Dic₁₃ #259

order	52
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₃) (order 312) · Inn(G) = D₁₃ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = #72214210
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

Dic₁₃

C₁₃ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃
commutator, hypocenter	C₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(52, 1);