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C₂₇ #79

order	27
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₉ (order 18) · Inn(G) = C₁ · Out(G) = C₂×C₉
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/27)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₂₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₃
socle, prime radical	C₃	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₉ ▷ C₃ ▷ C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 1 generator on 27 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₂₇, C₈₁⋊C₃, #6903, …
derived subgroup of	D₂₇, Dic₂₇, C₂×D₂₇, …
Frattini subgroup of	C₈₁, D₈₁, C₂×C₈₁, …
Fitting subgroup of	D₂₇, C₂₇×A₅
maximal subgroup of	D₂₇, C₂×C₂₇, C₈₁, …
maximal normal subgroup of	D₂₇, C₂×C₂₇, C₈₁, …
maximal quotient of	C₂×C₂₇, C₈₁, C₃×C₂₇, …
outer automorphism group of	D₈₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27, 1);