order	192
automorphism	Aut(G) = S₄×AGL(1,ℤ/8) (order 768) · Inn(G) = D₄×A₄ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×Q₁₆
unambiguous semidirect product	(C₂²×Q₁₆)⋊C₃

Splittings

C₂²×Q₁₆ ⋊ C₃

73 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1016);