C₅×#4745 #11750244

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#4745

Splittings

Fitting	C₃×C₅×#100 ⋊ C₂
Hall	C₃×C₅ ⋊ #413
Sylow	C₃ ⋊ C₅×#413

Subgroups

	subgroup	G/H
center	C₂×C₅	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#100	C₂
socle	C₂×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#237	C₂
cosocle	C₂×C₃×C₄	C₂³×C₅
hypercenter	C₅×#100	D₃
hypocenter	C₃	C₅×#413
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#100

300 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 240

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

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#69136, C₅×#69144, #14313168, …
maximal normal subgroup of	C₅×#69136, C₅×#69144, #14313168, …
maximal quotient of	C₅×#68100, C₅×#68119, C₅×#68102, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4439);