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C₂₇×#120 #11720209

order	864
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₇ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₂₇	C₂³
commutator	C₂²	C₂³×C₂₇
Frattini, cosocle	C₂²×C₉	C₂³×C₃
socle	C₂²×C₃	C₂³×C₉
prime radical	C₃×#120	C₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇ ◁ G

260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29)(30,32)(31,33)(34,35);(29,32)(33,35);(29,33)(32,35)

Conjugacy classes & character table

378 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13134032, #13134537, #13136324, …
maximal subgroup of	#13134032, C₂₇×AΣL(1,𝔽₁₆), #13134537, …
maximal normal subgroup of	#13134032, C₂₇×AΣL(1,𝔽₁₆), #13134537, …
maximal quotient of	C₂₇×#379, C₂₇×#385, C₂₇×#386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 176);