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C₃×#953 #64076

order	288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

Splittings

Fitting	C₂⁴×C₃² ⋊ C₂
Sylow	C₃² ⋊ #120

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₃
Fitting	C₂⁴×C₃²	C₂
socle	C₂²×C₃²	C₂³
hypercenter	C₂⁴×C₃	D₃
hypocenter	C₃	C₃×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃

327 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 24

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

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591725, C₃×#5032, #10595494, …
maximal normal subgroup of	#10591725, C₃×#5032, #10595494, …
maximal quotient of	C₃×#5210, C₃×#5213, C₃×#5217, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 724);