C₃⋊#4169 #92292

order	486
isoclinism	isoclinic class of C₃⋊#4169
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃²

Aliases

Fitting, Sylow	C₃×C₃²⋊C₉ ⋊ C₂
derived	C₃²×C₉ ⋊ C₂×C₃

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

177 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 137);