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C₁₉ #49

order	19
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,𝔽₁₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₁₉

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 19

A minimal faithful permutation representation — 1 generator on 19 points:

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

Conjugacy classes & character table

19 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₁₉, D₅×C₁₉, A₄×C₁₉, …
derived subgroup of	D₁₉, C₁₉⋊C₃, Dic₁₉, …
Frattini subgroup of	C₃₆₁, D₃₆₁, C₂×C₃₆₁, …
Fitting subgroup of	D₁₉, C₁₉⋊C₃, D₁₉⋊C₃, …
socle of	D₁₉, C₁₉⋊C₃, D₁₉⋊C₃, …
maximal subgroup of	D₁₉, C₂×C₁₉, C₁₉⋊C₃, …
maximal normal subgroup of	D₁₉, C₂×C₁₉, C₁₉⋊C₃, …
maximal quotient of	C₂×C₁₉, C₃×C₁₉, C₅×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19, 1);