C₃×#65468 #11750076

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

Aliases

direct product

C₃×#65468

Splittings

Fitting	C₂×C₃×C₅×Q₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #510
Sylow	C₅ ⋊ C₃×#510

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×#686
commutator, cosocle	C₅×C₈	C₂³×C₃
Frattini	C₈	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×Q₁₆	C₂
socle	C₂×C₃×C₅	C₂×D₈
prime radical	C₃×#4050	C₂
hypercenter	C₂×C₃×Q₁₆	D₅
hypocenter	C₅	C₃×#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×Q₁₆

164 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 70 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)(9,12,15,25)(10,17,19,33)(11,20,14,23)(13,27,29,40)(16,31,30,32)(18,34,35,36)(21,37,28,24)(22,38,39,26);(9,11,15,14)(10,16,19,30)(12,23,25,20)(13,21,29,28)(17,32,33,31)(18,26,35,38)(22,34,39,36)(24,40,37,27);(5,6)(7,8)(9,10)(11,16)(12,22)(13,26)(14,30)(15,19)(17,24)(18,28)(20,36)(21,35)(23,34)(25,39)(27,31)(29,38)(32,40)(33,37)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10601710, C₃×#10600057, #14315793, …
maximal normal subgroup of	C₃×#10601710, C₃×#10600057, #14315793, …
maximal quotient of	C₃×#10601391, C₃×#10602854, C₃×#10601389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4271);