D₃×S₄ #3840

order	144
isoclinism	isoclinic class of D₃×S₄
automorphism	Aut(G) = D₃×S₄ (order 144) · Inn(G) = G · Out(G) = C₁
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

direct product	D₃×S₄
unambiguous semidirect product	(C₃×S₄)⋊C₂
unambiguous semidirect product	(C₃⋊S₄)⋊C₂
unambiguous semidirect product	(D₃×A₄)⋊C₂

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

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

70 conjugacy classes of subgroups.

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 7 points:

(1,2,4,3)(5,6,7);(2,3,4)(6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 183);