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D₁₉ #164

order	38
isoclinism	isoclinic class of D₁₉
automorphism	Aut(G) = AGL(1,𝔽₁₉) (order 342) · Inn(G) = G · Out(G) = C₉
holomorph	G ⋊ Aut(G) = #20266599⋊C₂
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₉
unambiguous semidirect product	C₁₉⋊C₂

Splittings

Fitting, socle, derived, Sylow

C₁₉ ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2,19|b^a=b^18

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	2	2	2	2	2	2	2	2	2	19
order	1	19	19	19	19	19	19	19	19	19	2
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	1	1	-1
χ₃	2	ζ+ζ¹⁸	ζ²+ζ¹⁷	ζ³+ζ¹⁶	ζ⁴+ζ¹⁵	ζ⁵+ζ¹⁴	ζ⁶+ζ¹³	ζ⁷+ζ¹²	ζ⁸+ζ¹¹	ζ⁹+ζ¹⁰	0
χ₄	2	ζ²+ζ¹⁷	ζ⁴+ζ¹⁵	ζ⁶+ζ¹³	ζ⁸+ζ¹¹	ζ⁹+ζ¹⁰	ζ⁷+ζ¹²	ζ⁵+ζ¹⁴	ζ³+ζ¹⁶	ζ+ζ¹⁸	0
χ₅	2	ζ³+ζ¹⁶	ζ⁶+ζ¹³	ζ⁹+ζ¹⁰	ζ⁷+ζ¹²	ζ⁴+ζ¹⁵	ζ+ζ¹⁸	ζ²+ζ¹⁷	ζ⁵+ζ¹⁴	ζ⁸+ζ¹¹	0
χ₆	2	ζ⁴+ζ¹⁵	ζ⁸+ζ¹¹	ζ⁷+ζ¹²	ζ³+ζ¹⁶	ζ+ζ¹⁸	ζ⁵+ζ¹⁴	ζ⁹+ζ¹⁰	ζ⁶+ζ¹³	ζ²+ζ¹⁷	0
χ₇	2	ζ⁵+ζ¹⁴	ζ⁹+ζ¹⁰	ζ⁴+ζ¹⁵	ζ+ζ¹⁸	ζ⁶+ζ¹³	ζ⁸+ζ¹¹	ζ³+ζ¹⁶	ζ²+ζ¹⁷	ζ⁷+ζ¹²	0
χ₈	2	ζ⁶+ζ¹³	ζ⁷+ζ¹²	ζ+ζ¹⁸	ζ⁵+ζ¹⁴	ζ⁸+ζ¹¹	ζ²+ζ¹⁷	ζ⁴+ζ¹⁵	ζ⁹+ζ¹⁰	ζ³+ζ¹⁶	0
χ₉	2	ζ⁷+ζ¹²	ζ⁵+ζ¹⁴	ζ²+ζ¹⁷	ζ⁹+ζ¹⁰	ζ³+ζ¹⁶	ζ⁴+ζ¹⁵	ζ⁸+ζ¹¹	ζ+ζ¹⁸	ζ⁶+ζ¹³	0
χ₁₀	2	ζ⁸+ζ¹¹	ζ³+ζ¹⁶	ζ⁵+ζ¹⁴	ζ⁶+ζ¹³	ζ²+ζ¹⁷	ζ⁹+ζ¹⁰	ζ+ζ¹⁸	ζ⁷+ζ¹²	ζ⁴+ζ¹⁵	0
χ₁₁	2	ζ⁹+ζ¹⁰	ζ+ζ¹⁸	ζ⁸+ζ¹¹	ζ²+ζ¹⁷	ζ⁷+ζ¹²	ζ³+ζ¹⁶	ζ⁶+ζ¹³	ζ⁴+ζ¹⁵	ζ⁵+ζ¹⁴	0

Here ζ is a primitive root of unity of order 19 (ζ¹⁹ = 1).

Relationships

maximal subgroup of	C₂×D₁₉, D₁₉⋊C₃, C₃×D₁₉, …
maximal normal subgroup of	C₂×D₁₉, D₁₉⋊C₃, C₃×D₁₉, …
maximal quotient of	Dic₁₉, C₂×D₁₉, C₃×D₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38, 1);