S₄ #71

order	24
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = S₄ (order 24) · Inn(G) = G · Out(G) = C₁
holomorph	G ⋊ Aut(G) = S₄²
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

projective general linear	PGL(2,𝔽₃)
symmetric	S₄
affine general linear	AGL(2,𝔽₂)
affine general semilinear	AΓL(1,𝔽₄)
special orthogonal	SO(3,𝔽₃)
unambiguous semidirect product	A₄⋊C₂

Fitting, socle	C₂² ⋊ D₃
derived	A₄ ⋊ C₂

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

11 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:

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

Polycyclic presentation:

2,3,2^2|b^a=b^2,c^a=d,d^a=c,c^b=d,d^b=cd

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24, 12);