C₂×D₃×A₄ #3847

order	144
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = C₂×D₃×S₄ (order 288) · Inn(G) = D₃×A₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×D₃×A₄
unambiguous semidirect product	C₃⋊(C₂²×A₄)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×A₄)
unambiguous semidirect product	(C₂³×D₃)⋊C₃

Fitting, socle	C₂³×C₃ ⋊ C₂×C₃
derived	C₂²×C₃ ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₂	D₃×A₄
commutator, cosocle, hypocenter	C₂²×C₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₃	C₂×C₃
LCS	G ▷ C₂²×C₃
UCS	C₁ ◁ C₂

82 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 9 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 190);