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Cayley diagram

C2×C8×D5 #4045

order160
isoclinismisoclinic class of D5
automorphismAut(G) = C2×AGL(1,𝔽5)×#120 (order 1280) · Inn(G) = D5 · Out(G) = C22×#120
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×C8×D5

Splittings

FittingC2×C5×C8 ⋊ C2
derived, SylowC5 ⋊ C22×C8

Subgroups

subgroupG/H
center, hypercenterC2×C8D5
commutator, hypocenterC5C22×C8
FrattiniC4C22×D5
FittingC2×C5×C8C2
socleC22×C5C2×C4
prime radicalC22×D5C4
cosocleC4×C5C23
LCSG ▷ C5
UCSC1 ◁ C2×C8

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64727, #64887, (C2×AGL(1,𝔽5))⋊C8, …
maximal normal subgroup of#64727, #64887, (C2×AGL(1,𝔽5))⋊C8, …
maximal quotient ofC8×Dic10, C4×C8×D5, #64974, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 120);