C₃×D₃×A₄ #6329

order	216
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = D₃²×S₄ (order 864) · Inn(G) = D₃×A₄ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃×D₃×A₄
unambiguous semidirect product	C₃⋊(C₂×C₃×A₄)
unambiguous semidirect product	(C₂²×D₃)⋊C₃²
unambiguous semidirect product	(C₂²×C₃×D₃)⋊C₃

Fitting, socle	C₂²×C₃² ⋊ C₂×C₃
derived	C₂²×C₃ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃	D₃×A₄
commutator, cosocle, hypocenter	C₂²×C₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃²	C₂×C₃
LCS	G ▷ C₂²×C₃
UCS	C₁ ◁ C₃

81 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 10 points:

(2,3,4)(9,10);(1,4)(2,3)(5,6,7)(8,9,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 166);