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## G = Dic7order 28 = 22·7

### Dicyclic group

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

Series: Derived Chief Lower central Upper central

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

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

Character table of Dic7

 class 1 2 4A 4B 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 -i i 1 1 1 -1 -1 -1 linear of order 4 ρ4 1 -1 i -i 1 1 1 -1 -1 -1 linear of order 4 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 lifted from D7 ρ8 2 -2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 symplectic faithful, Schur index 2 ρ9 2 -2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 symplectic faithful, Schur index 2 ρ10 2 -2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 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 22 8 15)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)

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

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

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

G:=TransitiveGroup(28,3);

Dic7 is a maximal subgroup of
C4×D7  C7⋊D4  C7⋊C12  C7⋊F5  C7⋊Dic7  C32⋊Dic7  C91⋊C4  C17⋊Dic7
Dic7p: Dic14  Dic21  Dic35  Dic49  Dic77  Dic91  Dic119 ...
Dic7 is a maximal quotient of
C7⋊F5  C32⋊Dic7  C91⋊C4  C17⋊Dic7
C2p.D7: C7⋊C8  Dic21  Dic35  Dic49  C7⋊Dic7  Dic77  Dic91  Dic119 ...

Matrix representation of Dic7 in GL2(𝔽13) generated by

 5 11 11 1
,
 8 3 0 5
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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