Finite Group Databasebeta

← Prev (#1226) Next (#1228) →

Cayley diagram

C5×S4 #1227

order120
isoclinismisoclinic class of S4
automorphismAut(G) = C4×S4 (order 96) · Inn(G) = S4 · Out(G) = C4
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×S42
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×S4
unambiguous semidirect productA4⋊(C2×C5)

Splittings

Fitting, socleC22×C5 ⋊ D3
derivedA4 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC5S4
commutator, cosocle, hypocenterA4C2×C5
FrattiniC1G
Fitting, socleC22×C5D3
LCSG ▷ A4
UCSC1 ◁ C5

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2,3,4,5)(6,7,8,9);(6,7)

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×S4, C2×C5×S4, C3×C5×S4, …
maximal normal subgroup ofD5×S4, C2×C5×S4, C3×C5×S4, …
maximal quotient ofC5×#226, C5×GL(2,𝔽3), C5×SL(2,ℤ/4), …
outer automorphism group ofC22×D11

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 37);