order	108
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	Aut(G) order 606528
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃³

Aliases

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

Splittings

Fitting, socle	C₂×C₃³ ⋊ C₂
derived, Sylow	C₃³ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃³	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

140 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 4 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 44);