C₃⋊#4174 #92321

order	486
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

Fitting, Sylow	C₃×AΣL(1,𝔽₂₇) ⋊ C₂
derived	C₃×H(3,𝔽₃) ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₃	C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂×C₃
Frattini	C₃²	C₃×C₃⋊D₃
Fitting	C₃×AΣL(1,𝔽₂₇)	C₂
socle	C₃²	(C₃⋊D₃)⋊C₃
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₃

144 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 3 generators on 12 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 166);