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Cayley diagram

C2×D5⋊C8 #4125

order160
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = AGL(1,𝔽5)×B(4,𝔽2) (order 1280) · Inn(G) = AGL(1,𝔽5) · Out(G) = B(4,𝔽2)
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×D5⋊C8
unambiguous semidirect productD5⋊(C2×C8)

Splittings

derived, SylowC5 ⋊ C22×C8

Subgroups

subgroupG/H
center, hypercenterC2×C4AGL(1,𝔽5)
commutator, hypocenterC5C22×C8
FrattiniC2C22×AGL(1,𝔽5)
FittingC2×C4×C5C4
socleC22×C5C2×C4
prime radicalC22×D5C4
cosocleDic5C23
LCSG ▷ C5
UCSC1 ◁ C2×C4

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64874, (C2×AGL(1,𝔽5))⋊C8, #64901, …
maximal normal subgroup of#64874, (C2×AGL(1,𝔽5))⋊C8, #64901, …
maximal quotient ofC4×D5⋊C8, #65678, D5⋊(C4⋊C8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 200);