Dic₃₉ #3908

order	156
isoclinism	isoclinic class of D₃₉
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = D₃₉ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₉
unambiguous semidirect product	C₃⋊Dic₁₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₉
commutator, hypocenter	C₃×C₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₃	C₂
LCS	G ▷ C₃×C₁₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20);(1,3,2,4)(6,7)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(156, 5);