C₃⋊(C₃⋊(C₃⋊D₉)) #92402

order	486
isoclinism	isoclinic class of C₃⋊(C₃⋊(C₃⋊D₉))
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃⁶

Aliases

unambiguous semidirect product	C₃⋊(C₃⋊(C₃⋊D₉))
unambiguous semidirect product	C₉⋊(C₃⋊(C₃⋊D₃))

C₃³×C₉ ⋊ C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃³×C₉	C₂
Frattini	C₃	C₃⋊(C₃⋊(C₃⋊D₃))
socle	C₃⁴	D₃
LCS	G ▷ C₃³×C₉
UCS	C₁

792 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 5 generators on 18 points:

(2,3)(5,6)(8,9)(11,13)(12,16)(14,15)(17,18);(7,8,9);(4,5,6);(1,2,3);(10,18,13,16,15,14,12,11,17)

Polycyclic presentation:

2,3^5|b^3=f^2,b^a=b^2f,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 247);