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Cayley diagram

C2×C4×AGL(1,𝔽5) #4128

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) = C2×C42 · Schur multiplier (H2) = C22×C4

Aliases

direct productC2×C4×AGL(1,𝔽5)
unambiguous semidirect productD5⋊C42

Splittings

FittingC2×C4×C5 ⋊ C4
derived, SylowC5 ⋊ C2×C42

Subgroups

subgroupG/H
center, hypercenterC2×C4AGL(1,𝔽5)
commutator, hypocenterC5C2×C42
FrattiniC2C22×AGL(1,𝔽5)
FittingC2×C4×C5C4
socleC22×C5C2×C4
prime radicalC22×D5C22
cosocleC2×D5C23
LCSG ▷ C5
UCSC1 ◁ C2×C4

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 3 generators on 11 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64875, (C2×AGL(1,𝔽5))⋊C8, #64900, …
maximal normal subgroup of#64875, (C2×AGL(1,𝔽5))⋊C8, #64900, …
maximal quotient ofC4×D5⋊C8, #65676, C4×#700, …
automorphism group of#672, C5×Dic5, C2×C5×D5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 203);