Dic₃ #20

order	12
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃ (order 12) · Inn(G) = D₃ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #3811
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃
unambiguous semidirect product	C₃⋊C₄

C₃ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃
commutator, hypocenter	C₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 7 points:

(1,2)(3,4)(5,6,7);(1,3,2,4)(6,7)

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12, 1);