C₄⋊GL(2,𝔽₃) #5393

order	192
isoclinism	isoclinic class of #5386
automorphism	Aut(G) = S₄×#120 (order 768) · Inn(G) = C₂×S₄ · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

C₄×Q₈ ⋊ D₃

91 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,3,4)(5,6,10)(8,11,12);(2,4)(6,7)(9,11)(10,12)

Polycyclic presentation:

2^3,3,2^3|b^2=c,e^2=g,f^2=g,b^a=bc,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 952);