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C₁₉⋊C₃ #295

order	57
isoclinism	isoclinic class of C₁₉⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₁₉) (order 342) · Inn(G) = G · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72214746
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₉⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₁₉ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉	C₃
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:

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

Polycyclic presentation:

3,19|b^a=b^7

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉
size	1	3	3	3	3	3	3	19	19
order	1	19	19	19	19	19	19	3	3
χ₁	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	−ζ²−ζ¹⁴−ζ²²−ζ⁴⁰−ζ⁴¹−ζ⁵²	−ζ⁸−ζ³¹−ζ³⁷−ζ⁴⁶−ζ⁵⁰−ζ⁵⁶	−ζ⁴−ζ²³−ζ²⁵−ζ²⁸−ζ⁴⁴−ζ⁴⁷	−ζ−ζ⁷−ζ¹¹−ζ²⁰−ζ²⁶−ζ⁴⁹	−ζ¹⁰−ζ¹³−ζ²⁹−ζ³²−ζ³⁴−ζ⁵³	−ζ⁵−ζ¹⁶−ζ¹⁷−ζ³⁵−ζ⁴³−ζ⁵⁵	0	0
χ₅	3	−ζ⁸−ζ³¹−ζ³⁷−ζ⁴⁶−ζ⁵⁰−ζ⁵⁶	−ζ¹⁰−ζ¹³−ζ²⁹−ζ³²−ζ³⁴−ζ⁵³	−ζ⁵−ζ¹⁶−ζ¹⁷−ζ³⁵−ζ⁴³−ζ⁵⁵	−ζ⁴−ζ²³−ζ²⁵−ζ²⁸−ζ⁴⁴−ζ⁴⁷	−ζ²−ζ¹⁴−ζ²²−ζ⁴⁰−ζ⁴¹−ζ⁵²	−ζ−ζ⁷−ζ¹¹−ζ²⁰−ζ²⁶−ζ⁴⁹	0	0
χ₆	3	−ζ⁴−ζ²³−ζ²⁵−ζ²⁸−ζ⁴⁴−ζ⁴⁷	−ζ⁵−ζ¹⁶−ζ¹⁷−ζ³⁵−ζ⁴³−ζ⁵⁵	−ζ⁸−ζ³¹−ζ³⁷−ζ⁴⁶−ζ⁵⁰−ζ⁵⁶	−ζ²−ζ¹⁴−ζ²²−ζ⁴⁰−ζ⁴¹−ζ⁵²	−ζ−ζ⁷−ζ¹¹−ζ²⁰−ζ²⁶−ζ⁴⁹	−ζ¹⁰−ζ¹³−ζ²⁹−ζ³²−ζ³⁴−ζ⁵³	0	0
χ₇	3	−ζ−ζ⁷−ζ¹¹−ζ²⁰−ζ²⁶−ζ⁴⁹	−ζ⁴−ζ²³−ζ²⁵−ζ²⁸−ζ⁴⁴−ζ⁴⁷	−ζ²−ζ¹⁴−ζ²²−ζ⁴⁰−ζ⁴¹−ζ⁵²	−ζ¹⁰−ζ¹³−ζ²⁹−ζ³²−ζ³⁴−ζ⁵³	−ζ⁵−ζ¹⁶−ζ¹⁷−ζ³⁵−ζ⁴³−ζ⁵⁵	−ζ⁸−ζ³¹−ζ³⁷−ζ⁴⁶−ζ⁵⁰−ζ⁵⁶	0	0
χ₈	3	−ζ¹⁰−ζ¹³−ζ²⁹−ζ³²−ζ³⁴−ζ⁵³	−ζ²−ζ¹⁴−ζ²²−ζ⁴⁰−ζ⁴¹−ζ⁵²	−ζ−ζ⁷−ζ¹¹−ζ²⁰−ζ²⁶−ζ⁴⁹	−ζ⁵−ζ¹⁶−ζ¹⁷−ζ³⁵−ζ⁴³−ζ⁵⁵	−ζ⁸−ζ³¹−ζ³⁷−ζ⁴⁶−ζ⁵⁰−ζ⁵⁶	−ζ⁴−ζ²³−ζ²⁵−ζ²⁸−ζ⁴⁴−ζ⁴⁷	0	0
χ₉	3	−ζ⁵−ζ¹⁶−ζ¹⁷−ζ³⁵−ζ⁴³−ζ⁵⁵	−ζ−ζ⁷−ζ¹¹−ζ²⁰−ζ²⁶−ζ⁴⁹	−ζ¹⁰−ζ¹³−ζ²⁹−ζ³²−ζ³⁴−ζ⁵³	−ζ⁸−ζ³¹−ζ³⁷−ζ⁴⁶−ζ⁵⁰−ζ⁵⁶	−ζ⁴−ζ²³−ζ²⁵−ζ²⁸−ζ⁴⁴−ζ⁴⁷	−ζ²−ζ¹⁴−ζ²²−ζ⁴⁰−ζ⁴¹−ζ⁵²	0	0

Here ζ is a primitive root of unity of order 57 (ζ⁵⁷ = 1); i = ζ¹⁴.

Relationships

maximal subgroup of	D₁₉⋊C₃, C₂×C₁₉⋊C₃, #4296, …
maximal normal subgroup of	D₁₉⋊C₃, C₂×C₁₉⋊C₃, #4296, …
maximal quotient of	C₂×C₁₉⋊C₃, #4294, C₃×C₁₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(57, 1);