A₄² #3841

order	144
isoclinism	isoclinic class of A₄²
automorphism	Aut(G) = #10596770⋊C₂ (order 1152) · Inn(G) = G · Out(G) = D₄
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product	A₄²
unambiguous semidirect product	(C₂²×A₄)⋊C₃
unambiguous semidirect product	(C₂²⋊A₄)⋊C₃

C₂⁴ ⋊ C₃²

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂⁴	C₃²
LCS	G ▷ C₂⁴
UCS	C₁

41 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 8 points:

(1,4)(2,3)(6,7,8);(2,3,4)(5,8)(6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 184);