C₃×#876 #63999

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

Aliases

direct product

C₃×#876

Splittings

Fitting	C₃²×C₄² ⋊ C₂
Sylow	C₃² ⋊ #126

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
prime radical	C₂²×C₃×D₃	C₂²
hypercenter	C₃×C₄²	D₃
hypocenter	C₃	C₃×#126
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

135 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)(15,22,18,17)(16,21,19,20);(7,13,10,9)(8,14,12,11)(16,19)(20,21);(5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,21)(18,19)(20,22)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10593914, #10593919, #10593924, …
maximal normal subgroup of	#10593914, #10593919, #10593924, …
maximal quotient of	C₃×#4658, C₃×C₃⋊#401, C₃×#4675, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 647);

C3×#876 #63999

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×#876 #63999