C₃×#4183 #92320

order	486
isoclinism	isoclinic class of #4183
automorphism	Aut(G) = C₂×(#4174⋊C₂)⋊C₃ (order 1944) · Inn(G) = #4183 · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

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

	subgroup	G/H
center, hypercenter	C₃	#4183
commutator, cosocle, hypocenter	AΣL(1,𝔽₂₇)	C₂×C₃
Frattini	C₃²	C₃×C₃⋊D₃
Fitting	C₃×AΣL(1,𝔽₂₇)	C₂
socle	C₃²	#272
LCS	G ▷ AΣL(1,𝔽₂₇)
UCS	C₁ ◁ C₃

150 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 165);