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Cayley diagram

C2×C4×D5 #707

order80
isoclinismisoclinic class of D5
automorphismAut(G) = AGL(1,𝔽5)×#120 (order 640) · Inn(G) = D5 · Out(G) = C2×#120
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4×D5

Splittings

FittingC2×C4×C5 ⋊ C2
derived, SylowC5 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4D5
commutator, hypocenterC5C22×C4
FrattiniC2C22×D5
FittingC2×C4×C5C2
socleC22×C5C22
prime radicalC22×D5C2
cosocleC2×C5C23
LCSG ▷ C5
UCSC1 ◁ C2×C4

54 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:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#3952, #4003, (C2×AGL(1,𝔽5))⋊C4, …
maximal normal subgroup of#3952, #4003, (C2×AGL(1,𝔽5))⋊C4, …
maximal quotient ofC4×Dic10, C42×D5, #4018, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 36);