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Cayley diagram

D5×C25 #6949

order250
isoclinismisoclinic class of D5
automorphismAut(G) = C4×C5×AGL(1,𝔽5) (order 400) · Inn(G) = D5 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C2×C25 · Schur multiplier (H2) = C1

Aliases

direct productD5×C25
unambiguous semidirect productC5⋊(C2×C25)

Splittings

Fitting, SylowC5×C25 ⋊ C2
derivedC5 ⋊ C2×C25

Subgroups

subgroupG/H
center, hypercenterC25D5
commutator, hypocenterC5C2×C25
FrattiniC5C5×D5
FittingC5×C25C2
socle, cosocleC52C2×C5
prime radicalC5×D5C5
LCSG ▷ C5
UCSC1 ◁ C25

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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C25, #92533, D5×D25, …
maximal normal subgroup ofAGL(1,𝔽5)×C25, #92533, D5×D25, …
maximal quotient ofDic5×C25, C2×D5×C25, C3×D5×C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 4);