Finite Group Databasebeta

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Cayley diagram

C3×#3951 #91021

order480
isoclinismisoclinic class of #677
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#3951

Splittings

HallC3×C5 ⋊ #108
SylowC5 ⋊ C3×#108

Subgroups

subgroupG/H
centerC3×C4D20
commutatorC4×C5C2×C3×C4
FrattiniC2×C4C2×C3×D5
FittingC2×C3×C5×C8C2
socleC2×C3×C5C4⋊C4
prime radicalC22×C3×C5Q8
cosocleC2×C4×C5C22×C3
hypercenterC2×C3×C8D5
hypocenterC5C3×#108
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

60 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)(6,17,9,8,11,16,10,18)(7,15,19,12,14,21,13,20)(22,23,24);(2,3)(4,5)(6,13,10,7,11,19,9,14)(8,20,17,15,18,12,16,21)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745905, #11745907, #11745917, …
maximal normal subgroup of#11745905, #11745907, #11745917, …
maximal quotient ofC3×#64677, C3×#64678, C3×#64772, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 97);