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G = D8  order 16 = 24

Dihedral group

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

Aliases: D8, D4⋊C2, C8⋊1C2, 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

Derived series
C1 — C4 — D8
Chief series
C1 — C2 — C4 — D4 — D8
Lower central
C1 — C2 — C4 — D8
Upper central
C1 — C2 — C4 — D8
Jennings
C1 — C2 — C2 — C4 — D8

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

C1
C2
4C2
4C2
C4
2C22
2C22
D4
C8
D4
D8

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
ρ62-2000√2-√2    orthogonal faithful
ρ72-2000-√2√2    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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10)]])
 
G:=TransitiveGroup(16,13);
 

D8 is a maximal subgroup of
 SD32  C32⋊D8  PGL2(𝔽7)  C52⋊D8
 D8p: D16  D24  D40  D56  D88  D104  D136  D152 ...
 D4p⋊C2: C4○D8  C8⋊C22  D4⋊S3  D4⋊D5  D4⋊D7  D4⋊D11  D4⋊D13  D4⋊D17 ...
D8 is a maximal quotient of
 C2.D8  C32⋊D8  C52⋊D8
 D8p: D16  D24  D40  D56  D88  D104  D136  D152 ...
 C4.D2p: D4⋊C4  SD32  Q32  D4⋊S3  D4⋊D5  D4⋊D7  D4⋊D11  D4⋊D13 ...

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
 

Export

Subgroup lattice of D8 in TeX
Character table of D8 in TeX

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