C₃×#88130 #11928766

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

supersolvable
metabelian
solvable
monomial

Aliases

direct product

C₃×#88130

Splittings

Fitting	C₃×C₄²×C₁₃ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #126
Sylow	C₁₃ ⋊ C₃×#126

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₁₃
Fitting	C₃×C₄²×C₁₃	C₂
socle	C₂²×C₃×C₁₃	C₂³
prime radical	C₂²×C₃×D₁₃	C₂²
hypercenter	C₃×C₄²	D₁₃
hypocenter	C₁₃	C₃×#126
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 624

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

330 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 878);

C3×#88130 #11928766