C₅×#67042 #13183421

order	1760
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#67042

Splittings

Fitting	C₅×C₁₁×#41 ⋊ C₂
Hall	C₅×C₁₁ ⋊ H(5,𝔽₂)
Sylow	C₁₁ ⋊ C₅×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂³×D₁₁
commutator, cosocle	C₂×C₁₁	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₁₁
Fitting	C₅×C₁₁×#41	C₂
socle	C₂×C₅×C₁₁	C₂⁴
hypercenter	C₅×#41	D₁₁
hypocenter	C₁₁	C₅×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41

332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 4 generators on 24 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1760, 1230);