C₇×S₄ #4275

order	168
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂×C₃×S₄ (order 144) · Inn(G) = S₄ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = S₄²×AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×S₄
unambiguous semidirect product	A₄⋊(C₂×C₇)

Fitting, socle	C₂²×C₇ ⋊ D₃
derived	A₄ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	S₄
commutator, cosocle, hypocenter	A₄	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₇	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₇

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2,3,4,5,6,7)(8,9,10,11);(8,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 45);