C₅×#66985 #13183064

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

Aliases

direct product

C₅×#66985

Splittings

Fitting	C₂×C₅×D₄×C₁₁ ⋊ C₂
Hall	C₅×C₁₁ ⋊ AGL(1,ℤ/8)
Sylow	C₁₁ ⋊ C₅×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₁₁
Fitting	C₂×C₅×D₄×C₁₁	C₂
socle	C₂×C₅×C₁₁	C₂×D₄
hypercenter	C₂×C₅×D₄	D₁₁
hypocenter	C₁₁	C₅×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×D₄

136 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 136 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) = 24
transitive degree	μ_t(G) = 440

A minimal faithful permutation representation — 3 generators on 24 points:

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

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, 873);