Finite Group Databasebeta

← Prev (#10620241) Next (#10620243) →

D325 #10620242

order650
isoclinismisoclinic class of D325
automorphismAut(G) order 78000
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD325
unambiguous semidirect productC13⋊D25
unambiguous semidirect productC25⋊D13

Splittings

Fitting, derived, HallC13×C25 ⋊ C2
SylowC25 ⋊ D13
SylowC13 ⋊ D25

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC13×C25C2
FrattiniC5D65
socleC5×C13D5
LCSG ▷ C13×C25
UCSC1

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 325

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)

Polycyclic presentation:

2,5,13,5|b^5=d,b^a=b^4d^4,c^a=c^12,d^a=d^4

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13049907, #13049908, D13×D25, …
maximal normal subgroup of#13049907, #13049908, D13×D25, …
maximal quotient ofDic325, C2×D325, C3×D325, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(650, 3);