Finite Group Databasebeta

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Cayley diagram

C5×#1035 #92530

order500
isoclinismisoclinic class of #1035
automorphismAut(G) = C4×AGL(1,ℤ/25) (order 2000) · Inn(G) = #1035 · Out(G) = C4×C5
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#1035

Splittings

Fitting, SylowC5×C25 ⋊ C4
derivedC25 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC5#1035
commutator, hypocenterC25C4×C5
FrattiniC5C5×AGL(1,𝔽5)
FittingC5×C25C4
socleC52AGL(1,𝔽5)
prime radicalC5×D25C2
cosocleD25C2×C5
LCSG ▷ C25
UCSC1 ◁ C5

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5)(7,9,10,15)(8,13,14,25)(11,19,20,24)(12,22,23,16)(17,27,28,30)(18,21,26,29);(6,7,11,21,18,8,12,9,16,27,13,24,17,28,19,25,30,22,15,23,14,26,29,20,10)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#1035, C2×C5×#1035, C3×C5×#1035, …
maximal normal subgroup ofD5×#1035, C2×C5×#1035, C3×C5×#1035, …
maximal quotient ofC5×#6016, C2×C5×#1035, C3×C5×#1035, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 16);