C₃×#65085 #11749693

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

Aliases

direct product

C₃×#65085

Splittings

Hall	C₃×C₅ ⋊ #479
Sylow	C₅ ⋊ C₃×#479

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×#103	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₃×#103	D₅
hypocenter	C₅	C₃×#479
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#103

180 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 82 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)(10,13,14,15)(18,23)(19,25)(20,28)(22,29)(26,27)(30,32);(17,19,21,27)(18,22,23,30)(20,25,28,26)(24,32,31,29);(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,24)(21,23)(25,32)(26,29)(27,30)(28,31)

Conjugacy classes & character table

141 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₅×#479, C₂×C₃×#65085, #14430154, …
maximal normal subgroup of	C₃×D₅×#479, C₂×C₃×#65085, #14430154, …
maximal quotient of	C₃×#10601015, C₃×#10601029, C₃×#10601030, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 3888);