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Cayley diagram

C4×A5 #6731

order240
isoclinismisoclinic class of A5
automorphismAut(G) = C2×S5 (order 240) · Inn(G) = A5 · Out(G) = C22
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×A5

Subgroups

subgroupG/H
center, Fitting, hypercenterC4A5
commutator, hypocenterA5C4
Frattini, cosocleC2C2×A5
socle, prime radicalC2×A5C2
LCSG ▷ A5
UCSC1 ◁ C4

35 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:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11751487, #11751488, #11751489, …
maximal subgroup ofA5⋊C8, C8×A5, C4×S5, …
maximal normal subgroup ofA5⋊C8, C8×A5, C4×S5, …
maximal quotient ofC8×A5, #91145, C4×SL(2,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 92);