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C₃×#870 #63993

order	288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

C₃² ⋊ #125

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₃
Fitting	C₃²×C₄²	C₂
socle, prime radical	C₂²×C₃²	C₂³
hypercenter	C₃×C₄²	D₃
hypocenter	C₃	C₃×#125
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3)(4,5,6)(8,12)(11,14)(15,21,18,17)(16,22,19,20);(7,13,10,9)(8,14,12,11)(16,19)(20,22);(5,6)(7,8)(9,11)(10,12)(13,14)(15,19,18,16)(17,22,21,20)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589088, C₃×#4465, #10590270, …
maximal normal subgroup of	#10589088, C₃×#4465, #10590270, …
maximal quotient of	C₃×#4657, C₃×#4660, C₃×#4928, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 641);