C₃⋊#4175 #92322

order	486
isoclinism	isoclinic class of C₃⋊#4175
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃²

Aliases

Fitting, Sylow	C₃×AΣL(1,𝔽₂₇) ⋊ C₂
derived	C₃⁴ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃⁴	C₂×C₃
Frattini	C₃²	C₃×C₃⋊D₃
Fitting	C₃×AΣL(1,𝔽₂₇)	C₂
socle	C₃²	(C₃⋊D₃)⋊C₃
LCS	G ▷ C₃⁴
UCS	C₁

204 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 27

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 167);