C₃×#88189 #11928824

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#88189

Splittings

Hall	C₃×C₁₃ ⋊ #123
Sylow	C₁₃ ⋊ C₃×#123

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₁₃
Fitting, prime radical	C₂×C₃×D₄×C₁₃	C₂
socle	C₂²×C₃×C₁₃	C₂³
hypercenter	C₂×C₃×D₄	D₁₃
hypocenter	C₁₃	C₃×#123
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×D₄

156 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 28
transitive degree	μ_t(G) = 624

A minimal faithful permutation representation — 3 generators on 28 points:

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

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, 936);