Finite Group Databasebeta

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Cayley diagram

C5×#890 #91693

order480
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×#890

Splittings

HallC3×C5 ⋊ #125
SylowC3 ⋊ C5×#125

Subgroups

subgroupG/H
centerC22×C5C22×D3
commutator, cosocleC22×C3C23×C5
FrattiniC22C22×C5×D3
FittingC3×C5×C4⋊C4C2
socle, prime radicalC22×C3×C5C23
hypercenterC5×C4⋊C4D3
hypocenterC3C5×#125
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 480

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11747148, #11747149, #11747161, …
maximal normal subgroup of#11747148, #11747149, #11747161, …
maximal quotient ofC5×#4654, C5×#4655, C5×#4660, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 769);