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C₃×#874 #63997

order	288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

Splittings

Fitting	C₃²×C₄² ⋊ C₂
Sylow	C₃² ⋊ #127

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	C₂²×C₃²	C₂³
hypercenter	C₃×C₄²	D₃
hypocenter	C₃	C₃×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 14 points:

(1,2,3)(4,5,6)(11,14,13,12);(7,10,9,8);(5,6)(8,10)(12,14)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589090, C₃×#4485, #10590273, …
maximal normal subgroup of	#10589090, C₃×#4485, #10590273, …
maximal quotient of	C₃×#4671, C₃×#4677, C₃×#4693, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 645);