A₄×C₃⋊C₈ #63760

order	288
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = C₂²×D₃×S₄ (order 576) · Inn(G) = D₃×A₄ · Out(G) = C₂³
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₃⋊C₈
unambiguous semidirect product	C₃⋊(C₈×A₄)
unambiguous semidirect product	(C₂²×C₃⋊C₈)⋊C₃

C₂²×C₃ ⋊ C₃×C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×A₄
commutator, hypocenter	C₂²×C₃	C₃×C₈
Fitting, cosocle	C₂²×C₃×C₄	C₂×C₃
socle	C₂³×C₃	C₃×C₄
prime radical	C₂×C₃×A₄	C₄
LCS	G ▷ C₂²×C₃
UCS	C₁ ◁ C₄

58 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 15 points:

(2,3)(5,6,7)(8,12,14,9,11,15,10,13);(1,2,3)(4,7)(5,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 408);