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
| C7 — D14 |
Generators and relations for D14
G = < a,b | a14=b2=1, bab=a-1 >
Character table of D14
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 14A | 14B | 14C | |
| size | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ6 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ7 | 2 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal faithful |
| ρ8 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ9 | 2 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | orthogonal faithful |
| ρ10 | 2 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal faithful |
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21)]])
G:=TransitiveGroup(28,4);
D14 is a maximal subgroup of
D28 C7⋊D4
D14 is a maximal quotient of Dic14 D28 C7⋊D4
| action | f(x) | Disc(f) |
|---|---|---|
| 14T3 | x14+7x13+16x12+5x11-19x10+26x9+33x8-255x7-282x6+401x5+782x4+422x3+1018x2+943x-911 | 391·74·112·717 |
Matrix representation of D14 ►in GL2(𝔽13) generated by
| 0 | 12 |
| 1 | 6 |
| 6 | 9 |
| 12 | 7 |
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
Export
Subgroup lattice of D14 in TeX
Character table of D14 in TeX