C₅×#5000 #11750499

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#5000

Splittings

Hall	C₃×C₅ ⋊ #433
Sylow	C₃ ⋊ C₅×#433

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×C₅	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₅×#117	D₃
hypocenter	C₃	C₅×#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#117

192 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 110 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)(13,16)(19,25)(22,28)(26,31)(29,32);(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#433, C₂×C₅×#5000, D₅×#5000, …
maximal normal subgroup of	C₅×D₃×#433, C₂×C₅×#5000, D₅×#5000, …
maximal quotient of	C₅×#69435, C₅×#71582, C₅×#69436, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4694);