C₅×#66995 #13183071

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

Aliases

direct product

C₅×#66995

Splittings

Fitting	C₂×C₅×Q₈×C₁₁ ⋊ C₂
Hall	C₅×C₁₁ ⋊ #137
Sylow	C₁₁ ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₁₁
Fitting	C₂×C₅×Q₈×C₁₁	C₂
socle	C₂×C₅×C₁₁	C₂×D₄
prime radical	C₅×#4339	C₂
hypercenter	C₂×C₅×Q₈	D₁₁
hypocenter	C₁₁	C₅×#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×Q₈

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1760, 880);