Finite Group Databasebeta

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Cayley diagram

C5×#901 #91704

order480
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#901

Splittings

FittingC2×C3×C5×C8 ⋊ C2
HallC3×C5 ⋊ #131
SylowC3 ⋊ C5×#131

Subgroups

subgroupG/H
centerC5×C8C2×D3
commutatorC2×C3C22×C4×C5
FrattiniC4C22×C5×D3
FittingC2×C3×C5×C8C2
socleC2×C3×C5C22×C4
prime radicalC5×#235C2
cosocleC3×C4C23×C5
hypercenterC2×C5×C8D3
hypocenterC3C5×#131
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C5×C8 ◁ 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)(4,13,7,18,8,6,14,12)(5,15,11,19,9,10,16,17)(20,21,22,23,24);(5,9)(10,15)(11,16)(17,19);(2,3)(4,5)(6,10)(7,11)(8,9)(12,17)(13,15)(14,16)(18,19)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745887, #11745888, C5×#4508, …
maximal normal subgroup of#11745887, #11745888, C5×#4508, …
maximal quotient ofC5×C8×Dic6, C5×#4679, C5×C8×D12, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 780);