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Cayley diagram

D5⋊C16 #3989

order160
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = #41×AGL(1,𝔽5) (order 320) · Inn(G) = AGL(1,𝔽5) · Out(G) = #41
homologyAbelianization (H1) = C2×C16 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD5⋊C16

Splittings

derived, SylowC5 ⋊ C2×C16

Subgroups

subgroupG/H
center, hypercenterC8AGL(1,𝔽5)
commutator, hypocenterC5C2×C16
FrattiniC4C2×AGL(1,𝔽5)
FittingC5×C8C4
socleC2×C5C2×C8
prime radicalC2×D5C8
cosocle#168C22
LCSG ▷ C5
UCSC1 ◁ C8

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3,5,4)(6,8,10,17,11,9,16,18)(7,12,14,20,15,13,19,21);(2,3,4,5)(6,7,9,13,10,14,18,21,11,15,8,12,16,19,17,20)

Polycyclic presentation:

2^5,5|a^2=c,b^2=d,c^2=d,d^2=e,f^a=f^2,f^c=f^4

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC16×AGL(1,𝔽5), #64844, #64904, …
maximal normal subgroup ofC16×AGL(1,𝔽5), #64844, #64904, …
maximal quotient ofD5⋊C32, #64842, #64885, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 64);