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C₂×C₁₉ #165

order	38
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,𝔽₁₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₈
abelian	C₂×C₁₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 38

A minimal faithful permutation representation — 1 generator on 21 points:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₁₉, Q₈×C₁₉, Dic₃×C₁₉, …
derived subgroup of	Dic₃₈, D₇₆, C₂²⋊D₁₉, …
Frattini subgroup of	Dic₃₆₁, C₄×C₃₆₁, C₃×C₄×C₃₆₁, …
Fitting subgroup of	Dic₁₉, C₂×D₁₉, C₂×C₁₉⋊C₃, …
socle of	Dic₁₉, C₄×C₁₉, C₂×D₁₉, …
maximal subgroup of	Dic₁₉, C₄×C₁₉, C₂×D₁₉, …
maximal normal subgroup of	Dic₁₉, C₄×C₁₉, C₂×D₁₉, …
maximal quotient of	C₄×C₁₉, C₂²×C₁₉, D₃×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38, 2);