Finite Group Databasebeta

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Cayley diagram

C3×#4065 #91636

order480
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#4065

Splittings

FittingC3×C5×Q16 ⋊ C2
HallC3×C5 ⋊ #135
SylowC5 ⋊ C3×#135

Subgroups

subgroupG/H
centerC2×C3D4×D5
commutator, cosocleC4×C5C23×C3
FrattiniC4C22×C3×D5
FittingC3×C5×Q16C2
socleC2×C3×C5C2×D4
hypercenterC3×Q16D5
hypocenterC5C3×#135
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q16

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#64907, C3×#64909, #11746778, …
maximal normal subgroup ofC3×#64907, C3×#64909, #11746778, …
maximal quotient ofC3×#65077, C3×#65085, C3×#65089, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 712);