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C₃×C₉ #80

order	27
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) = #72210336
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

C₃×C₉

Subgroups

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

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

center of	C₃×D₃×C₉, #6864, C₃²⋊C₂₇, …
derived subgroup of	C₃⋊D₉, C₃⋊Dic₉, D₃×D₉, …
Frattini subgroup of	C₉×C₂₇, #6864, C₃²⋊C₂₇, …
Fitting subgroup of	C₃×D₉, D₃×C₉, C₃⋊D₉, …
maximal subgroup of	C₃×D₉, D₃×C₉, C₃⋊D₉, …
maximal normal subgroup of	C₃×D₉, D₃×C₉, C₃⋊D₉, …
maximal quotient of	C₂×C₃×C₉, C₉², C₃²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27, 2);