Finite Group Databasebeta

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Cayley diagram

C3×#3930 #91000

order480
isoclinismisoclinic class of Dic10
automorphismAut(G) = C23×C4×AGL(1,𝔽5) (order 640) · Inn(G) = C2×D5 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1

Aliases

direct productC3×#3930

Splittings

FittingC3×C5×C16 ⋊ C2
HallC3×C5 ⋊ M32
SylowC5 ⋊ C3×M32

Subgroups

subgroupG/H
centerC3×C8C2×D5
commutatorC2×C5C2×C3×C8
FrattiniC8C2×C3×D5
FittingC3×C5×C16C2
socleC2×C3×C5C2×C8
prime radicalC2×C3×D5C8
cosocleC5×C8C22×C3
hypercenterC3×C16D5
hypocenterC5C3×M32
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C8 ◁ C3×C16

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#64845, C3×#64846, C3×#64847, …
maximal normal subgroup ofC3×#64845, C3×#64846, C3×#64847, …
maximal quotient ofC3×#64721, C3×#64722, C3×#64727, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 76);