C₃⋊(C₉⋊D₉) #92335

order	486
isoclinism	isoclinic class of C₃⋊(C₉⋊D₉)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃²×C₉

Aliases

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

C₃×C₉² ⋊ C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₉²	C₂
Frattini	C₃²	C₃⋊(C₃⋊D₃)
socle	C₃³	C₃⋊D₃
LCS	G ▷ C₃×C₉²
UCS	C₁

252 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 4 generators on 21 points:

(2,3)(5,7)(6,10)(8,9)(11,12)(14,16)(15,19)(17,18)(20,21);(1,2,3);(13,21,16,19,18,17,15,14,20);(4,12,7,10,9,8,6,5,11)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 180);