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Cayley diagram

C4×AGL(1,𝔽5) #701

order80
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = D4×AGL(1,𝔽5) (order 160) · Inn(G) = AGL(1,𝔽5) · Out(G) = D4
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C4

Aliases

BorelB(2,𝔽5)
direct productC4×AGL(1,𝔽5)

Splittings

FittingC4×C5 ⋊ C4
derived, SylowC5 ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC4AGL(1,𝔽5)
commutator, hypocenterC5C42
FrattiniC2C2×AGL(1,𝔽5)
FittingC4×C5C4
socleC2×C5C2×C4
prime radical, cosocleC2×D5C22
LCSG ▷ C5
UCSC1 ◁ C4

30 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:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×AGL(1,𝔽5), #3992, #4008, …
maximal normal subgroup ofC8×AGL(1,𝔽5), #3992, #4008, …
maximal quotient ofC8×AGL(1,𝔽5), #3992, C4×#170, …
automorphism group ofC5×D5, #674, C5×AGL(1,𝔽5)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 30);