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Cayley diagram

C4×S5 #91867

order480
isoclinismisoclinic class of S5
automorphismAut(G) = C22×S5 (order 480) · Inn(G) = S5 · Out(G) = C22
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

direct productC4×S5

Splittings

derivedA5 ⋊ C2×C4

Subgroups

subgroupG/H
center, Fitting, hypercenterC4S5
commutator, hypocenterA5C2×C4
FrattiniC2C2×S5
socle, cosocleC2×A5C22
prime radicalC2×S5C2
LCSG ▷ A5
UCSC1 ◁ C4

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 9 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×S5, #11751491, C2×C4×S5, …
maximal normal subgroup ofC8×S5, #11751491, C2×C4×S5, …
maximal quotient ofC8×S5, #11751490, #11751491, …
automorphism group ofC5×A5
outer automorphism group ofC5×2−1+4, C5×C22⋊A4

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 943);