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Cayley diagram

C5×A5 #64479

order300
isoclinismisoclinic class of A5
automorphismAut(G) = C4×S5 (order 480) · Inn(G) = A5 · Out(G) = C2×C4
homologyAbelianization (H1) = C5 · Schur multiplier (H2) = C2

Aliases

general unitaryGU(2,𝔽4)
direct productC5×A5

Subgroups

subgroupG/H
center, Fitting, hypercenterC5A5
commutator, hypocenterA5C5
FrattiniC1G
LCSG ▷ A5
UCSC1 ◁ C5

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 25

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊S5, D5×A5, A5⋊Dic5, …
socle ofC5×S5, C5⋊S5, D5×A5, …
maximal subgroup ofC5×S5, C5⋊S5, D5×A5, …
maximal normal subgroup ofC5×S5, C5⋊S5, D5×A5, …
maximal quotient ofC5×SL(2,𝔽5), C2×C5×A5, C3×C5×A5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 22);