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Cayley diagram

C4×#168 #3934

order160
isoclinismisoclinic class of D5
automorphismAut(G) = AGL(1,𝔽5)×#2021 (order 2560) · Inn(G) = D5 · Out(G) = C2×#2021
homologyAbelianization (H1) = C4×C8 · Schur multiplier (H2) = C4

Aliases

direct productC4×#168

Splittings

derived, SylowC5 ⋊ C4×C8

Subgroups

subgroupG/H
center, hypercenterC42D5
commutator, hypocenterC5C4×C8
FrattiniC2×C4C2×D5
FittingC42×C5C2
socle, prime radicalC22×C5C2×C4
cosocleC2×C4×C5C22
LCSG ▷ C5
UCSC1 ◁ C42

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 17 points:

(2,3)(4,5)(10,11,12,14,13,15,16,17);(1,2,4,5,3)(6,7,8,9)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#168, #64674, #64675, …
maximal normal subgroup ofC8×#168, #64674, #64675, …
maximal quotient ofC8×#168, #64675, C4×#672, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 9);