C₃⋊(C₃⋊D₃) #278

order	54
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	Aut(G) order 303264
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃³

Aliases

C₃⋊(C₃⋊D₃)

C₃³ ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³	C₂
LCS	G ▷ C₃³
UCS	C₁

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 4 generators on 9 points:

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

Polycyclic presentation:

2,3^3|b^a=b^2,c^a=c^2,d^a=d^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 14);