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Cayley diagram

C8×AGL(1,𝔽5) #3991

order160
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2×D4
homologyAbelianization (H1) = C4×C8 · Schur multiplier (H2) = C4

Aliases

direct productC8×AGL(1,𝔽5)

Splittings

FittingC5×C8 ⋊ C4
derived, SylowC5 ⋊ C4×C8

Subgroups

subgroupG/H
center, hypercenterC8AGL(1,𝔽5)
commutator, hypocenterC5C4×C8
FrattiniC4C2×AGL(1,𝔽5)
FittingC5×C8C4
socleC2×C5C42
prime radicalC2×D5C2×C4
cosocleC4×D5C22
LCSG ▷ C5
UCSC1 ◁ C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC16×AGL(1,𝔽5), #64844, C2×C8×AGL(1,𝔽5), …
maximal normal subgroup ofC16×AGL(1,𝔽5), #64844, C2×C8×AGL(1,𝔽5), …
maximal quotient ofC16×AGL(1,𝔽5), #64844, C8×#170, …
automorphism group of#3928

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 66);