Finite Group Databasebeta

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Cayley diagram

C4⋊(C5⋊S4) #91950

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

Aliases

unambiguous semidirect productC4⋊(C5⋊S4)

Splittings

FittingC22×C4×C5 ⋊ D3
SylowC5 ⋊ C4⋊S4

Subgroups

subgroupG/H
center, FrattiniC2C2×C5⋊S4
commutator, cosocleC2×C5×A4C22
FittingC22×C4×C5D3
socleC23×C5C2×D3
hypercenterC4C5⋊S4
hypocenterC5×A4D4
LCSG ▷ C2×C5×A4 ▷ C5×A4
UCSC1 ◁ C2 ◁ 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:

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

Polycyclic presentation:

2^3,3,5,2^2|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=g,g^a=f,f^d=g,g^d=fg

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751574, #11751575, #11751580, …
maximal normal subgroup of#11751574, #11751575, #11751580, …
maximal quotient ofC4⋊(C5⋊#226), C4⋊(C5⋊GL(2,𝔽3)), #11752114, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1026);