C₃×C₇×A₄ #7000

order	252
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₂×C₃×D₃×S₄ (order 864) · Inn(G) = A₄ · Out(G) = C₂²×C₃×D₃
homology	Abelianization (H₁) = C₃²×C₇ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃×C₇×A₄
unambiguous semidirect product	C₂²⋊(C₃²×C₇)
unambiguous semidirect product	(C₂²×C₃)⋊(C₃×C₇)

	subgroup	G/H
center, hypercenter	C₃×C₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₇

28 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)(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(252, 39);