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G = D28order 56 = 23·7

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D28, C4⋊D7, C71D4, C281C2, D141C2, C2.4D14, C14.3C22, sometimes denoted D56 or Dih28 or Dih56, SmallGroup(56,5)

Series: Derived Chief Lower central Upper central

C1C14 — D28
C1C7C14D14 — D28
C7C14 — D28
C1C2C4

Generators and relations for D28
 G = < a,b | a28=b2=1, bab=a-1 >

14C2
14C2
7C22
7C22
2D7
2D7
7D4

Character table of D28

 class 12A2B2C47A7B7C14A14B14C28A28B28C28D28E28F
 size 1114142222222222222
ρ111111111111111111    trivial
ρ211-11-1111111-1-1-1-1-1-1    linear of order 2
ρ311-1-11111111111111    linear of order 2
ρ4111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ52-2000222-2-2-2000000    orthogonal lifted from D4
ρ62200-2ζ7572ζ7473ζ767ζ7572ζ7473ζ7677677677473757275727473    orthogonal lifted from D14
ρ72200-2ζ767ζ7572ζ7473ζ767ζ7572ζ74737473747375727677677572    orthogonal lifted from D14
ρ822002ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ7473ζ7473ζ7572ζ767ζ767ζ7572    orthogonal lifted from D7
ρ92200-2ζ7473ζ767ζ7572ζ7473ζ767ζ75727572757276774737473767    orthogonal lifted from D14
ρ1022002ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7572ζ7572ζ767ζ7473ζ7473ζ767    orthogonal lifted from D7
ρ1122002ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ767ζ767ζ7473ζ7572ζ7572ζ7473    orthogonal lifted from D7
ρ122-2000ζ7572ζ7473ζ767757274737674ζ764ζ7ζ4ζ764ζ7ζ4ζ744ζ73ζ43ζ7543ζ7243ζ7543ζ724ζ744ζ73    orthogonal faithful
ρ132-2000ζ7473ζ767ζ757274737677572ζ43ζ7543ζ7243ζ7543ζ724ζ764ζ7ζ4ζ744ζ734ζ744ζ73ζ4ζ764ζ7    orthogonal faithful
ρ142-2000ζ7572ζ7473ζ76775727473767ζ4ζ764ζ74ζ764ζ74ζ744ζ7343ζ7543ζ72ζ43ζ7543ζ72ζ4ζ744ζ73    orthogonal faithful
ρ152-2000ζ7473ζ767ζ75727473767757243ζ7543ζ72ζ43ζ7543ζ72ζ4ζ764ζ74ζ744ζ73ζ4ζ744ζ734ζ764ζ7    orthogonal faithful
ρ162-2000ζ767ζ7572ζ7473767757274734ζ744ζ73ζ4ζ744ζ7343ζ7543ζ72ζ4ζ764ζ74ζ764ζ7ζ43ζ7543ζ72    orthogonal faithful
ρ172-2000ζ767ζ7572ζ747376775727473ζ4ζ744ζ734ζ744ζ73ζ43ζ7543ζ724ζ764ζ7ζ4ζ764ζ743ζ7543ζ72    orthogonal faithful

Permutation representations of D28
On 28 points - transitive group 28T10
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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)

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

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

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

G:=TransitiveGroup(28,10);

Matrix representation of D28 in GL2(𝔽29) generated by

2724
512
,
2724
182
G:=sub<GL(2,GF(29))| [27,5,24,12],[27,18,24,2] >;

D28 in GAP, Magma, Sage, TeX

D_{28}
% in TeX

G:=Group("D28");
// GroupNames label

G:=SmallGroup(56,5);
// by ID

G=gap.SmallGroup(56,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-7,49,21,771]);
// Polycyclic

G:=Group<a,b|a^28=b^2=1,b*a*b=a^-1>;
// generators/relations

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