C₃×#65185 #11749793

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

Aliases

direct product

C₃×#65185

Splittings

Fitting	C₃×C₅×#108 ⋊ C₂
Hall	C₃×C₅ ⋊ #473
Sylow	C₅ ⋊ C₃×#473

Subgroups

	subgroup	G/H
center	C₂×C₃	#4041
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#108	C₂
socle	C₂×C₃×C₅	#121
prime radical	C₃×#709	C₂²
hypercenter	C₃×#108	D₅
hypocenter	C₅	C₃×#473
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#108

200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 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)(9,12,13,26,15,27,14,29)(10,17,18,24,19,35,28,34)(11,20,21,25,23,37,22,39)(16,30,31,36,32,40,38,33);(9,11,14,22,15,23,13,21)(10,16,18,31,19,32,28,38)(12,25,26,37,27,39,29,20)(17,33,34,40,35,36,24,30);(5,6)(7,8)(9,10)(11,16)(12,24)(13,28)(14,18)(15,19)(17,26)(20,36)(21,38)(22,31)(23,32)(25,30)(27,34)(29,35)(33,37)(39,40)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10600037, C₃×#10600052, #14316547, …
maximal normal subgroup of	C₃×#10600037, C₃×#10600052, #14316547, …
maximal quotient of	C₃×#10599370, C₃×#10599380, C₃×#10599371, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 3988);