C₅×#5065 #11750564

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#5065

Splittings

Hall	C₃×C₅ ⋊ #499
Sylow	C₃ ⋊ C₅×#499

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#125	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₅×#125	D₃
hypocenter	C₃	C₅×#499
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#125

172 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 960

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,4,5)(6,7,8)(9,11,12,16)(10,13,14,15)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(9,10,12,14)(11,15,16,13)(17,19)(18,22)(20,27)(21,26)(23,31)(24,29)(25,30)(28,32);(7,8)(17,18,20,24,21,25,28,23)(19,22,27,29,26,30,32,31)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#499, C₂×C₅×#5065, D₅×#5065, …
maximal normal subgroup of	C₅×D₃×#499, C₂×C₅×#5065, D₅×#5065, …
maximal quotient of	C₅×#69549, C₅×#69618, C₅×#69619, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4759);