C₃×#65334 #11749942

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

Aliases

direct product

C₃×#65334

Splittings

Hall	C₃×C₅ ⋊ #485
Sylow	C₅ ⋊ C₃×#485

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×C₅	D₄
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×#485
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×C₂²⋊Q₈

192 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 480

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

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

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₅×#485, C₂×C₃×#65334, #14431621, …
maximal normal subgroup of	C₃×D₅×#485, C₂×C₃×#65334, #14431621, …
maximal quotient of	C₃×#10600296, C₃×#10600299, C₃×#10602486, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4137);