D₁₃ #77

order	26
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = AGL(1,𝔽₁₃) (order 156) · Inn(G) = G · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72210302
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃
unambiguous semidirect product	C₁₃⋊C₂

C₁₃ ⋊ C₂

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

4 conjugacy classes of subgroups.

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

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

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

Polycyclic presentation:

2,13|b^a=b^12

Here ζ is a primitive root of unity of order 13 (ζ¹³ = 1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26, 1);