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C₂×C₃×C₉ #273

order	54
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×#272 (order 108) · Inn(G) = C₁ · Out(G) = C₂×#272
holomorph	G ⋊ Aut(G) = C₂×#72210336
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

C₂×C₃×C₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂×C₃²
socle, prime radical	C₂×C₃²	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

center of	C₃×D₄×C₉, C₃×Q₈×C₉, C₃×C₉×Dic₃, …
derived subgroup of	#6189, #6192, #6194, …
Frattini subgroup of	C₉×Dic₂₇, C₂₇×Dic₉, #6864⋊C₄, …
Fitting subgroup of	C₃×Dic₉, C₉×Dic₃, C₃⋊Dic₉, …
maximal subgroup of	C₃×Dic₉, C₉×Dic₃, C₃⋊Dic₉, …
maximal normal subgroup of	C₃×Dic₉, C₉×Dic₃, C₃⋊Dic₉, …
maximal quotient of	C₃×C₄×C₉, C₂²×C₃×C₉, C₂×C₉², …
outer automorphism group of	D₇×C₁₉, D₉×C₁₉, D₇×C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 9);