C₅×#4915 #11750414

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

Aliases

direct product

C₅×#4915

Splittings

Fitting	C₃×C₅×SD₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #510
Sylow	C₃ ⋊ C₅×#510

Subgroups

	subgroup	G/H
center	C₂×C₅	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³×C₅
Frattini	C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×SD₃₂	C₂
socle	C₂×C₃×C₅	C₂×D₈
prime radical	C₅×#912	C₂
hypercenter	C₅×SD₃₂	D₃
hypocenter	C₃	C₅×#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×SD₃₂

164 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4,5)(6,7,8)(10,16)(11,19)(12,23)(13,24)(15,27)(17,34)(20,21)(22,38)(25,39)(26,37)(28,36)(29,40)(30,33)(31,35);(9,11,14,22)(10,15,16,30)(12,19,25,38)(13,26,24,21)(17,33,31,27)(18,36,32,29)(20,39,37,23)(28,34,40,35);(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,33)(20,28)(21,29)(22,30)(23,35)(24,32)(25,31)(26,36)(27,38)(34,39)(37,40)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×D₃×#510, C₂×C₅×#4915, D₅×#4915, …
maximal normal subgroup of	C₅×D₃×#510, C₂×C₅×#4915, D₅×#4915, …
maximal quotient of	C₅×#69189, C₅×#69222, C₅×#69346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4609);