metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14, C7⋊Q8, C4.D7, C28.1C2, C2.3D14, Dic7.C2, C14.1C22, SmallGroup(56,3)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14
G = < a,b | a28=1, b2=a14, bab-1=a-1 >
Character table of Dic14
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 14A | 14B | 14C | 28A | 28B | 28C | 28D | 28E | 28F | |
| size | 1 | 1 | 2 | 14 | 14 | 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 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ6 | 2 | 2 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 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 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ8 | 2 | 2 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 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 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ4ζ75-ζ4ζ72 | -ζ4ζ75+ζ4ζ72 | -ζ43ζ76+ζ43ζ7 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | ζ43ζ76-ζ43ζ7 | symplectic faithful, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | -ζ4ζ75+ζ4ζ72 | ζ43ζ76-ζ43ζ7 | -ζ43ζ76+ζ43ζ7 | ζ4ζ75-ζ4ζ72 | symplectic faithful, Schur index 2 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ4ζ75+ζ4ζ72 | ζ4ζ75-ζ4ζ72 | ζ43ζ76-ζ43ζ7 | ζ4ζ74-ζ4ζ73 | -ζ4ζ74+ζ4ζ73 | -ζ43ζ76+ζ43ζ7 | symplectic faithful, Schur index 2 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ4ζ74+ζ4ζ73 | ζ4ζ74-ζ4ζ73 | ζ4ζ75-ζ4ζ72 | -ζ43ζ76+ζ43ζ7 | ζ43ζ76-ζ43ζ7 | -ζ4ζ75+ζ4ζ72 | symplectic faithful, Schur index 2 |
| ρ16 | 2 | -2 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ43ζ76-ζ43ζ7 | -ζ43ζ76+ζ43ζ7 | ζ4ζ74-ζ4ζ73 | -ζ4ζ75+ζ4ζ72 | ζ4ζ75-ζ4ζ72 | -ζ4ζ74+ζ4ζ73 | symplectic faithful, Schur index 2 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ43ζ76+ζ43ζ7 | ζ43ζ76-ζ43ζ7 | -ζ4ζ74+ζ4ζ73 | ζ4ζ75-ζ4ζ72 | -ζ4ζ75+ζ4ζ72 | ζ4ζ74-ζ4ζ73 | symplectic faithful, Schur index 2 |
►Regular action on 56 points
Generators in S56
(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)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)
(1 45 15 31)(2 44 16 30)(3 43 17 29)(4 42 18 56)(5 41 19 55)(6 40 20 54)(7 39 21 53)(8 38 22 52)(9 37 23 51)(10 36 24 50)(11 35 25 49)(12 34 26 48)(13 33 27 47)(14 32 28 46)
G:=sub<Sym(56)| (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,45,15,31)(2,44,16,30)(3,43,17,29)(4,42,18,56)(5,41,19,55)(6,40,20,54)(7,39,21,53)(8,38,22,52)(9,37,23,51)(10,36,24,50)(11,35,25,49)(12,34,26,48)(13,33,27,47)(14,32,28,46)>;
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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,45,15,31)(2,44,16,30)(3,43,17,29)(4,42,18,56)(5,41,19,55)(6,40,20,54)(7,39,21,53)(8,38,22,52)(9,37,23,51)(10,36,24,50)(11,35,25,49)(12,34,26,48)(13,33,27,47)(14,32,28,46) );
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),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(1,45,15,31),(2,44,16,30),(3,43,17,29),(4,42,18,56),(5,41,19,55),(6,40,20,54),(7,39,21,53),(8,38,22,52),(9,37,23,51),(10,36,24,50),(11,35,25,49),(12,34,26,48),(13,33,27,47),(14,32,28,46)]])
Dic14 is a maximal subgroup of
C56⋊C2 Dic28 D4.D7 C7⋊Q16 C4○D28 D4⋊2D7 Q8×D7 C4.F7 C21⋊Q8 Dic42 C35⋊Q8 Dic70 Dic98 C72⋊2Q8 C72⋊4Q8
Dic14 is a maximal quotient of
Dic7⋊C4 C4⋊Dic7 C21⋊Q8 Dic42 C35⋊Q8 Dic70 Dic98 C72⋊2Q8 C72⋊4Q8
Matrix representation of Dic14 ►in GL2(𝔽29) generated by
| 7 | 16 |
| 13 | 5 |
| 17 | 0 |
| 16 | 12 |
G:=sub<GL(2,GF(29))| [7,13,16,5],[17,16,0,12] >;
Dic14 in GAP, Magma, Sage, TeX
{\rm Dic}_{14} % in TeX
G:=Group("Dic14"); // GroupNames label
G:=SmallGroup(56,3);
// by ID
G=gap.SmallGroup(56,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-7,16,49,21,771]);
// Polycyclic
G:=Group<a,b|a^28=1,b^2=a^14,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic14 in TeX
Character table of Dic14 in TeX