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G = D8order 16 = 24

Dihedral group

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: D8, D4⋊C2, C81C2, C2.3D4, C4.1C22, 2-Sylow(PGL(2,7)), sometimes denoted D16 or Dih8 or Dih16, SmallGroup(16,7)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D8
C1C2C4D4 — D8
C1C2C4 — D8
C1C2C4 — D8
C1C2C2C4 — D8

Generators and relations for D8
 G = < a,b | a8=b2=1, bab=a-1 >

4C2
4C2
2C22
2C22

Character table of D8

 class 12A2B2C48A8B
 size 1144222
ρ11111111    trivial
ρ211-111-1-1    linear of order 2
ρ3111-11-1-1    linear of order 2
ρ411-1-1111    linear of order 2
ρ52200-200    orthogonal lifted from D4
ρ62-20002-2    orthogonal faithful
ρ72-2000-22    orthogonal faithful

Permutation representations of D8
On 8 points - transitive group 8T6
Generators in S8
(1 2 3 4 5 6 7 8)
(1 8)(2 7)(3 6)(4 5)

G:=sub<Sym(8)| (1,2,3,4,5,6,7,8), (1,8)(2,7)(3,6)(4,5)>;

G:=Group( (1,2,3,4,5,6,7,8), (1,8)(2,7)(3,6)(4,5) );

G=PermutationGroup([(1,2,3,4,5,6,7,8)], [(1,8),(2,7),(3,6),(4,5)])

G:=TransitiveGroup(8,6);

Regular action on 16 points - transitive group 16T13
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)

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

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

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

G:=TransitiveGroup(16,13);

Polynomial with Galois group D8 over ℚ
actionf(x)Disc(f)
8T6x8-2x7-14x4-16x+4-220·38·77

Matrix representation of D8 in GL2(𝔽7) generated by

06
13
,
31
64
G:=sub<GL(2,GF(7))| [0,1,6,3],[3,6,1,4] >;

D8 in GAP, Magma, Sage, TeX

D_8
% in TeX

G:=Group("D8");
// GroupNames label

G:=SmallGroup(16,7);
// by ID

G=gap.SmallGroup(16,7);
# by ID

G:=PCGroup([4,-2,2,-2,-2,49,146,78,34]);
// Polycyclic

G:=Group<a,b|a^8=b^2=1,b*a*b=a^-1>;
// generators/relations

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