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D₁₃⋊C₃ #665

order	78
isoclinism	isoclinic class of D₁₃⋊C₃
automorphism	Aut(G) = AGL(1,𝔽₁₃) (order 156) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72220623
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₁₃⋊C₃
unambiguous semidirect product	(C₁₃⋊C₃)⋊C₂

Splittings

Fitting, socle, derived, Sylow	C₁₃ ⋊ C₂×C₃
Hall	C₁₃⋊C₃ ⋊ C₂
Hall	D₁₃ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃	C₂×C₃
LCS	G ▷ C₁₃
UCS	C₁

8 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:

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

Polycyclic presentation:

2,3,13|c^a=c^12,c^b=c^3

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈
size	1	6	6	13	13	13	13	13
order	1	13	13	2	3	3	6	6
χ₁	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	-1	1	1	-1	-1
χ₅	1	1	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	1	1	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	6	$\frac{- 1 + \sqrt{13}}{2}$	$\frac{- 1 - \sqrt{13}}{2}$	0	0	0	0	0
χ₈	6	$\frac{- 1 - \sqrt{13}}{2}$	$\frac{- 1 + \sqrt{13}}{2}$	0	0	0	0	0

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup of	AGL(1,𝔽₁₃), C₂×D₁₃⋊C₃, C₃×D₁₃⋊C₃, …
maximal normal subgroup of	AGL(1,𝔽₁₃), C₂×D₁₃⋊C₃, C₃×D₁₃⋊C₃, …
maximal quotient of	Dic₁₃⋊C₃, C₂×D₁₃⋊C₃, D₁₃⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(78, 1);