C₃×#65257 #11749865

order	960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#65257

Splittings

Fitting	C₂×C₃×C₅×C₄⋊C₄ ⋊ C₂
Hall	C₃×C₅ ⋊ #481
Sylow	C₅ ⋊ C₃×#481

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃×C₅	C₂×D₄
prime radical	C₃×#4076	C₂
hypercenter	C₂×C₃×C₄⋊C₄	D₅
hypocenter	C₅	C₃×#481
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₄

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

normal subgroups only

This group has 228 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) = 24
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

177 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10601961, C₃×#10600507, #14288537, …
maximal normal subgroup of	C₃×#10601961, C₃×#10600507, #14288537, …
maximal quotient of	C₃×#10602347, C₃×#10602349, C₃×#10602353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 4060);