C₅×#66932 #13183007

order	1760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#66932

Splittings

Hall	C₅×C₁₁ ⋊ #126
Sylow	C₁₁ ⋊ C₅×#126

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂²×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂³×C₅
Frattini	C₂²	C₂²×C₅×D₁₁
Fitting	C₅×C₁₁×C₂²⋊C₄	C₂
socle	C₂²×C₅×C₁₁	C₂³
prime radical	C₂³×C₅×C₁₁	C₂²
hypercenter	C₅×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	C₅×#126
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊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) = 880

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1760, 816);