C₅×#5055 #11750554

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

Aliases

direct product

C₅×#5055

Splittings

Hall	C₃×C₅ ⋊ #489
Sylow	C₃ ⋊ C₅×#489

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×#489
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#124

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

normal subgroups only

This group has 200 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)(17,18)(19,23)(20,21)(22,24);(9,11,12,15)(10,13,14,16)(18,21)(22,23);(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,19,22,20,21,24,23)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#71949, C₅×#68352, #14291019, …
maximal normal subgroup of	C₅×#71949, C₅×#68352, #14291019, …
maximal quotient of	C₅×#69548, C₅×#69675, C₅×#69645, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4749);