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C₉×(C₃⋊D₃)⋊C₃ #92253

order	486
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = C₃×D₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 1944) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃²

Aliases

C₉×(C₃⋊D₃)⋊C₃

Splittings

Fitting, Sylow	C₉×H(3,𝔽₃) ⋊ C₂
derived	C₃² ⋊ C₂×C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	(C₃⋊D₃)⋊C₃
commutator, hypocenter	C₃²	C₂×C₃×C₉
Frattini, socle	C₃²	C₃²×D₃
Fitting	C₉×H(3,𝔽₃)	C₂
prime radical	C₃×(C₃⋊D₃)⋊C₃	C₃
cosocle	C₃³	C₂×C₃²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₉

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 54

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

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×(C₃⋊D₃)⋊C₃, C₉×((C₃⋊D₃)⋊C₃)⋊C₂, ((C₃⋊D₃)⋊C₃)⋊D₉, …
maximal normal subgroup of	D₉×(C₃⋊D₃)⋊C₃, C₉×((C₃⋊D₃)⋊C₃)⋊C₂, ((C₃⋊D₃)⋊C₃)⋊D₉, …
maximal quotient of	C₉×(C₃⋊Dic₃)⋊C₃, C₂×C₉×(C₃⋊D₃)⋊C₃, C₉×#4169, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 98);