C₂³×C₃×D₃ #3852

order	144
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×AGL(3,𝔽₂) (order 16128) · Inn(G) = D₃ · Out(G) = C₂×AGL(3,𝔽₂)
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂³×C₃×D₃
unambiguous semidirect product	C₃⋊(C₂⁴×C₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₂³×C₃)

	subgroup	G/H
center, hypercenter	C₂³×C₃	D₃
commutator, cosocle, hypocenter	C₃	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₃

284 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 195);