C₃×#88205 #11928841

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

Aliases

direct product

C₃×#88205

Splittings

Hall	C₃×C₁₃ ⋊ #137
Sylow	C₁₃ ⋊ C₃×#137

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₁₃
Fitting, prime radical	C₃×C₁₃×#41	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₃×#41	D₁₃
hypocenter	C₁₃	C₃×#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#41

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 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)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(17,19)(18,22)(20,27)(21,26)(23,31)(24,29)(25,30)(28,32);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,31,32,29)

Conjugacy classes & character table

213 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 953);