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## G = Dic10order 40 = 23·5

### Dicyclic group

Aliases: Dic10, C5⋊Q8, C4.D5, C20.1C2, C2.3D10, C10.1C22, Dic5.1C2, SmallGroup(40,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic10
 Chief series C1 — C5 — C10 — Dic5 — Dic10
 Lower central C5 — C10 — Dic10
 Upper central C1 — C2 — C4

Generators and relations for Dic10
G = < a,b | a20=1, b2=a10, bab-1=a-1 >

Character table of Dic10

 class 1 2 4A 4B 4C 5A 5B 10A 10B 20A 20B 20C 20D size 1 1 2 10 10 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ6 2 2 -2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ7 2 2 -2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ8 2 2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 -2 0 0 0 2 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ10 2 -2 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 symplectic faithful, Schur index 2 ρ11 2 -2 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 symplectic faithful, Schur index 2 ρ12 2 -2 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 symplectic faithful, Schur index 2 ρ13 2 -2 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 symplectic faithful, Schur index 2

Smallest permutation representation of Dic10
Regular action on 40 points
Generators in S40
(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)
(1 25 11 35)(2 24 12 34)(3 23 13 33)(4 22 14 32)(5 21 15 31)(6 40 16 30)(7 39 17 29)(8 38 18 28)(9 37 19 27)(10 36 20 26)

G:=sub<Sym(40)| (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), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)], [(1,25,11,35),(2,24,12,34),(3,23,13,33),(4,22,14,32),(5,21,15,31),(6,40,16,30),(7,39,17,29),(8,38,18,28),(9,37,19,27),(10,36,20,26)])

Dic10 is a maximal subgroup of
C40⋊C2  Dic20  D4.D5  C5⋊Q16  C4○D20  D42D5  Q8×D5  C15⋊Q8  Dic30  Dic50  C522Q8  C524Q8  C35⋊Q8  Dic70  C32⋊Dic10  C55⋊Q8  Dic110
Dic10 is a maximal quotient of
C10.D4  C4⋊Dic5  C15⋊Q8  Dic30  Dic50  C522Q8  C524Q8  C35⋊Q8  Dic70  C32⋊Dic10  C55⋊Q8  Dic110

Matrix representation of Dic10 in GL2(𝔽19) generated by

 8 17 17 3
,
 0 18 1 0
G:=sub<GL(2,GF(19))| [8,17,17,3],[0,1,18,0] >;

Dic10 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}
% in TeX

G:=Group("Dic10");
// GroupNames label

G:=SmallGroup(40,4);
// by ID

G=gap.SmallGroup(40,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,16,49,21,515]);
// Polycyclic

G:=Group<a,b|a^20=1,b^2=a^10,b*a*b^-1=a^-1>;
// generators/relations

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