A₄ #22

order	12
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = S₄ (order 24) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = A₄⋊S₄
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

alternating	A₄
projective special linear	PSL(2,𝔽₃)
projective symplectic	PSp(2,𝔽₃)
affine general linear	AGL(1,𝔽₄)
omega	Ω(3,𝔽₃)
projective special unitary	PSU(2,𝔽₃)
unambiguous semidirect product	C₂²⋊C₃

C₂² ⋊ C₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²	C₃
LCS	G ▷ C₂²
UCS	C₁

5 conjugacy classes of subgroups.

minimal degree	μ(G) = 4
transitive degree	μ_t(G) = 4

A minimal faithful permutation representation — 2 generators on 4 points:

(2,3,4);(1,2,3)

Polycyclic presentation:

3,2^2|b^a=c,c^a=bc

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12, 3);