C₃×#65357 #11749965

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

Aliases

direct product

C₃×#65357

Splittings

Fitting	C₃×C₅×#125 ⋊ C₂
Hall	C₃×C₅ ⋊ #488
Sylow	C₅ ⋊ C₃×#488

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	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×D₂₀	C₂²
hypercenter	C₃×#125	D₅
hypocenter	C₅	C₃×#488
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#125

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598740, #14279540, C₃×#10598741, …
maximal normal subgroup of	C₃×#10598740, #14279540, C₃×#10598741, …
maximal quotient of	C₃×#10600340, C₃×#10600407, C₃×#10600408, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4160);