D₃³ #6325

order	216
isoclinism	isoclinic class of D₃³
automorphism	Aut(G) = #10620184⋊C₂ (order 1296) · Inn(G) = G · Out(G) = D₃
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	D₃³
unambiguous semidirect product	(D₃×C₃⋊D₃)⋊C₂

C₃³ ⋊ C₂³

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³	C₂³
LCS	G ▷ C₃³
UCS	C₁

162 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 3 generators on 9 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 162);