C₃⋊(C₃⋊D₉) #4209

order	162
isoclinism	isoclinic class of C₃⋊(C₃⋊D₉)
automorphism	Aut(G) order 1889568
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃³

Aliases

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

C₃²×C₉ ⋊ C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃²×C₉	C₂
Frattini	C₃	C₃⋊(C₃⋊D₃)
socle	C₃³	D₃
LCS	G ▷ C₃²×C₉
UCS	C₁

100 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 81

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

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

Polycyclic presentation:

2,3^4|b^3=e^2,b^a=b^2e,c^a=c^2,d^a=d^2,e^a=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 45);