order	36
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×AGL(2,𝔽₃) (order 864) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×S₄
holomorph	G ⋊ Aut(G) = #72219734
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

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

Splittings

Fitting, socle	C₂×C₃² ⋊ C₂
derived, Sylow	C₃² ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃²	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 3 generators on 8 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36, 13);