C₃×#65696 #11752229

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#65696

Splittings

Hall	C₃×C₅ ⋊ #424
Sylow	C₅ ⋊ C₃×#424

Subgroups

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₂²⋊C₄	C₄
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³×C₃
hypercenter	C₃×C₂²⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#424
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊C₄

228 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 104 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)(4,5,6,8,7)(9,10,11,12)(15,21)(18,24)(22,27)(25,28);(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(5,6,7,8)(10,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28)

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#10603921, #14300395, C₂×C₃×#65696, …
maximal normal subgroup of	C₃×#10603921, #14300395, C₂×C₃×#65696, …
maximal quotient of	C₃×#10603897, C₃×#10603898, C₃×#10603899, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 6424);