C₃×#65371 #11749979

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#65371

Splittings

Hall	C₃×C₅ ⋊ #491
Sylow	C₅ ⋊ C₃×#491

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×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#491
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

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

normal subgroups only

This group has 180 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) = 960

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

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598752, #14278373, C₃×#10598042, …
maximal normal subgroup of	C₃×#10598752, #14278373, C₃×#10598042, …
maximal quotient of	C₃×#10600368, C₃×#10600468, C₃×#10602579, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4174);