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Cayley diagram

D205 #88015

order410
isoclinismisoclinic class of D205
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD205
unambiguous semidirect productC5⋊D41
unambiguous semidirect productC41⋊D5

Splittings

Fitting, socle, derived, HallC5×C41 ⋊ C2
SylowC41 ⋊ D5
SylowC5 ⋊ D41

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C41C2
LCSG ▷ C5×C41
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 46
transitive degreeμt(G) = 205

A minimal faithful permutation representation — 2 generators on 46 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46);(2,5)(3,4)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Polycyclic presentation:

2,5,41|b^a=b^4,c^a=c^40

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11717961, #11717962, D5×D41, …
maximal normal subgroup of#11717961, #11717962, D5×D41, …
maximal quotient ofDic205, C2×D205, C3×D205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(410, 5);