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## G = Dic14order 56 = 23·7

### Dicyclic group

Aliases: Dic14, C7⋊Q8, C4.D7, C28.1C2, C2.3D14, Dic7.C2, C14.1C22, SmallGroup(56,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Dic14
 Chief series C1 — C7 — C14 — Dic7 — Dic14
 Lower central C7 — C14 — Dic14
 Upper central C1 — C2 — C4

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

Smallest permutation representation of Dic14
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 35 15 49)(2 34 16 48)(3 33 17 47)(4 32 18 46)(5 31 19 45)(6 30 20 44)(7 29 21 43)(8 56 22 42)(9 55 23 41)(10 54 24 40)(11 53 25 39)(12 52 26 38)(13 51 27 37)(14 50 28 36)

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), (1,35,15,49)(2,34,16,48)(3,33,17,47)(4,32,18,46)(5,31,19,45)(6,30,20,44)(7,29,21,43)(8,56,22,42)(9,55,23,41)(10,54,24,40)(11,53,25,39)(12,52,26,38)(13,51,27,37)(14,50,28,36)>;

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), (1,35,15,49)(2,34,16,48)(3,33,17,47)(4,32,18,46)(5,31,19,45)(6,30,20,44)(7,29,21,43)(8,56,22,42)(9,55,23,41)(10,54,24,40)(11,53,25,39)(12,52,26,38)(13,51,27,37)(14,50,28,36) );

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)], [(1,35,15,49),(2,34,16,48),(3,33,17,47),(4,32,18,46),(5,31,19,45),(6,30,20,44),(7,29,21,43),(8,56,22,42),(9,55,23,41),(10,54,24,40),(11,53,25,39),(12,52,26,38),(13,51,27,37),(14,50,28,36)])

Dic14 is a maximal subgroup of
C56⋊C2  Dic28  D4.D7  C7⋊Q16  C4○D28  D42D7  Q8×D7  C4.F7  C21⋊Q8  Dic42  C35⋊Q8  Dic70  Dic98  C722Q8  C724Q8
Dic14 is a maximal quotient of
Dic7⋊C4  C4⋊Dic7  C21⋊Q8  Dic42  C35⋊Q8  Dic70  Dic98  C722Q8  C724Q8

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

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