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C₄×D₃×C₉ #6210

order	216
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂³×C₃×D₃ (order 144) · Inn(G) = D₃ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₃×C₉

Splittings

Fitting	C₃×C₄×C₉ ⋊ C₂
derived	C₃ ⋊ C₂×C₄×C₉
Sylow	C₃×C₉ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₉	D₃
commutator, hypocenter	C₃	C₂×C₄×C₉
Frattini	C₂×C₃	C₂×C₃×D₃
Fitting	C₃×C₄×C₉	C₂
socle, cosocle	C₂×C₃²	C₂²×C₃
prime radical	C₂×C₃×D₃	C₂×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₉

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₉⋊C₈, #88420, D₃×C₈×C₉, …
maximal normal subgroup of	D₃×C₉⋊C₈, #88420, D₃×C₈×C₉, …
maximal quotient of	D₃×C₈×C₉, C₉×ΓL(1,𝔽₂₅), C₄×C₉×Dic₃, …
automorphism group of	C₂²×C₃₇, D₃×C₃₇
outer automorphism group of	C₂²×C₃₇, Q₈×C₃₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 47);