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Cayley diagram

C4×C5×S4 #91938

order480
isoclinismisoclinic class of S4
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = S4 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC4×C5×S4

Splittings

FittingC22×C4×C5 ⋊ D3
derivedA4 ⋊ C2×C4×C5

Subgroups

subgroupG/H
center, hypercenterC4×C5S4
commutator, hypocenterA4C2×C4×C5
FrattiniC2C2×C5×S4
FittingC22×C4×C5D3
socleC23×C5C2×D3
prime radicalC2×C5×S4C2
cosocleC2×A4C22×C5
LCSG ▷ A4
UCSC1 ◁ 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×#168, #11751571, C5×C8×S4, …
maximal normal subgroup ofS4×#168, #11751571, C5×C8×S4, …
maximal quotient ofC4×C5×#226, C5×#5388, C4×C5×GL(2,𝔽3), …
automorphism group ofQ8×C25, A4×C25

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1014);