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G = D7order 14 = 2·7

Dihedral group

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

Aliases: D7, C7⋊C2, sometimes denoted D14 or Dih7 or Dih14, SmallGroup(14,1)

Series: Derived Chief Lower central Upper central

C1C7 — D7
C1C7 — D7
C7 — D7
C1

Generators and relations for D7
 G = < a,b | a7=b2=1, bab=a-1 >

7C2

Character table of D7

 class 127A7B7C
 size 17222
ρ111111    trivial
ρ21-1111    linear of order 2
ρ320ζ767ζ7572ζ7473    orthogonal faithful
ρ420ζ7473ζ767ζ7572    orthogonal faithful
ρ520ζ7572ζ7473ζ767    orthogonal faithful

Permutation representations of D7
On 7 points: primitive - transitive group 7T2
Generators in S7
(1 2 3 4 5 6 7)
(1 7)(2 6)(3 5)

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

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

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

G:=TransitiveGroup(7,2);

Regular action on 14 points - transitive group 14T2
Generators in S14
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)

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

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

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

G:=TransitiveGroup(14,2);

Polynomial with Galois group D7 over ℚ
actionf(x)Disc(f)
7T2x7-2x6-x5+x4+x3+x2-x-1-713
14T2x14+9x12+53x10+333x8+1251x6+731x4+5415x2+8591-2182·74·112·717

Matrix representation of D7 in GL2(𝔽13) generated by

118
512
,
120
81
G:=sub<GL(2,GF(13))| [11,5,8,12],[12,8,0,1] >;

D7 in GAP, Magma, Sage, TeX

D_7
% in TeX

G:=Group("D7");
// GroupNames label

G:=SmallGroup(14,1);
// by ID

G=gap.SmallGroup(14,1);
# by ID

G:=PCGroup([2,-2,-7,49]);
// Polycyclic

G:=Group<a,b|a^7=b^2=1,b*a*b=a^-1>;
// generators/relations

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