C₂³×C₃⋊(C₃⋊D₃) #89127

order	432
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₃³

Aliases

Fitting, socle	C₂³×C₃³ ⋊ C₂
derived, Sylow	C₃³ ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂³	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂³×C₃³	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂³

1876 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 4 generators on 15 points:

(1,2)(3,4)(5,6)(8,9)(11,12)(14,15);(5,6)(7,8,9);(3,4)(10,11,12);(1,2)(13,14,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 774);