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Cayley diagram

C5×D25 #6948

order250
isoclinismisoclinic class of D25
automorphismAut(G) = C4×AGL(1,ℤ/25) (order 2000) · Inn(G) = D25 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D25

Splittings

Fitting, SylowC5×C25 ⋊ C2
derivedC25 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC5D25
commutator, cosocle, hypocenterC25C2×C5
FrattiniC5C5×D5
FittingC5×C25C2
socleC52D5
LCSG ▷ C25
UCSC1 ◁ C5

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 50

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#1035, #92534, D5×D25, …
maximal normal subgroup ofC5×#1035, #92534, D5×D25, …
maximal quotient ofC5×Dic25, C2×C5×D25, C3×C5×D25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 3);