C₃×D₃² #1111

order	108
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = C₂×GO⁺(4,𝔽₂) (order 144) · Inn(G) = D₃² · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×D₃²
unambiguous semidirect product	(C₃×D₃)⋊(C₂×C₃)

Fitting, socle, Sylow	C₃³ ⋊ C₂²
derived	C₃² ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₃	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃³	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃

54 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)(4,5,6)(8,9);(5,6)(7,8,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 38);