C₃×C₄×Q₁₆ #5313

order	192
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂³×#3003 (order 1024) · Inn(G) = D₄ · Out(G) = C₂⁴×D₄
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₃×C₄×Q₁₆

110 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 872);