C₃×#88143 #11928780

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

Aliases

direct product

C₃×#88143

Splittings

Hall	C₃×C₁₃ ⋊ #125
Sylow	C₁₃ ⋊ C₃×#125

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₁₃
Fitting	C₃×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃×C₁₃	C₂³
hypercenter	C₃×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₃×#125
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊C₄

112 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1248

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

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 892);