C₁₃×#909 #11928885

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#909

Splittings

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

Subgroups

	subgroup	G/H
center	C₂×C₁₃	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂³×C₁₃
Frattini	C₄	C₂²×D₃×C₁₃
Fitting	C₃×C₁₃×M₁₆	C₂
socle	C₂×C₃×C₁₃	C₂×D₄
prime radical	C₁₃×#235	C₂
hypercenter	C₁₃×M₁₆	D₃
hypocenter	C₃	C₁₃×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₁₃×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:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16)(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);(15,16)(17,18)(19,23)(20,27)(21,30)(22,25)(24,28)(26,32)(29,31)

Conjugacy classes & character table

273 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 997);