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Cayley diagram

C42×AGL(1,𝔽5) #65685

order320
isoclinismisoclinic class of AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C43 · Schur multiplier (H2) = C43

Aliases

direct productC42×AGL(1,𝔽5)

Splittings

FittingC42×C5 ⋊ C4
derived, SylowC5 ⋊ C43

Subgroups

subgroupG/H
center, hypercenterC42AGL(1,𝔽5)
commutator, hypocenterC5C43
FrattiniC22C22×AGL(1,𝔽5)
FittingC42×C5C4
socleC22×C5C22×C4
prime radical, cosocleC22×D5C23
LCSG ▷ C5
UCSC1 ◁ C42

258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 13 points:

(2,3,4,5);(1,2,3,5,4)(10,11,12,13);(6,7,8,9)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10598583, C4×C8×AGL(1,𝔽5), C4×#3992, …
maximal normal subgroup of#10598583, C4×C8×AGL(1,𝔽5), C4×#3992, …
maximal quotient of#10603773, AGL(1,𝔽5)×#95, C4×C8×AGL(1,𝔽5), …
automorphism group ofC5×#674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1023);