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G = Dic7order 28 = 22·7

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic7, C7⋊C4, C2.D7, C14.C2, SmallGroup(28,1)

Series: Derived Chief Lower central Upper central

C1C7 — Dic7
C1C7C14 — Dic7
C7 — Dic7
C1C2

Generators and relations for Dic7
 G = < a,b | a14=1, b2=a7, bab-1=a-1 >

7C4

Character table of Dic7

 class 124A4B7A7B7C14A14B14C
 size 1177222222
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111-1-1-1    linear of order 4
ρ41-1i-i111-1-1-1    linear of order 4
ρ52200ζ7572ζ7473ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ62200ζ767ζ7572ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ72200ζ7473ζ767ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ82-200ζ7473ζ767ζ757275727473767    symplectic faithful, Schur index 2
ρ92-200ζ7572ζ7473ζ76776775727473    symplectic faithful, Schur index 2
ρ102-200ζ767ζ7572ζ747374737677572    symplectic faithful, Schur index 2

Permutation representations of Dic7
Regular action on 28 points - transitive group 28T3
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 25 8 18)(2 24 9 17)(3 23 10 16)(4 22 11 15)(5 21 12 28)(6 20 13 27)(7 19 14 26)

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), (1,25,8,18)(2,24,9,17)(3,23,10,16)(4,22,11,15)(5,21,12,28)(6,20,13,27)(7,19,14,26)>;

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), (1,25,8,18)(2,24,9,17)(3,23,10,16)(4,22,11,15)(5,21,12,28)(6,20,13,27)(7,19,14,26) );

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)], [(1,25,8,18),(2,24,9,17),(3,23,10,16),(4,22,11,15),(5,21,12,28),(6,20,13,27),(7,19,14,26)])

G:=TransitiveGroup(28,3);

Matrix representation of Dic7 in GL2(𝔽13) generated by

511
111
,
83
05
G:=sub<GL(2,GF(13))| [5,11,11,1],[8,0,3,5] >;

Dic7 in GAP, Magma, Sage, TeX

{\rm Dic}_7
% in TeX

G:=Group("Dic7");
// GroupNames label

G:=SmallGroup(28,1);
// by ID

G=gap.SmallGroup(28,1);
# by ID

G:=PCGroup([3,-2,-2,-7,6,218]);
// Polycyclic

G:=Group<a,b|a^14=1,b^2=a^7,b*a*b^-1=a^-1>;
// generators/relations

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