C₇×C₃⋊S₄ #92750

order	504
isoclinism	isoclinic class of C₃⋊S₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₇×C₃⋊S₄
unambiguous semidirect product	C₃⋊(C₇×S₄)
unambiguous semidirect product	A₄⋊(D₃×C₇)

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

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

60 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 176);