C₅×#5966 #11757128

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₅×#5966

Splittings

Fitting	C₃×C₅×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₅ ⋊ #585
Sylow	C₃ ⋊ C₅×#585

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂⁴×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₅
Frattini	C₂	C₂⁴×C₅×D₃
Fitting	C₃×C₅×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₅	C₂⁵
hypercenter	C₅×H(5,𝔽₂)	D₃
hypocenter	C₃	C₅×#585
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×H(5,𝔽₂)

1620 conjugacy classes of subgroups; only the 886 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 886 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) = 240

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#80277, C₅×#80291, #14425238, …
maximal normal subgroup of	C₅×#80277, C₅×#80291, #14425238, …
maximal quotient of	C₅×#85856, C₅×#85885, C₅×#85857, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 11323);