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G = D19order 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

C1C19 — D19
C1C19 — D19
C19 — D19
C1

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

19C2

Character table of D19

 class 1219A19B19C19D19E19F19G19H19I
 size 119222222222
ρ111111111111    trivial
ρ21-1111111111    linear of order 2
ρ320ζ1915194ζ1913196ζ1911198ζ1910199ζ1912197ζ1914195ζ1916193ζ191819ζ1917192    orthogonal faithful
ρ420ζ191819ζ1911198ζ1917192ζ1912197ζ1916193ζ1913196ζ1915194ζ1914195ζ1910199    orthogonal faithful
ρ520ζ1916193ζ1914195ζ1913196ζ1917192ζ1910199ζ191819ζ1912197ζ1915194ζ1911198    orthogonal faithful
ρ620ζ1912197ζ191819ζ1914195ζ1911198ζ1917192ζ1915194ζ1910199ζ1916193ζ1913196    orthogonal faithful
ρ720ζ1913196ζ1910199ζ1912197ζ1915194ζ191819ζ1917192ζ1914195ζ1911198ζ1916193    orthogonal faithful
ρ820ζ1917192ζ1916193ζ1915194ζ1914195ζ1913196ζ1912197ζ1911198ζ1910199ζ191819    orthogonal faithful
ρ920ζ1910199ζ1915194ζ191819ζ1913196ζ1911198ζ1916193ζ1917192ζ1912197ζ1914195    orthogonal faithful
ρ1020ζ1914195ζ1917192ζ1910199ζ1916193ζ1915194ζ1911198ζ191819ζ1913196ζ1912197    orthogonal faithful
ρ1120ζ1911198ζ1912197ζ1916193ζ191819ζ1914195ζ1910199ζ1913196ζ1917192ζ1915194    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);

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

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