Finite Group Databasebeta

← Prev (#91938) Next (#91940) →

Cayley diagram

C5×C4⋊S4 #91939

order480
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×C4⋊S4
unambiguous semidirect productC4⋊(C5×S4)

Splittings

FittingC22×C4×C5 ⋊ D3

Subgroups

subgroupG/H
centerC2×C5C2×S4
commutator, cosocleC2×A4C22×C5
FrattiniC2C2×C5×S4
FittingC22×C4×C5D3
socleC23×C5C2×D3
hypercenterC4×C5S4
hypocenterA4C5×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C5 ◁ C4×C5

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2,3,4)(5,6,8)(9,10,11,12,13);(2,4)(6,7)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751573, #11751575, #11751579, …
maximal normal subgroup of#11751573, #11751575, #11751579, …
maximal quotient ofC5×C4⋊#226, C5×C4⋊GL(2,𝔽3), C5×#5395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1015);