order	324
isoclinism	isoclinic class of C₃⋊(C₃⋊(C₃⋊D₃))
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃⁶

Aliases

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

Splittings

Fitting, socle	C₂×C₃⁴ ⋊ C₂
derived, Sylow	C₃⁴ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₃⋊(C₃⋊D₃))
commutator, cosocle, hypocenter	C₃⁴	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃⁴	C₂
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₂

1060 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 5 generators on 14 points:

(1,2)(4,5)(7,8)(10,11)(13,14);(1,2)(3,4,5);(6,7,8);(9,10,11);(12,13,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 175);