C₅×#5056 #11750555

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#5056

Splittings

Hall	C₃×C₅ ⋊ #488
Sylow	C₃ ⋊ C₅×#488

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×#124	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₅×#124	D₃
hypocenter	C₃	C₅×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#124

192 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 480

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

(1,2,3,4,5)(6,7,8)(10,14)(11,15)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,27)(21,26)(23,31)(24,29)(25,30)(28,32);(7,8)(9,10)(11,13)(12,14)(15,16)(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×#67543, C₅×#67547, #14279549, …
maximal normal subgroup of	C₅×#67543, C₅×#67547, #14279549, …
maximal quotient of	C₅×#69547, C₅×#69674, C₅×#69644, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4750);