ ⋊ D₄
Sylow	C₆₁ ⋊ D₂₀
Sylow	C₅ ⋊ D₂₄₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₀₅
commutator, socle, cosocle	C₂×C₅×C₆₁	C₂²
Fitting	C₄×C₅×C₆₁	C₂
hypercenter	C₄	D₃₀₅
hypocenter	C₅×C₆₁	D₄
LCS	G ▷ C₂×C₅×C₆₁ ▷ C₅×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 1220

A minimal faithful permutation representation — 2 generators on 70 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);(2,4)(6,9)(7,8)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2^3,5,61|b^2=c,b^a=bc,d^a=d^4,e^a=e^60

Conjugacy classes & character table

613 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₄₄₀, D₃₆₆₀, D₆₁₀₀, …
maximal normal subgroup of	D₂₄₄₀
maximal quotient of	D₂₄₄₀, D₃₆₆₀, Dic₁₂₂₀, …

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, 61]);;
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.5^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1220 #72210141