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Cayley diagram

C5×C4⋊AGL(1,𝔽5) #87844

order400
isoclinismisoclinic class of #700
automorphismAut(G) = C4×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×C4⋊AGL(1,𝔽5)
unambiguous semidirect productC4⋊(C5×AGL(1,𝔽5))

Splittings

FittingC4×C52 ⋊ C4
SylowC52 ⋊ C4⋊C4

Subgroups

subgroupG/H
centerC2×C5C2×AGL(1,𝔽5)
commutatorC2×C5C2×C4×C5
FrattiniC2C2×C5×AGL(1,𝔽5)
FittingC4×C52C4
socleC2×C52C2×C4
prime radicalC2×C5×D5C22
cosocleC2×D5C22×C5
hypercenterC4×C5AGL(1,𝔽5)
hypocenterC5C5×C4⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C5 ◁ C4×C5

55 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11716734, #11716736, #11716742, …
maximal normal subgroup of#11716734, #11716736, #11716742, …
maximal quotient ofC5×#3993, C5×#3994, C5×#3995, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 138);