Finite Group Databasebeta

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Cayley diagram

D5×D25 #92540

order500
isoclinismisoclinic class of D5×D25
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productD5×D25
unambiguous semidirect product(C5×D25)⋊C2
unambiguous semidirect product(D5×C25)⋊C2
unambiguous semidirect product(C5⋊D25)⋊C2

Splittings

Fitting, derived, SylowC5×C25 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC5×C25C22
FrattiniC5D52
socleC52C2×D5
LCSG ▷ C5×C25
UCSC1

38 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:

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

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×D25, D5×#1035, D5⋊#1035, …
maximal normal subgroup ofAGL(1,𝔽5)×D25, D5×#1035, D5⋊#1035, …
maximal quotient of#11758650, Dic5×D25, #11758652, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 26);