C₂⁴×A₄ #5980

order	192
isoclinism	isoclinic class of A₄
automorphism	Aut(G) order 483840
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

C₂⁴×A₄

Fitting, socle, Sylow	C₂⁶ ⋊ C₃
derived	C₂² ⋊ C₂⁴×C₃

	subgroup	G/H
center, hypercenter	C₂⁴	A₄
commutator, cosocle, hypocenter	C₂²	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂⁶	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂⁴

1165 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1539);