D₃×C₇×A₄ #92757

order	504
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = C₂×C₃×D₃×S₄ (order 864) · Inn(G) = D₃×A₄ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×C₇×A₄
unambiguous semidirect product	C₃⋊(C₂×C₇×A₄)
unambiguous semidirect product	(C₂²×D₃)⋊(C₃×C₇)

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

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

52 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 183);