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G = D16order 32 = 25

Dihedral group

p-group, metacyclic, nilpotent (class 4), monomial

Aliases: D16, C161C2, D81C2, C2.3D8, C4.1D4, C8.2C22, 2-Sylow(PGL(2,17)), sometimes denoted D32 or Dih16 or Dih32, SmallGroup(32,18)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — D16
C1C2C4C8D8 — D16
C1C2C4C8 — D16
C1C2C4C8 — D16
C1C2C2C2C2C4C4C8 — D16

Generators and relations for D16
 G = < a,b | a16=b2=1, bab=a-1 >

8C2
8C2
4C22
4C22
2D4
2D4

Character table of D16

 class 12A2B2C48A8B16A16B16C16D
 size 11882222222
ρ111111111111    trivial
ρ211-1-11111111    linear of order 2
ρ3111-1111-1-1-1-1    linear of order 2
ρ411-11111-1-1-1-1    linear of order 2
ρ522002-2-20000    orthogonal lifted from D4
ρ62200-200-2-222    orthogonal lifted from D8
ρ72200-20022-2-2    orthogonal lifted from D8
ρ82-2000-2216716ζ16716ζ165163165163    orthogonal faithful
ρ92-2000-22ζ1671616716165163ζ165163    orthogonal faithful
ρ102-20002-2165163ζ16516316716ζ16716    orthogonal faithful
ρ112-20002-2ζ165163165163ζ1671616716    orthogonal faithful

Permutation representations of D16
On 16 points - transitive group 16T56
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)])

G:=TransitiveGroup(16,56);

Matrix representation of D16 in GL2(𝔽17) generated by

114
1311
,
10
016
G:=sub<GL(2,GF(17))| [11,13,4,11],[1,0,0,16] >;

D16 in GAP, Magma, Sage, TeX

D_{16}
% in TeX

G:=Group("D16");
// GroupNames label

G:=SmallGroup(32,18);
// by ID

G=gap.SmallGroup(32,18);
# by ID

G:=PCGroup([5,-2,2,-2,-2,-2,61,182,97,102,483,248,58]);
// Polycyclic

G:=Group<a,b|a^16=b^2=1,b*a*b=a^-1>;
// generators/relations

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