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Cayley diagram

C22×S5 #92110

order480
isoclinismisoclinic class of S5
automorphismAut(G) = S4×S5 (order 2880) · Inn(G) = S5 · Out(G) = S4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

direct productC22×S5
unambiguous semidirect productA5⋊C23
unambiguous semidirect product(C2×A5)⋊C22

Splittings

socleC22×A5 ⋊ C2
derivedA5 ⋊ C23

Subgroups

subgroupG/H
center, Fitting, hypercenterC22S5
commutator, cosocle, hypocenterA5C23
FrattiniC1G
socleC22×A5C2
LCSG ▷ A5
UCSC1 ◁ C22

225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 9 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×S5)⋊C4, C2×C4×S5, C2×C4⋊S5, …
maximal normal subgroup of(C2×S5)⋊C4, C2×C4×S5, C2×C4⋊S5, …
maximal quotient ofC2×C4×S5, #11756669, C2×C4⋊S5, …
automorphism group ofA5⋊C8, GL(2,𝔽5), #91143, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1186);