C₅×#66939 #13183013

order	1760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#66939

Splittings

Fitting	C₅×C₁₁×C₂²⋊C₄ ⋊ C₂
Hall	C₅×C₁₁ ⋊ #123
Sylow	C₁₁ ⋊ C₅×#123

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	C₂×C₅×C₂²⋊D₁₁	C₂
hypercenter	C₅×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	C₅×#123
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

156 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 28
transitive degree	μ_t(G) = 880

A minimal faithful permutation representation — 3 generators on 28 points:

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

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, 822);