C₃×#65167 #11749775

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#65167

Splittings

Hall	C₃×C₅ ⋊ #499
Sylow	C₅ ⋊ C₃×#499

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#107	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₃×#107	D₅
hypocenter	C₅	C₃×#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#107

172 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598615, #14278495, C₃×#10598616, …
maximal normal subgroup of	C₃×#10598615, #14278495, C₃×#10598616, …
maximal quotient of	C₃×#10601400, C₃×#10599651, C₃×#10601404, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 3970);