C₃×#65205 #11749813

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#65205

Splittings

Fitting	C₃×C₅×SD₃₂ ⋊ C₂
Hall	C₃×C₅ ⋊ #508
Sylow	C₅ ⋊ C₃×#508

Subgroups

	subgroup	G/H
center	C₂×C₃	D₅×D₈
commutator, cosocle	C₅×C₈	C₂³×C₃
Frattini	C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×SD₃₂	C₂
socle	C₂×C₃×C₅	C₂×D₈
hypercenter	C₃×SD₃₂	D₅
hypocenter	C₅	C₃×#508
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×SD₃₂

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

normal subgroups only

This group has 168 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)(4,5,7,8,6)(9,11,14,22)(10,16,15,19)(12,24,26,39)(13,27,25,21)(17,34,28,40)(18,36,29,32)(20,37,38,23)(30,35,31,33);(10,15)(11,20)(12,23)(13,25)(16,30)(17,33)(19,31)(21,39)(22,38)(24,27)(26,37)(28,35)(32,34)(36,40);(5,6)(7,8)(9,10)(11,19)(12,17)(13,18)(14,15)(16,22)(20,30)(21,36)(23,35)(24,40)(25,29)(26,28)(27,32)(31,38)(33,37)(34,39)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10604128, #14315636, C₃×#10604129, …
maximal normal subgroup of	C₃×#10604128, #14315636, C₃×#10604129, …
maximal quotient of	C₃×#10599981, C₃×#10600013, C₃×#10600138, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4008);