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C₂×C₉ #45

order	18
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃ (order 6) · Inn(G) = C₁ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = C₂×AGL(1,ℤ/9)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₈
abelian	C₂×C₉
multiplicative	(ℤ/19)ˣ, (ℤ/27)ˣ, (ℤ/38)ˣ, (ℤ/54)ˣ

Subgroups

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

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 1 generator on 11 points:

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

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₉, Q₈×C₉, C₉×Dic₃, …
derived subgroup of	Dic₁₈, D₃₆, #611, …
Frattini subgroup of	Dic₂₇, C₄×C₂₇, Q₈⋊C₂₇, …
Fitting subgroup of	Dic₉, C₂×D₉, C₉×SL(2,𝔽₅), …
maximal subgroup of	Dic₉, C₄×C₉, C₂×D₉, …
maximal normal subgroup of	Dic₉, C₄×C₉, C₂×D₉, …
maximal quotient of	C₄×C₉, C₂²×C₉, C₂×C₂₇, …
automorphism group of	C₁₉, C₂₇, C₂×C₁₉, …
outer automorphism group of	C₁₉, C₂₇, C₂×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18, 2);