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## G = D29order 58 = 2·29

### Dihedral group

Aliases: D29, C29⋊C2, sometimes denoted D58 or Dih29 or Dih58, SmallGroup(58,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C29 — D29
 Chief series C1 — C29 — D29
 Lower central C29 — D29
 Upper central C1

Generators and relations for D29
G = < a,b | a29=b2=1, bab=a-1 >

Character table of D29

 class 1 2 29A 29B 29C 29D 29E 29F 29G 29H 29I 29J 29K 29L 29M 29N size 1 29 2 2 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 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 ζ2920+ζ299 ζ2928+ζ29 ζ2918+ζ2911 ζ2921+ζ298 ζ2927+ζ292 ζ2917+ζ2912 ζ2922+ζ297 ζ2926+ζ293 ζ2916+ζ2913 ζ2923+ζ296 ζ2925+ζ294 ζ2915+ζ2914 ζ2924+ζ295 ζ2919+ζ2910 orthogonal faithful ρ4 2 0 ζ2925+ζ294 ζ2923+ζ296 ζ2921+ζ298 ζ2919+ζ2910 ζ2917+ζ2912 ζ2915+ζ2914 ζ2916+ζ2913 ζ2918+ζ2911 ζ2920+ζ299 ζ2922+ζ297 ζ2924+ζ295 ζ2926+ζ293 ζ2928+ζ29 ζ2927+ζ292 orthogonal faithful ρ5 2 0 ζ2926+ζ293 ζ2919+ζ2910 ζ2923+ζ296 ζ2922+ζ297 ζ2920+ζ299 ζ2925+ζ294 ζ2917+ζ2912 ζ2928+ζ29 ζ2915+ζ2914 ζ2927+ζ292 ζ2918+ζ2911 ζ2924+ζ295 ζ2921+ζ298 ζ2916+ζ2913 orthogonal faithful ρ6 2 0 ζ2928+ζ29 ζ2916+ζ2913 ζ2927+ζ292 ζ2917+ζ2912 ζ2926+ζ293 ζ2918+ζ2911 ζ2925+ζ294 ζ2919+ζ2910 ζ2924+ζ295 ζ2920+ζ299 ζ2923+ζ296 ζ2921+ζ298 ζ2922+ζ297 ζ2915+ζ2914 orthogonal faithful ρ7 2 0 ζ2916+ζ2913 ζ2924+ζ295 ζ2926+ζ293 ζ2918+ζ2911 ζ2919+ζ2910 ζ2927+ζ292 ζ2923+ζ296 ζ2915+ζ2914 ζ2922+ζ297 ζ2928+ζ29 ζ2920+ζ299 ζ2917+ζ2912 ζ2925+ζ294 ζ2921+ζ298 orthogonal faithful ρ8 2 0 ζ2922+ζ297 ζ2925+ζ294 ζ2915+ζ2914 ζ2926+ζ293 ζ2921+ζ298 ζ2919+ζ2910 ζ2928+ζ29 ζ2917+ζ2912 ζ2923+ζ296 ζ2924+ζ295 ζ2916+ζ2913 ζ2927+ζ292 ζ2920+ζ299 ζ2918+ζ2911 orthogonal faithful ρ9 2 0 ζ2917+ζ2912 ζ2918+ζ2911 ζ2924+ζ295 ζ2928+ζ29 ζ2922+ζ297 ζ2916+ζ2913 ζ2919+ζ2910 ζ2925+ζ294 ζ2927+ζ292 ζ2921+ζ298 ζ2915+ζ2914 ζ2920+ζ299 ζ2926+ζ293 ζ2923+ζ296 orthogonal faithful ρ10 2 0 ζ2923+ζ296 ζ2920+ζ299 ζ2917+ζ2912 ζ2915+ζ2914 ζ2918+ζ2911 ζ2921+ζ298 ζ2924+ζ295 ζ2927+ζ292 ζ2928+ζ29 ζ2925+ζ294 ζ2922+ζ297 ζ2919+ζ2910 ζ2916+ζ2913 ζ2926+ζ293 orthogonal faithful ρ11 2 0 ζ2918+ζ2911 ζ2927+ζ292 ζ2922+ζ297 ζ2916+ζ2913 ζ2925+ζ294 ζ2924+ζ295 ζ2915+ζ2914 ζ2923+ζ296 ζ2926+ζ293 ζ2917+ζ2912 ζ2921+ζ298 ζ2928+ζ29 ζ2919+ζ2910 ζ2920+ζ299 orthogonal faithful ρ12 2 0 ζ2924+ζ295 ζ2922+ζ297 ζ2919+ζ2910 ζ2927+ζ292 ζ2915+ζ2914 ζ2926+ζ293 ζ2920+ζ299 ζ2921+ζ298 ζ2925+ζ294 ζ2916+ζ2913 ζ2928+ζ29 ζ2918+ζ2911 ζ2923+ζ296 ζ2917+ζ2912 orthogonal faithful ρ13 2 0 ζ2921+ζ298 ζ2917+ζ2912 ζ2916+ζ2913 ζ2920+ζ299 ζ2924+ζ295 ζ2928+ζ29 ζ2926+ζ293 ζ2922+ζ297 ζ2918+ζ2911 ζ2915+ζ2914 ζ2919+ζ2910 ζ2923+ζ296 ζ2927+ζ292 ζ2925+ζ294 orthogonal faithful ρ14 2 0 ζ2915+ζ2914 ζ2921+ζ298 ζ2928+ζ29 ζ2923+ζ296 ζ2916+ζ2913 ζ2920+ζ299 ζ2927+ζ292 ζ2924+ζ295 ζ2917+ζ2912 ζ2919+ζ2910 ζ2926+ζ293 ζ2925+ζ294 ζ2918+ζ2911 ζ2922+ζ297 orthogonal faithful ρ15 2 0 ζ2919+ζ2910 ζ2915+ζ2914 ζ2920+ζ299 ζ2925+ζ294 ζ2928+ζ29 ζ2923+ζ296 ζ2918+ζ2911 ζ2916+ζ2913 ζ2921+ζ298 ζ2926+ζ293 ζ2927+ζ292 ζ2922+ζ297 ζ2917+ζ2912 ζ2924+ζ295 orthogonal faithful ρ16 2 0 ζ2927+ζ292 ζ2926+ζ293 ζ2925+ζ294 ζ2924+ζ295 ζ2923+ζ296 ζ2922+ζ297 ζ2921+ζ298 ζ2920+ζ299 ζ2919+ζ2910 ζ2918+ζ2911 ζ2917+ζ2912 ζ2916+ζ2913 ζ2915+ζ2914 ζ2928+ζ29 orthogonal faithful

Permutation representations of D29
On 29 points: primitive - transitive group 29T2
Generators in S29
```(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)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)```

`G:=sub<Sym(29)| (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)>;`

`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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16) );`

`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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16)])`

`G:=TransitiveGroup(29,2);`

D29 is a maximal subgroup of   C29⋊C4  D87  D145  C29⋊C14  D203
D29 is a maximal quotient of   Dic29  D87  D145  D203

Matrix representation of D29 in GL2(𝔽59) generated by

 14 58 1 0
,
 14 58 18 45
`G:=sub<GL(2,GF(59))| [14,1,58,0],[14,18,58,45] >;`

D29 in GAP, Magma, Sage, TeX

`D_{29}`
`% in TeX`

`G:=Group("D29");`
`// GroupNames label`

`G:=SmallGroup(58,1);`
`// by ID`

`G=gap.SmallGroup(58,1);`
`# by ID`

`G:=PCGroup([2,-2,-29,225]);`
`// Polycyclic`

`G:=Group<a,b|a^29=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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