Finite Group Databasebeta

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Cayley diagram

C3×#4094 #91665

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

Aliases

direct productC3×#4094

Splittings

HallC3×C5 ⋊ #131
SylowC5 ⋊ C3×#131

Subgroups

subgroupG/H
centerC3×C4C22×D5
commutatorC2×C5C22×C3×C4
FrattiniC4C22×C3×D5
Fitting, prime radicalC3×C5×#41C2
socleC2×C3×C5C22×C4
cosocleC4×C5C23×C3
hypercenterC3×#41D5
hypocenterC5C3×#131
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C3×#41

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#64932, #11746803, #11749432, …
maximal normal subgroup ofC3×#64932, #11746803, #11749432, …
maximal quotient ofC3×#65286, C3×#65288, C3×#65289, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 741);