Finite Group Databasebeta

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Cayley diagram

C3×#4050 #91621

order480
isoclinismisoclinic class of #677
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#4050

Splittings

FittingC2×C3×C5×C8 ⋊ C2
HallC3×C5 ⋊ #135
SylowC5 ⋊ C3×#135

Subgroups

subgroupG/H
centerC3×C4D20
commutator, cosocleC4×C5C23×C3
FrattiniC4C22×C3×D5
FittingC2×C3×C5×C8C2
socleC2×C3×C5C2×D4
hypercenterC2×C3×C8D5
hypocenterC5C3×#135
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×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:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745903, #11745904, #11745909, …
maximal normal subgroup of#11745903, #11745904, #11745909, …
maximal quotient ofC3×#64972, C3×C4×#677, C3×C4×D40, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 697);