C₁₃×#898 #11928874

order	1248
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃×#898

Splittings

Fitting	C₃×C₁₃×C₄⋊C₄ ⋊ C₂
Hall	C₃×C₁₃ ⋊ #126
Sylow	C₃ ⋊ C₁₃×#126

Subgroups

	subgroup	G/H
center	C₂²×C₁₃	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₁₃
Frattini	C₂²	C₂²×D₃×C₁₃
Fitting	C₃×C₁₃×C₄⋊C₄	C₂
socle	C₂²×C₃×C₁₃	C₂³
prime radical	C₂²×D₃×C₁₃	C₂²
hypercenter	C₁₃×C₄⋊C₄	D₃
hypocenter	C₃	C₁₃×#126
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×C₄⋊C₄

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)(18,22)(21,24)(25,27,28,31)(26,30,29,32);(17,19,20,23)(18,21,22,24)(26,29)(27,31);(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,29)(31,32)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 986);