D₃²⋊D₃ #6321

order	216
isoclinism	isoclinic class of D₃²⋊D₃
automorphism	Aut(G) = D₃×GO⁺(4,𝔽₂) (order 432) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₃²⋊D₃

C₃³ ⋊ D₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₃×C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃³	D₄
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁

60 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 9 points:

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

Polycyclic presentation:

2^3,3^3|b^a=bc,d^a=e,e^a=d,f^a=f^2,d^b=d^2,d^c=d^2,e^c=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 158);