Dic₁₃⋊C₃ #3904

order	156
isoclinism	isoclinic class of D₁₃⋊C₃
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₃) (order 312) · Inn(G) = D₁₃⋊C₃ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #72242057
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

Dic₁₃⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃⋊C₃
commutator, hypocenter	C₁₃	C₃×C₄
Fitting, socle, cosocle	C₂×C₁₃	C₂×C₃
prime radical	C₂×C₁₃⋊C₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂

12 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(156, 1);