Finite Group Databasebeta

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Cayley diagram

C3×#4053 #91624

order480
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#4053

Splittings

FittingC3×C5×M16 ⋊ C2
HallC3×C5 ⋊ #131
SylowC5 ⋊ C3×#131

Subgroups

subgroupG/H
centerC3×C4C22×D5
commutatorC2×C5C22×C3×C4
FrattiniC4C22×C3×D5
FittingC3×C5×M16C2
socleC2×C3×C5C22×C4
prime radicalC3×#709C2
cosocleC4×C5C23×C3
hypercenterC3×M16D5
hypocenterC5C3×#131
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C3×M16

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#64898, #11746769, #11748202, …
maximal normal subgroup ofC3×#64898, #11746769, #11748202, …
maximal quotient ofC3×#64990, C3×#64995, C3×#64997, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 700);