Dic₃×A₄ #3786

order	144
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = C₂×D₃×S₄ (order 288) · Inn(G) = D₃×A₄ · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	Dic₃×A₄
unambiguous semidirect product	C₃⋊(C₄×A₄)
unambiguous semidirect product	(C₂²×Dic₃)⋊C₃

C₂²×C₃ ⋊ C₃×C₄

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

42 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 129);