D₁₁₈₀ #72210134

order	2360
automorphism	Aut(G) order 547520
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₈₀

Splittings

Fitting	C₄×C₅×C₅₉ ⋊ C₂
Hall	C₅×C₅₉ ⋊ D₄
Sylow	C₅₉ ⋊ D₂₀
Sylow	C₅ ⋊ D₂₃₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₅
commutator, socle, cosocle	C₂×C₅×C₅₉	C₂²
Fitting	C₄×C₅×C₅₉	C₂
hypercenter	C₄	D₂₉₅
hypocenter	C₅×C₅₉	D₄
LCS	G ▷ C₂×C₅×C₅₉ ▷ C₅×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 1180

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

Polycyclic presentation:

2^3,5,59|b^2=c,b^a=bc,d^a=d^4,e^a=e^58

Conjugacy classes & character table

593 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₃₆₀, D₃₅₄₀, D₅₉₀₀, …
maximal normal subgroup of	D₂₃₆₀
maximal quotient of	D₂₃₆₀, D₃₅₄₀, Dic₁₁₈₀, …

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, 59]);;
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.5^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1180 #72210134