C₃×#88170 #11928806

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

Aliases

direct product

C₃×#88170

Splittings

Fitting	C₃×C₁₃×SD₁₆ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #135
Sylow	C₁₃ ⋊ C₃×#135

Subgroups

	subgroup	G/H
center	C₂×C₃	D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₁₃
Fitting	C₃×C₁₃×SD₁₆	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
hypercenter	C₃×SD₁₆	D₁₃
hypocenter	C₁₃	C₃×#135
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×SD₁₆

124 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 54 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)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,31,30,32);(18,22)(19,26)(20,28)(23,31)(25,32)(27,29);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,30)(29,32)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 918);