Finite Group Databasebeta

← Prev (#4128) Next (#4130) →

Cayley diagram

C2×C4⋊AGL(1,𝔽5) #4129

order160
isoclinismisoclinic class of #700
automorphismAut(G) = AGL(1,𝔽5)×B(4,𝔽2) (order 1280) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = #120
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

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

Splittings

FittingC2×C4×C5 ⋊ C4
SylowC5 ⋊ C2×C4⋊C4

Subgroups

subgroupG/H
centerC22C2×AGL(1,𝔽5)
commutatorC2×C5C22×C4
FrattiniC2C22×AGL(1,𝔽5)
FittingC2×C4×C5C4
socleC22×C5C2×C4
prime radicalC22×D5C22
cosocleC2×D5C23
hypercenterC2×C4AGL(1,𝔽5)
hypocenterC5C2×C4⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4

92 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:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64874, #64893, #64899, …
maximal normal subgroup of#64874, #64893, #64899, …
maximal quotient ofD5⋊(C4⋊C8), #65681, #65682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 204);