C₃×C₇×S₄ #92748

order	504
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂²×C₃×S₄ (order 288) · Inn(G) = S₄ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×C₇×S₄
unambiguous semidirect product	A₄⋊(C₂×C₃×C₇)

Fitting, socle	C₂²×C₃×C₇ ⋊ D₃
derived	A₄ ⋊ C₂×C₃×C₇

	subgroup	G/H
center, hypercenter	C₃×C₇	S₄
commutator, cosocle, hypocenter	A₄	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₃×C₇

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

(1,2,4,3)(5,6,7,8,9,10,11)(12,13,14);(2,3,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 174);