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Cayley diagram

C4×C5⋊S4 #91949

order480
isoclinismisoclinic class of C5⋊S4
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

direct productC4×C5⋊S4

Splittings

FittingC22×C4×C5 ⋊ D3
derivedC5×A4 ⋊ C2×C4
SylowC5 ⋊ C4×S4

Subgroups

subgroupG/H
center, hypercenterC4C5⋊S4
commutator, hypocenterC5×A4C2×C4
FrattiniC2C2×C5⋊S4
FittingC22×C4×C5D3
socleC23×C5C2×D3
prime radicalC2×C5⋊S4C2
cosocleC2×C5×A4C22
LCSG ▷ C5×A4
UCSC1 ◁ C4

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751570, #11751572, C8×C5⋊S4, …
maximal normal subgroup of#11751570, #11751572, C8×C5⋊S4, …
maximal quotient ofC4×C5⋊#226, #11752107, C4×C5⋊GL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1025);