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C₁₃⋊C₃ #166

order	39
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₁₃) (order 156) · Inn(G) = G · Out(G) = C₄
holomorph	G ⋊ Aut(G) = #14523144⋊C₃
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₃⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₁₃ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃	C₃
LCS	G ▷ C₁₃
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 13

A minimal faithful permutation representation — 2 generators on 13 points:

(2,3,5)(4,7,10)(6,9,8)(11,13,12);(1,2,4,3,6,10,7,11,13,5,8,12,9)

Polycyclic presentation:

3,13|b^a=b^3

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇
size	1	3	3	3	3	13	13
order	1	13	13	13	13	3	3
χ₁	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	−ζ²−ζ⁵−ζ¹⁹−ζ²⁸−ζ³¹−ζ³²	−ζ−ζ¹⁴−ζ¹⁶−ζ²²−ζ²⁹−ζ³⁵	−ζ⁴−ζ¹⁰−ζ¹⁷−ζ²³−ζ²⁵−ζ³⁸	−ζ⁷−ζ⁸−ζ¹¹−ζ²⁰−ζ³⁴−ζ³⁷	0	0
χ₅	3	−ζ−ζ¹⁴−ζ¹⁶−ζ²²−ζ²⁹−ζ³⁵	−ζ⁷−ζ⁸−ζ¹¹−ζ²⁰−ζ³⁴−ζ³⁷	−ζ²−ζ⁵−ζ¹⁹−ζ²⁸−ζ³¹−ζ³²	−ζ⁴−ζ¹⁰−ζ¹⁷−ζ²³−ζ²⁵−ζ³⁸	0	0
χ₆	3	−ζ⁴−ζ¹⁰−ζ¹⁷−ζ²³−ζ²⁵−ζ³⁸	−ζ²−ζ⁵−ζ¹⁹−ζ²⁸−ζ³¹−ζ³²	−ζ⁷−ζ⁸−ζ¹¹−ζ²⁰−ζ³⁴−ζ³⁷	−ζ−ζ¹⁴−ζ¹⁶−ζ²²−ζ²⁹−ζ³⁵	0	0
χ₇	3	−ζ⁷−ζ⁸−ζ¹¹−ζ²⁰−ζ³⁴−ζ³⁷	−ζ⁴−ζ¹⁰−ζ¹⁷−ζ²³−ζ²⁵−ζ³⁸	−ζ−ζ¹⁴−ζ¹⁶−ζ²²−ζ²⁹−ζ³⁵	−ζ²−ζ⁵−ζ¹⁹−ζ²⁸−ζ³¹−ζ³²	0	0

Here ζ is a primitive root of unity of order 39 (ζ³⁹ = 1); i = ζ⁹.

Relationships

maximal subgroup of	D₁₃⋊C₃, C₂×C₁₃⋊C₃, C₃×C₁₃⋊C₃, …
maximal normal subgroup of	D₁₃⋊C₃, C₂×C₁₃⋊C₃, C₃×C₁₃⋊C₃, …
maximal quotient of	C₂×C₁₃⋊C₃, C₁₃⋊C₉, C₃×C₁₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(39, 1);