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D₉×Dic₅ #67078

order	360
isoclinism	isoclinic class of D₅×D₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

D₉×Dic₅

Splittings

derived, Hall	C₅×C₉ ⋊ C₂×C₄
Sylow	C₉ ⋊ C₂×Dic₅
Sylow	C₅ ⋊ C₄×D₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅×D₉
commutator, hypocenter	C₅×C₉	C₂×C₄
Frattini	C₂×C₃	D₃×D₅
Fitting, cosocle	C₂×C₅×C₉	C₂²
socle	C₂×C₃×C₅	C₂×D₃
prime radical	C₂×C₅×D₉	C₂
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₂

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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×#170, #10623628, D₉×Dic₁₀, …
maximal normal subgroup of	D₉×#170, #10623628, D₉×Dic₁₀, …
maximal quotient of	D₉×#168, #10623540, Dic₅×Dic₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 8);