Finite Group Databasebeta

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Cayley diagram

C3×#4072 #91643

order480
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×#4072

Splittings

HallC3×C5 ⋊ #117
SylowC5 ⋊ C3×#117

Subgroups

subgroupG/H
centerC2×C3×C4C2×D5
commutatorC2×C5C22×C3×C4
FrattiniC22C22×C3×D5
FittingC22×C3×C4×C5C2
socleC22×C3×C5C23
prime radicalC23×C3×C5C22
cosocleC22×C5C23×C3
hypercenterC22×C3×C4D5
hypocenterC5C3×#117
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745969, #11745970, #11745973, …
maximal normal subgroup of#11745969, #11745970, #11745973, …
maximal quotient ofC3×#65214, C3×#65215, C3×#65221, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 719);