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Cayley diagram

C5×C22⋊AGL(1,𝔽5) #87847

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) = C22

Aliases

direct productC5×C22⋊AGL(1,𝔽5)
unambiguous semidirect productC22⋊(C5×AGL(1,𝔽5))
unambiguous semidirect product(C2×AGL(1,𝔽5))⋊(C2×C5)

Splittings

FittingC22×C52 ⋊ C4
SylowC52 ⋊ C22⋊C4

Subgroups

subgroupG/H
centerC2×C5C2×AGL(1,𝔽5)
commutatorC2×C5C2×C4×C5
FrattiniC2C2×C5×AGL(1,𝔽5)
FittingC22×C52C4
socleC2×C52C2×C4
prime radicalC22×C5×D5C2
cosocleC2×D5C22×C5
hypercenterC22×C5AGL(1,𝔽5)
hypocenterC5C5×C22⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C5 ◁ C22×C5

73 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 40

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11716755, #11716756, C5×#3999, …
maximal normal subgroup of#11716755, #11716756, C5×#3999, …
maximal quotient ofC5×#3999, C5×#4003, C5×#4005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 141);