C₃×#65358 #11749966

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

Aliases

direct product

C₃×#65358

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×#125	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₃×#125	D₅
hypocenter	C₅	C₃×#491
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#125

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598736, #14278375, C₃×#10598737, …
maximal normal subgroup of	C₃×#10598736, #14278375, C₃×#10598737, …
maximal quotient of	C₃×#10600341, C₃×#10600409, C₃×#10600345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4161);