C₃×#65349 #11749957

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#65349

Splittings

Hall	C₃×C₅ ⋊ #497
Sylow	C₅ ⋊ C₃×#497

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×#124	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₃×#124	D₅
hypocenter	C₅	C₃×#497
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#124

248 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 28
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10598036, C₃×#10598040, #14279422, …
maximal normal subgroup of	C₃×#10598036, C₃×#10598040, #14279422, …
maximal quotient of	C₃×#10601977, C₃×#10602010, C₃×#10602012, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4152);