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G = D19  order 38 = 2·19

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D19, C19⋊C2, sometimes denoted D38 or Dih19 or Dih38, SmallGroup(38,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C19 — D19
Chief series
C1 — C19 — D19
Lower central
C19 — D19
Upper central
C1

Generators and relations for D19
 G = < a,b | a19=b2=1, bab=a-1 >

C1
19C2
C19
D19

Character table of D19

 class 1219A19B19C19D19E19F19G19H19I
 size 119222222222
ρ111111111111    trivial
ρ21-1111111111    linear of order 2
ρ320ζ1915+ζ194ζ1913+ζ196ζ1911+ζ198ζ1910+ζ199ζ1912+ζ197ζ1914+ζ195ζ1916+ζ193ζ1918+ζ19ζ1917+ζ192    orthogonal faithful
ρ420ζ1918+ζ19ζ1911+ζ198ζ1917+ζ192ζ1912+ζ197ζ1916+ζ193ζ1913+ζ196ζ1915+ζ194ζ1914+ζ195ζ1910+ζ199    orthogonal faithful
ρ520ζ1916+ζ193ζ1914+ζ195ζ1913+ζ196ζ1917+ζ192ζ1910+ζ199ζ1918+ζ19ζ1912+ζ197ζ1915+ζ194ζ1911+ζ198    orthogonal faithful
ρ620ζ1912+ζ197ζ1918+ζ19ζ1914+ζ195ζ1911+ζ198ζ1917+ζ192ζ1915+ζ194ζ1910+ζ199ζ1916+ζ193ζ1913+ζ196    orthogonal faithful
ρ720ζ1913+ζ196ζ1910+ζ199ζ1912+ζ197ζ1915+ζ194ζ1918+ζ19ζ1917+ζ192ζ1914+ζ195ζ1911+ζ198ζ1916+ζ193    orthogonal faithful
ρ820ζ1917+ζ192ζ1916+ζ193ζ1915+ζ194ζ1914+ζ195ζ1913+ζ196ζ1912+ζ197ζ1911+ζ198ζ1910+ζ199ζ1918+ζ19    orthogonal faithful
ρ920ζ1910+ζ199ζ1915+ζ194ζ1918+ζ19ζ1913+ζ196ζ1911+ζ198ζ1916+ζ193ζ1917+ζ192ζ1912+ζ197ζ1914+ζ195    orthogonal faithful
ρ1020ζ1914+ζ195ζ1917+ζ192ζ1910+ζ199ζ1916+ζ193ζ1915+ζ194ζ1911+ζ198ζ1918+ζ19ζ1913+ζ196ζ1912+ζ197    orthogonal faithful
ρ1120ζ1911+ζ198ζ1912+ζ197ζ1916+ζ193ζ1918+ζ19ζ1914+ζ195ζ1910+ζ199ζ1913+ζ196ζ1917+ζ192ζ1915+ζ194    orthogonal faithful

Permutation representations of D19
►On 19 points: primitive - transitive group 19T2
Generators in S19
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)

(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)
 
G:=sub<Sym(19)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11)]])
 
G:=TransitiveGroup(19,2);
 

D19 is a maximal subgroup of
 C19⋊C6
 D19p: D57  D95  D133  D209  D247 ...
D19 is a maximal quotient of
 Dic19
 D19p: D57  D95  D133  D209  D247 ...

Matrix representation of D19 ►in GL2(𝔽37) generated by

926
1136
,
360
261
G:=sub<GL(2,GF(37))| [9,11,26,36],[36,26,0,1] >;
 

D19 in GAP, Magma, Sage, TeX 

D_{19}
 
% in TeX
 
G:=Group("D19");
 
// GroupNames label
 
G:=SmallGroup(38,1);
 
// by ID
 
G=gap.SmallGroup(38,1);
 
# by ID
 
G:=PCGroup([2,-2,-19,145]);
 
// Polycyclic
 
G:=Group<a,b|a^19=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D19 in TeX
Character table of D19 in TeX

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