C₃×#65707 #11752240

order	960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#65707

Splittings

Hall	C₃×C₅ ⋊ #446
Sylow	C₅ ⋊ C₃×#446

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×C₄⋊C₄	C₄
socle, prime radical	C₂²×C₃×C₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂³×C₃
hypercenter	C₃×C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#446
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊C₄

188 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 960

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

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃×#65707, C₃×D₅⋊#446, D₃×#65707, …
maximal normal subgroup of	C₂×C₃×#65707, C₃×D₅⋊#446, D₃×#65707, …
maximal quotient of	C₃×#10603780, C₃×#10604316, C₃×#10603781, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 6435);