C₃×#88163 #11928799

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

Aliases

direct product

C₃×#88163

Splittings

Fitting	C₃×C₁₃×M₁₆ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #137
Sylow	C₁₃ ⋊ C₃×#137

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×D₅₂
commutator, cosocle	C₄×C₁₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₁₃
Fitting	C₃×C₁₃×M₁₆	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
prime radical	C₃×#6127	C₂
hypercenter	C₃×M₁₆	D₁₃
hypocenter	C₁₃	C₃×#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×M₁₆

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

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, 911);