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Cayley diagram

C22×C5×S4 #92122

order480
isoclinismisoclinic class of S4
automorphism?
homologyAbelianization (H1) = C23×C5 · Schur multiplier (H2) = C24

Aliases

direct productC22×C5×S4
unambiguous semidirect productA4⋊(C23×C5)
unambiguous semidirect product(C2×A4)⋊(C22×C5)

Splittings

Fitting, socleC24×C5 ⋊ D3
derivedA4 ⋊ C23×C5

Subgroups

subgroupG/H
center, hypercenterC22×C5S4
commutator, cosocle, hypocenterA4C23×C5
FrattiniC1G
Fitting, socleC24×C5D3
LCSG ▷ A4
UCSC1 ◁ C22×C5

262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×S4)⋊Dic5, C5×(C2×S4)⋊C4, C2×Dic5×S4, …
maximal normal subgroup of(C2×S4)⋊Dic5, C5×(C2×S4)⋊C4, C2×Dic5×S4, …
maximal quotient ofC2×C5×A4⋊Q8, C2×C4×C5×S4, C2×C5×C4⋊S4, …
automorphism group ofC11×#41, C3×Q8×C11, C11×#226
outer automorphism group ofC2×Q8×C11

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1198);