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Cayley diagram

D209 #88273

order418
isoclinismisoclinic class of D209
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD209
unambiguous semidirect productC11⋊D19
unambiguous semidirect productC19⋊D11

Splittings

Fitting, socle, derived, HallC11×C19 ⋊ C2
SylowC19 ⋊ D11
SylowC11 ⋊ D19

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC11×C19C2
LCSG ▷ C11×C19
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 209

A minimal faithful permutation representation — 2 generators on 30 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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

2,11,19|b^a=b^10,c^a=c^18

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×D19, C2×D209, C11⋊(D19⋊C3), …
maximal normal subgroup ofD11×D19, C2×D209, C11⋊(D19⋊C3), …
maximal quotient ofDic209, C2×D209, C3×D209, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(418, 3);