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## G = D99order 198 = 2·32·11

### Dihedral group

Aliases: D99, C9⋊D11, C11⋊D9, C991C2, C3.D33, C33.1S3, sometimes denoted D198 or Dih99 or Dih198, SmallGroup(198,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C99 — D99
 Chief series C1 — C3 — C33 — C99 — D99
 Lower central C99 — D99
 Upper central C1

Generators and relations for D99
G = < a,b | a99=b2=1, bab=a-1 >

99C2
33S3
9D11
11D9
3D33

Smallest permutation representation of D99
On 99 points
Generators in S99
```(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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99)
(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)```

`G:=sub<Sym(99)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99), (2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99), (2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99)], [(2,99),(3,98),(4,97),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,76),(26,75),(27,74),(28,73),(29,72),(30,71),(31,70),(32,69),(33,68),(34,67),(35,66),(36,65),(37,64),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51)]])`

D99 is a maximal subgroup of   D9×D11
D99 is a maximal quotient of   Dic99

51 conjugacy classes

 class 1 2 3 9A 9B 9C 11A ··· 11E 33A ··· 33J 99A ··· 99AD order 1 2 3 9 9 9 11 ··· 11 33 ··· 33 99 ··· 99 size 1 99 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

51 irreducible representations

 dim 1 1 2 2 2 2 2 type + + + + + + + image C1 C2 S3 D9 D11 D33 D99 kernel D99 C99 C33 C11 C9 C3 C1 # reps 1 1 1 3 5 10 30

Matrix representation of D99 in GL2(𝔽199) generated by

 62 156 43 105
,
 1 0 198 198
`G:=sub<GL(2,GF(199))| [62,43,156,105],[1,198,0,198] >;`

D99 in GAP, Magma, Sage, TeX

`D_{99}`
`% in TeX`

`G:=Group("D99");`
`// GroupNames label`

`G:=SmallGroup(198,3);`
`// by ID`

`G=gap.SmallGroup(198,3);`
`# by ID`

`G:=PCGroup([4,-2,-3,-11,-3,561,533,722,2115]);`
`// Polycyclic`

`G:=Group<a,b|a^99=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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