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C₂×C₂₇ #266

order	54
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) = C₂×AGL(1,ℤ/27)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₅₄
abelian	C₂×C₂₇
multiplicative	(ℤ/81)ˣ, (ℤ/162)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂×C₃
socle, prime radical	C₂×C₃	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 1 generator on 29 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₂₇, Q₈×C₂₇, Dic₃×C₂₇, …
derived subgroup of	Dic₅₄, D₁₀₈, #6171, …
Frattini subgroup of	Dic₈₁, C₄×C₈₁, Q₈⋊C₈₁, …
Fitting subgroup of	Dic₂₇, C₂×D₂₇
maximal subgroup of	Dic₂₇, C₄×C₂₇, C₂×D₂₇, …
maximal normal subgroup of	Dic₂₇, C₄×C₂₇, C₂×D₂₇, …
maximal quotient of	C₄×C₂₇, C₂²×C₂₇, C₂×C₈₁, …
automorphism group of	C₈₁, C₂×C₈₁
outer automorphism group of	C₈₁, C₂×C₈₁, D₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 2);