C₂×A₄² #64381

order	288
isoclinism	isoclinic class of A₄²
automorphism	Aut(G) = #10596770⋊C₂ (order 1152) · Inn(G) = A₄² · Out(G) = D₄
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product	C₂×A₄²
unambiguous semidirect product	(C₂³×A₄)⋊C₃
unambiguous semidirect product	(C₂×C₂²⋊A₄)⋊C₃

Fitting, socle, Sylow	C₂⁵ ⋊ C₃²
derived	C₂⁴ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₂	A₄²
commutator, cosocle, hypocenter	C₂⁴	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂⁵	C₃²
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₂

122 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1029);