C₄⋊(C₃⋊C₁₆) #4462

order	192
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) = C₂×D₃×#525 (order 768) · Inn(G) = C₂×D₃ · Out(G) = C₂²×#41
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

C₄⋊(C₃⋊C₁₆)

C₃ ⋊ C₄⋊C₁₆

50 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3)(4,5,6,7);(2,3)(5,7)(8,9,10,13,11,14,16,19,12,15,17,20,18,21,22,23)

Polycyclic presentation:

2^6,3|a^2=c,b^2=d,c^2=e,e^2=f,b^a=bd,g^a=g^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 21);