Dic₁₂ #206

order	48
isoclinism	isoclinic class of #204
automorphism	Aut(G) = D₃×AGL(1,ℤ/8) (order 192) · Inn(G) = D₁₂ · Out(G) = C₂³
holomorph	G ⋊ Aut(G) = #72219802
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₁₂

C₃ ⋊ Q₁₆

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2^4,3|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 8);