C₃×#65156 #11749764

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

Aliases

direct product

C₃×#65156

Splittings

Fitting	C₃×C₅×#106 ⋊ C₂
Hall	C₃×C₅ ⋊ #485
Sylow	C₅ ⋊ C₃×#485

Subgroups

	subgroup	G/H
center	C₂²×C₃	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#106	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₅	D₄
hypercenter	C₃×#106	D₅
hypocenter	C₅	C₃×#485
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#106

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

normal subgroups only

This group has 192 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) = 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)(10,14)(13,16)(17,19,20,26,21,27,28,32)(18,23,24,31,22,25,29,30);(9,11,12,15)(10,13,14,16)(18,22)(19,26)(20,28)(23,30)(25,31)(27,32);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,30)(28,29)(31,32)

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₅×#485, C₂×C₃×#65156, #14431681, …
maximal normal subgroup of	C₃×D₅×#485, C₂×C₃×#65156, #14431681, …
maximal quotient of	C₃×#10599606, C₃×#10601373, C₃×#10599672, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 3959);