metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D28, C4⋊D7, C7⋊1D4, C28⋊1C2, D14⋊1C2, C2.4D14, C14.3C22, sometimes denoted D56 or Dih28 or Dih56, SmallGroup(56,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28
G = < a,b | a28=b2=1, bab=a-1 >
Character table of D28
| class | 1 | 2A | 2B | 2C | 4 | 7A | 7B | 7C | 14A | 14B | 14C | 28A | 28B | 28C | 28D | 28E | 28F | |
| size | 1 | 1 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 0 | 0 | -2 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ7 | 2 | 2 | 0 | 0 | -2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ9 | 2 | 2 | 0 | 0 | -2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ12 | 2 | -2 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ4ζ76+ζ4ζ7 | ζ4ζ76-ζ4ζ7 | ζ4ζ74-ζ4ζ73 | ζ43ζ75-ζ43ζ72 | -ζ43ζ75+ζ43ζ72 | -ζ4ζ74+ζ4ζ73 | orthogonal faithful |
| ρ13 | 2 | -2 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ43ζ75-ζ43ζ72 | -ζ43ζ75+ζ43ζ72 | -ζ4ζ76+ζ4ζ7 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | ζ4ζ76-ζ4ζ7 | orthogonal faithful |
| ρ14 | 2 | -2 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ4ζ76-ζ4ζ7 | -ζ4ζ76+ζ4ζ7 | -ζ4ζ74+ζ4ζ73 | -ζ43ζ75+ζ43ζ72 | ζ43ζ75-ζ43ζ72 | ζ4ζ74-ζ4ζ73 | orthogonal faithful |
| ρ15 | 2 | -2 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ43ζ75+ζ43ζ72 | ζ43ζ75-ζ43ζ72 | ζ4ζ76-ζ4ζ7 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | -ζ4ζ76+ζ4ζ7 | orthogonal faithful |
| ρ16 | 2 | -2 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | -ζ43ζ75+ζ43ζ72 | ζ4ζ76-ζ4ζ7 | -ζ4ζ76+ζ4ζ7 | ζ43ζ75-ζ43ζ72 | orthogonal faithful |
| ρ17 | 2 | -2 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | ζ43ζ75-ζ43ζ72 | -ζ4ζ76+ζ4ζ7 | ζ4ζ76-ζ4ζ7 | -ζ43ζ75+ζ43ζ72 | orthogonal faithful |
►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);
D28 is a maximal subgroup of
C56⋊C2 D56 D4⋊D7 Q8⋊D7 C4○D28 D4×D7 Q8⋊2D7 C4⋊F7 C3⋊D28 D84 C5⋊D28 D140 D196 C7⋊D28 C28⋊D7
D28 is a maximal quotient of
C56⋊C2 D56 Dic28 C4⋊Dic7 D14⋊C4 C3⋊D28 D84 C5⋊D28 D140 D196 C7⋊D28 C28⋊D7
Matrix representation of D28 ►in GL2(𝔽29) generated by
| 27 | 24 |
| 5 | 12 |
| 27 | 24 |
| 18 | 2 |
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
Export
Subgroup lattice of D28 in TeX
Character table of D28 in TeX