C₅×#5071 #11750570

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

Aliases

direct product

C₅×#5071

Splittings

Hall	C₃×C₅ ⋊ #490
Sylow	C₃ ⋊ C₅×#490

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#127	C₂
socle	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₅×#127	D₃
hypocenter	C₃	C₅×#490
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#127

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

normal subgroups only

This group has 220 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) = 24
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#67539, #14279506, C₅×#71327, …
maximal normal subgroup of	C₅×#67539, #14279506, C₅×#71327, …
maximal quotient of	C₅×#69575, C₅×#69645, C₅×#71765, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4765);