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G = D14order 28 = 22·7

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D14, C2×D7, C14⋊C2, C7⋊C22, sometimes denoted D28 or Dih14 or Dih28, SmallGroup(28,3)

Series: Derived Chief Lower central Upper central

C1C7 — D14
C1C7D7 — D14
C7 — D14
C1C2

Generators and relations for D14
 G = < a,b | a14=b2=1, bab=a-1 >

7C2
7C2
7C22

Character table of D14

 class 12A2B2C7A7B7C14A14B14C
 size 1177222222
ρ11111111111    trivial
ρ21-1-11111-1-1-1    linear of order 2
ρ311-1-1111111    linear of order 2
ρ41-11-1111-1-1-1    linear of order 2
ρ52200ζ7572ζ7473ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ62200ζ767ζ7572ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ72-200ζ7473ζ767ζ757275727473767    orthogonal faithful
ρ82200ζ7473ζ767ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ92-200ζ7572ζ7473ζ76776775727473    orthogonal faithful
ρ102-200ζ767ζ7572ζ747374737677572    orthogonal faithful

Permutation representations of D14
On 14 points - transitive group 14T3
Generators in S14
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)
(1 7)(2 6)(3 5)(8 14)(9 13)(10 12)

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

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

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

G:=TransitiveGroup(14,3);

Regular action on 28 points - transitive group 28T4
Generators in S28
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)
(1 15)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)

G:=sub<Sym(28)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28), (1,15)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28), (1,15)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28)], [(1,15),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16)])

G:=TransitiveGroup(28,4);

Polynomial with Galois group D14 over ℚ
actionf(x)Disc(f)
14T3x14+7x13+16x12+5x11-19x10+26x9+33x8-255x7-282x6+401x5+782x4+422x3+1018x2+943x-911391·74·112·717

Matrix representation of D14 in GL2(𝔽13) generated by

012
16
,
69
127
G:=sub<GL(2,GF(13))| [0,1,12,6],[6,12,9,7] >;

D14 in GAP, Magma, Sage, TeX

D_{14}
% in TeX

G:=Group("D14");
// GroupNames label

G:=SmallGroup(28,3);
// by ID

G=gap.SmallGroup(28,3);
# by ID

G:=PCGroup([3,-2,-2,-7,218]);
// Polycyclic

G:=Group<a,b|a^14=b^2=1,b*a*b=a^-1>;
// generators/relations

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