C₅×#5835 #11756220

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#5835

Splittings

Fitting	C₃×C₅×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₅ ⋊ #576
Sylow	C₃ ⋊ C₅×#576

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₅×H(5,𝔽₂)	D₃
hypocenter	C₃	C₅×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41 ◁ C₅×H(5,𝔽₂)

536 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 214 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 — 4 generators on 24 points:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#80280, C₅×#84015, C₅×#84009, …
maximal normal subgroup of	C₅×#80280, C₅×#84015, C₅×#84009, …
maximal quotient of	C₅×#82563, C₅×#84099, C₅×#82562, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10415);