Finite Group Databasebeta

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Cayley diagram

C5×#904 #91707

order480
isoclinismisoclinic class of #204
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#904

Splittings

FittingC2×C3×C5×C8 ⋊ C2
HallC3×C5 ⋊ #135
SylowC3 ⋊ C5×#135

Subgroups

subgroupG/H
centerC4×C5D12
commutator, cosocleC3×C4C23×C5
FrattiniC4C22×C5×D3
FittingC2×C3×C5×C8C2
socleC2×C3×C5C2×D4
hypercenterC2×C5×C8D3
hypocenterC3C5×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×C8

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745905, #11745906, #11745911, …
maximal normal subgroup of#11745905, #11745906, #11745911, …
maximal quotient ofC5×#4683, C4×C5×#204, C4×C5×D24, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 783);