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G = Dic14order 56 = 23·7

Dicyclic group

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

C1C14 — Dic14
C1C7C14Dic7 — Dic14
C7C14 — Dic14
C1C2C4

Generators and relations for Dic14
 G = < a,b | a28=1, b2=a14, bab-1=a-1 >

7C4
7C4
7Q8

Character table of Dic14

 class 124A4B4C7A7B7C14A14B14C28A28B28C28D28E28F
 size 1121414222222222222
ρ111111111111111111    trivial
ρ211-11-1111111-1-1-1-1-1-1    linear of order 2
ρ3111-1-1111111111111    linear of order 2
ρ411-1-11111111-1-1-1-1-1-1    linear of order 2
ρ522-200ζ7572ζ7473ζ767ζ7473ζ767ζ75727572757276774737473767    orthogonal lifted from D14
ρ622-200ζ767ζ7572ζ7473ζ7572ζ7473ζ7677677677473757275727473    orthogonal lifted from D14
ρ722200ζ767ζ7572ζ7473ζ7572ζ7473ζ767ζ767ζ767ζ7473ζ7572ζ7572ζ7473    orthogonal lifted from D7
ρ822-200ζ7473ζ767ζ7572ζ767ζ7572ζ74737473747375727677677572    orthogonal lifted from D14
ρ922200ζ7473ζ767ζ7572ζ767ζ7572ζ7473ζ7473ζ7473ζ7572ζ767ζ767ζ7572    orthogonal lifted from D7
ρ1022200ζ7572ζ7473ζ767ζ7473ζ767ζ7572ζ7572ζ7572ζ767ζ7473ζ7473ζ767    orthogonal lifted from D7
ρ112-2000222-2-2-2000000    symplectic lifted from Q8, Schur index 2
ρ122-2000ζ7572ζ7473ζ76774737677572ζ4ζ754ζ724ζ754ζ7243ζ7643ζ74ζ744ζ73ζ4ζ744ζ73ζ43ζ7643ζ7    symplectic faithful, Schur index 2
ρ132-2000ζ7473ζ767ζ757276775727473ζ4ζ744ζ734ζ744ζ734ζ754ζ72ζ43ζ7643ζ743ζ7643ζ7ζ4ζ754ζ72    symplectic faithful, Schur index 2
ρ142-2000ζ7572ζ7473ζ767747376775724ζ754ζ72ζ4ζ754ζ72ζ43ζ7643ζ7ζ4ζ744ζ734ζ744ζ7343ζ7643ζ7    symplectic faithful, Schur index 2
ρ152-2000ζ7473ζ767ζ7572767757274734ζ744ζ73ζ4ζ744ζ73ζ4ζ754ζ7243ζ7643ζ7ζ43ζ7643ζ74ζ754ζ72    symplectic faithful, Schur index 2
ρ162-2000ζ767ζ7572ζ747375727473767ζ43ζ7643ζ743ζ7643ζ7ζ4ζ744ζ734ζ754ζ72ζ4ζ754ζ724ζ744ζ73    symplectic faithful, Schur index 2
ρ172-2000ζ767ζ7572ζ74737572747376743ζ7643ζ7ζ43ζ7643ζ74ζ744ζ73ζ4ζ754ζ724ζ754ζ72ζ4ζ744ζ73    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 54 15 40)(2 53 16 39)(3 52 17 38)(4 51 18 37)(5 50 19 36)(6 49 20 35)(7 48 21 34)(8 47 22 33)(9 46 23 32)(10 45 24 31)(11 44 25 30)(12 43 26 29)(13 42 27 56)(14 41 28 55)

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

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

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

Matrix representation of Dic14 in GL2(𝔽29) generated by

716
135
,
170
1612
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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