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Cayley diagram

C2×C24⋊C5 #4160

order160
isoclinismisoclinic class of C24⋊C5
automorphismAut(G) = AΓL(1,𝔽16) (order 960) · Inn(G) = C24⋊C5 · Out(G) = Dic3
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C22

Aliases

affine special semilinearAΣL(1,𝔽32)
direct productC2×C24⋊C5
unambiguous semidirect productC25⋊C5

Splittings

Fitting, socle, SylowC25 ⋊ C5
derivedC24 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC2C24⋊C5
commutator, cosocle, hypocenterC24C2×C5
FrattiniC1G
Fitting, socleC25C5
LCSG ▷ C24
UCSC1 ◁ C2

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 10

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

derived subgroup of(C24⋊C5)⋊Q8, C4⋊((C24⋊C5)⋊C2), C22⋊((C24⋊C5)⋊C2), …
maximal subgroup of#66245, C4×C24⋊C5, C2×(C24⋊C5)⋊C2, …
maximal normal subgroup of#66245, C4×C24⋊C5, C2×(C24⋊C5)⋊C2, …
maximal quotient ofC4×C24⋊C5, C2×2−1+4⋊C5, #585⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 235);