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D₁₈₀ #67097

order	360
isoclinism	isoclinic class of Dic₉₀
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₈₀
unambiguous semidirect product	C₄⋊D₄₅

Splittings

Fitting	C₄×C₅×C₉ ⋊ C₂
Hall	C₅×C₉ ⋊ D₄
Sylow	C₉ ⋊ D₂₀
Sylow	C₅ ⋊ D₃₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₅
commutator, cosocle	C₂×C₅×C₉	C₂²
Frattini	C₂×C₃	C₂×D₁₅
Fitting	C₄×C₅×C₉	C₂
socle	C₂×C₃×C₅	C₂×D₃
hypercenter	C₄	D₄₅
hypocenter	C₅×C₉	D₄
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18);(2,4)(6,13)(7,12)(8,11)(9,10)(15,18)(16,17)

Polycyclic presentation:

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10623543, #10623544, #10623549, …
maximal normal subgroup of	#10623543, #10623544, #10623549, …
maximal quotient of	Dic₁₈₀, #10623599, D₃₆₀, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 27);