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## G = D49order 98 = 2·72

### Dihedral group

Aliases: D49, C49⋊C2, C7.D7, sometimes denoted D98 or Dih49 or Dih98, SmallGroup(98,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C49 — D49
 Chief series C1 — C7 — C49 — D49
 Lower central C49 — D49
 Upper central C1

Generators and relations for D49
G = < a,b | a49=b2=1, bab=a-1 >

Character table of D49

 class 1 2 7A 7B 7C 49A 49B 49C 49D 49E 49F 49G 49H 49I 49J 49K 49L 49M 49N 49O 49P 49Q 49R 49S 49T 49U size 1 49 2 2 2 2 2 2 2 2 2 2 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 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 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 2 2 2 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ4 2 0 2 2 2 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ5 2 0 2 2 2 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ6 2 0 ζ4935+ζ4914 ζ4928+ζ4921 ζ4942+ζ497 ζ4927+ζ4922 ζ4929+ζ4920 ζ4936+ζ4913 ζ4943+ζ496 ζ4948+ζ49 ζ4941+ζ498 ζ4930+ζ4919 ζ4937+ζ4912 ζ4944+ζ495 ζ4947+ζ492 ζ4940+ζ499 ζ4933+ζ4916 ζ4926+ζ4923 ζ4945+ζ494 ζ4946+ζ493 ζ4939+ζ4910 ζ4932+ζ4917 ζ4925+ζ4924 ζ4931+ζ4918 ζ4938+ζ4911 ζ4934+ζ4915 orthogonal faithful ρ7 2 0 ζ4942+ζ497 ζ4935+ζ4914 ζ4928+ζ4921 ζ4946+ζ493 ζ4925+ζ4924 ζ4945+ζ494 ζ4932+ζ4917 ζ4938+ζ4911 ζ4939+ζ4910 ζ4936+ζ4913 ζ4934+ζ4915 ζ4943+ζ496 ζ4927+ζ4922 ζ4948+ζ49 ζ4929+ζ4920 ζ4941+ζ498 ζ4944+ζ495 ζ4933+ζ4916 ζ4937+ζ4912 ζ4940+ζ499 ζ4930+ζ4919 ζ4947+ζ492 ζ4926+ζ4923 ζ4931+ζ4918 orthogonal faithful ρ8 2 0 ζ4942+ζ497 ζ4935+ζ4914 ζ4928+ζ4921 ζ4945+ζ494 ζ4932+ζ4917 ζ4938+ζ4911 ζ4939+ζ4910 ζ4931+ζ4918 ζ4946+ζ493 ζ4948+ζ49 ζ4929+ζ4920 ζ4941+ζ498 ζ4936+ζ4913 ζ4934+ζ4915 ζ4943+ζ496 ζ4927+ζ4922 ζ4926+ζ4923 ζ4944+ζ495 ζ4933+ζ4916 ζ4937+ζ4912 ζ4940+ζ499 ζ4930+ζ4919 ζ4947+ζ492 ζ4925+ζ4924 orthogonal faithful ρ9 2 0 ζ4928+ζ4921 ζ4942+ζ497 ζ4935+ζ4914 ζ4940+ζ499 ζ4926+ζ4923 ζ4937+ζ4912 ζ4947+ζ492 ζ4933+ζ4916 ζ4930+ζ4919 ζ4939+ζ4910 ζ4945+ζ494 ζ4931+ζ4918 ζ4932+ζ4917 ζ4946+ζ493 ζ4938+ζ4911 ζ4925+ζ4924 ζ4934+ζ4915 ζ4948+ζ49 ζ4936+ζ4913 ζ4927+ζ4922 ζ4941+ζ498 ζ4943+ζ496 ζ4929+ζ4920 ζ4944+ζ495 orthogonal faithful ρ10 2 0 ζ4935+ζ4914 ζ4928+ζ4921 ζ4942+ζ497 ζ4943+ζ496 ζ4948+ζ49 ζ4941+ζ498 ζ4934+ζ4915 ζ4927+ζ4922 ζ4929+ζ4920 ζ4926+ζ4923 ζ4930+ζ4919 ζ4937+ζ4912 ζ4944+ζ495 ζ4947+ζ492 ζ4940+ζ499 ζ4933+ζ4916 ζ4939+ζ4910 ζ4932+ζ4917 ζ4925+ζ4924 ζ4931+ζ4918 ζ4938+ζ4911 ζ4945+ζ494 ζ4946+ζ493 ζ4936+ζ4913 orthogonal faithful ρ11 2 0 ζ4942+ζ497 ζ4935+ζ4914 ζ4928+ζ4921 ζ4925+ζ4924 ζ4945+ζ494 ζ4932+ζ4917 ζ4938+ζ4911 ζ4939+ζ4910 ζ4931+ζ4918 ζ4943+ζ496 ζ4927+ζ4922 ζ4948+ζ49 ζ4929+ζ4920 ζ4941+ζ498 ζ4936+ζ4913 ζ4934+ζ4915 ζ4940+ζ499 ζ4930+ζ4919 ζ4947+ζ492 ζ4926+ζ4923 ζ4944+ζ495 ζ4933+ζ4916 ζ4937+ζ4912 ζ4946+ζ493 orthogonal faithful ρ12 2 0 ζ4935+ζ4914 ζ4928+ζ4921 ζ4942+ζ497 ζ4948+ζ49 ζ4941+ζ498 ζ4934+ζ4915 ζ4927+ζ4922 ζ4929+ζ4920 ζ4936+ζ4913 ζ4937+ζ4912 ζ4944+ζ495 ζ4947+ζ492 ζ4940+ζ499 ζ4933+ζ4916 ζ4926+ζ4923 ζ4930+ζ4919 ζ4931+ζ4918 ζ4938+ζ4911 ζ4945+ζ494 ζ4946+ζ493 ζ4939+ζ4910 ζ4932+ζ4917 ζ4925+ζ4924 ζ4943+ζ496 orthogonal faithful ρ13 2 0 ζ4928+ζ4921 ζ4942+ζ497 ζ4935+ζ4914 ζ4933+ζ4916 ζ4930+ζ4919 ζ4944+ζ495 ζ4940+ζ499 ζ4926+ζ4923 ζ4937+ζ4912 ζ4945+ζ494 ζ4931+ζ4918 ζ4932+ζ4917 ζ4946+ζ493 ζ4938+ζ4911 ζ4925+ζ4924 ζ4939+ζ4910 ζ4943+ζ496 ζ4929+ζ4920 ζ4934+ζ4915 ζ4948+ζ49 ζ4936+ζ4913 ζ4927+ζ4922 ζ4941+ζ498 ζ4947+ζ492 orthogonal faithful ρ14 2 0 ζ4935+ζ4914 ζ4928+ζ4921 ζ4942+ζ497 ζ4929+ζ4920 ζ4936+ζ4913 ζ4943+ζ496 ζ4948+ζ49 ζ4941+ζ498 ζ4934+ζ4915 ζ4944+ζ495 ζ4947+ζ492 ζ4940+ζ499 ζ4933+ζ4916 ζ4926+ζ4923 ζ4930+ζ4919 ζ4937+ζ4912 ζ4932+ζ4917 ζ4925+ζ4924 ζ4931+ζ4918 ζ4938+ζ4911 ζ4945+ζ494 ζ4946+ζ493 ζ4939+ζ4910 ζ4927+ζ4922 orthogonal faithful ρ15 2 0 ζ4928+ζ4921 ζ4942+ζ497 ζ4935+ζ4914 ζ4930+ζ4919 ζ4944+ζ495 ζ4940+ζ499 ζ4926+ζ4923 ζ4937+ζ4912 ζ4947+ζ492 ζ4932+ζ4917 ζ4946+ζ493 ζ4938+ζ4911 ζ4925+ζ4924 ζ4939+ζ4910 ζ4945+ζ494 ζ4931+ζ4918 ζ4948+ζ49 ζ4936+ζ4913 ζ4927+ζ4922 ζ4941+ζ498 ζ4943+ζ496 ζ4929+ζ4920 ζ4934+ζ4915 ζ4933+ζ4916 orthogonal faithful ρ16 2 0 ζ4942+ζ497 ζ4935+ζ4914 ζ4928+ζ4921 ζ4931+ζ4918 ζ4946+ζ493 ζ4925+ζ4924 ζ4945+ζ494 ζ4932+ζ4917 ζ4938+ζ4911 ζ4929+ζ4920 ζ4941+ζ498 ζ4936+ζ4913 ζ4934+ζ4915 ζ4943+ζ496 ζ4927+ζ4922 ζ4948+ζ49 ζ4930+ζ4919 ζ4947+ζ492 ζ4926+ζ4923 ζ4944+ζ495 ζ4933+ζ4916 ζ4937+ζ4912 ζ4940+ζ499 ζ4939+ζ4910 orthogonal faithful ρ17 2 0 ζ4942+ζ497 ζ4935+ζ4914 ζ4928+ζ4921 ζ4932+ζ4917 ζ4938+ζ4911 ζ4939+ζ4910 ζ4931+ζ4918 ζ4946+ζ493 ζ4925+ζ4924 ζ4941+ζ498 ζ4936+ζ4913 ζ4934+ζ4915 ζ4943+ζ496 ζ4927+ζ4922 ζ4948+ζ49 ζ4929+ζ4920 ζ4937+ζ4912 ζ4940+ζ499 ζ4930+ζ4919 ζ4947+ζ492 ζ4926+ζ4923 ζ4944+ζ495 ζ4933+ζ4916 ζ4945+ζ494 orthogonal faithful ρ18 2 0 ζ4928+ζ4921 ζ4942+ζ497 ζ4935+ζ4914 ζ4947+ζ492 ζ4933+ζ4916 ζ4930+ζ4919 ζ4944+ζ495 ζ4940+ζ499 ζ4926+ζ4923 ζ4925+ζ4924 ζ4939+ζ4910 ζ4945+ζ494 ζ4931+ζ4918 ζ4932+ζ4917 ζ4946+ζ493 ζ4938+ζ4911 ζ4936+ζ4913 ζ4927+ζ4922 ζ4941+ζ498 ζ4943+ζ496 ζ4929+ζ4920 ζ4934+ζ4915 ζ4948+ζ49 ζ4937+ζ4912 orthogonal faithful ρ19 2 0 ζ4928+ζ4921 ζ4942+ζ497 ζ4935+ζ4914 ζ4937+ζ4912 ζ4947+ζ492 ζ4933+ζ4916 ζ4930+ζ4919 ζ4944+ζ495 ζ4940+ζ499 ζ4946+ζ493 ζ4938+ζ4911 ζ4925+ζ4924 ζ4939+ζ4910 ζ4945+ζ494 ζ4931+ζ4918 ζ4932+ζ4917 ζ4929+ζ4920 ζ4934+ζ4915 ζ4948+ζ49 ζ4936+ζ4913 ζ4927+ζ4922 ζ4941+ζ498 ζ4943+ζ496 ζ4926+ζ4923 orthogonal faithful ρ20 2 0 ζ4928+ζ4921 ζ4942+ζ497 ζ4935+ζ4914 ζ4926+ζ4923 ζ4937+ζ4912 ζ4947+ζ492 ζ4933+ζ4916 ζ4930+ζ4919 ζ4944+ζ495 ζ4931+ζ4918 ζ4932+ζ4917 ζ4946+ζ493 ζ4938+ζ4911 ζ4925+ζ4924 ζ4939+ζ4910 ζ4945+ζ494 ζ4927+ζ4922 ζ4941+ζ498 ζ4943+ζ496 ζ4929+ζ4920 ζ4934+ζ4915 ζ4948+ζ49 ζ4936+ζ4913 ζ4940+ζ499 orthogonal faithful ρ21 2 0 ζ4935+ζ4914 ζ4928+ζ4921 ζ4942+ζ497 ζ4934+ζ4915 ζ4927+ζ4922 ζ4929+ζ4920 ζ4936+ζ4913 ζ4943+ζ496 ζ4948+ζ49 ζ4933+ζ4916 ζ4926+ζ4923 ζ4930+ζ4919 ζ4937+ζ4912 ζ4944+ζ495 ζ4947+ζ492 ζ4940+ζ499 ζ4925+ζ4924 ζ4931+ζ4918 ζ4938+ζ4911 ζ4945+ζ494 ζ4946+ζ493 ζ4939+ζ4910 ζ4932+ζ4917 ζ4941+ζ498 orthogonal faithful ρ22 2 0 ζ4942+ζ497 ζ4935+ζ4914 ζ4928+ζ4921 ζ4939+ζ4910 ζ4931+ζ4918 ζ4946+ζ493 ζ4925+ζ4924 ζ4945+ζ494 ζ4932+ζ4917 ζ4927+ζ4922 ζ4948+ζ49 ζ4929+ζ4920 ζ4941+ζ498 ζ4936+ζ4913 ζ4934+ζ4915 ζ4943+ζ496 ζ4933+ζ4916 ζ4937+ζ4912 ζ4940+ζ499 ζ4930+ζ4919 ζ4947+ζ492 ζ4926+ζ4923 ζ4944+ζ495 ζ4938+ζ4911 orthogonal faithful ρ23 2 0 ζ4928+ζ4921 ζ4942+ζ497 ζ4935+ζ4914 ζ4944+ζ495 ζ4940+ζ499 ζ4926+ζ4923 ζ4937+ζ4912 ζ4947+ζ492 ζ4933+ζ4916 ζ4938+ζ4911 ζ4925+ζ4924 ζ4939+ζ4910 ζ4945+ζ494 ζ4931+ζ4918 ζ4932+ζ4917 ζ4946+ζ493 ζ4941+ζ498 ζ4943+ζ496 ζ4929+ζ4920 ζ4934+ζ4915 ζ4948+ζ49 ζ4936+ζ4913 ζ4927+ζ4922 ζ4930+ζ4919 orthogonal faithful ρ24 2 0 ζ4935+ζ4914 ζ4928+ζ4921 ζ4942+ζ497 ζ4941+ζ498 ζ4934+ζ4915 ζ4927+ζ4922 ζ4929+ζ4920 ζ4936+ζ4913 ζ4943+ζ496 ζ4947+ζ492 ζ4940+ζ499 ζ4933+ζ4916 ζ4926+ζ4923 ζ4930+ζ4919 ζ4937+ζ4912 ζ4944+ζ495 ζ4946+ζ493 ζ4939+ζ4910 ζ4932+ζ4917 ζ4925+ζ4924 ζ4931+ζ4918 ζ4938+ζ4911 ζ4945+ζ494 ζ4948+ζ49 orthogonal faithful ρ25 2 0 ζ4935+ζ4914 ζ4928+ζ4921 ζ4942+ζ497 ζ4936+ζ4913 ζ4943+ζ496 ζ4948+ζ49 ζ4941+ζ498 ζ4934+ζ4915 ζ4927+ζ4922 ζ4940+ζ499 ζ4933+ζ4916 ζ4926+ζ4923 ζ4930+ζ4919 ζ4937+ζ4912 ζ4944+ζ495 ζ4947+ζ492 ζ4938+ζ4911 ζ4945+ζ494 ζ4946+ζ493 ζ4939+ζ4910 ζ4932+ζ4917 ζ4925+ζ4924 ζ4931+ζ4918 ζ4929+ζ4920 orthogonal faithful ρ26 2 0 ζ4942+ζ497 ζ4935+ζ4914 ζ4928+ζ4921 ζ4938+ζ4911 ζ4939+ζ4910 ζ4931+ζ4918 ζ4946+ζ493 ζ4925+ζ4924 ζ4945+ζ494 ζ4934+ζ4915 ζ4943+ζ496 ζ4927+ζ4922 ζ4948+ζ49 ζ4929+ζ4920 ζ4941+ζ498 ζ4936+ζ4913 ζ4947+ζ492 ζ4926+ζ4923 ζ4944+ζ495 ζ4933+ζ4916 ζ4937+ζ4912 ζ4940+ζ499 ζ4930+ζ4919 ζ4932+ζ4917 orthogonal faithful

Smallest permutation representation of D49
On 49 points
Generators in S49
```(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)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)```

`G:=sub<Sym(49)| (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)>;`

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

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

D49 is a maximal subgroup of   C49⋊C6  D147  D245
D49 is a maximal quotient of   Dic49  D147  D245

Matrix representation of D49 in GL2(𝔽197) generated by

 126 17 180 54
,
 109 106 189 88
`G:=sub<GL(2,GF(197))| [126,180,17,54],[109,189,106,88] >;`

D49 in GAP, Magma, Sage, TeX

`D_{49}`
`% in TeX`

`G:=Group("D49");`
`// GroupNames label`

`G:=SmallGroup(98,1);`
`// by ID`

`G=gap.SmallGroup(98,1);`
`# by ID`

`G:=PCGroup([3,-2,-7,-7,577,46,758]);`
`// Polycyclic`

`G:=Group<a,b|a^49=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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