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## G = D104order 208 = 24·13

### Dihedral group

Aliases: D104, C131D8, C81D13, C1041C2, D521C2, C2.4D52, C4.9D26, C26.2D4, C52.9C22, sometimes denoted D208 or Dih104 or Dih208, SmallGroup(208,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D104
 Chief series C1 — C13 — C26 — C52 — D52 — D104
 Lower central C13 — C26 — C52 — D104
 Upper central C1 — C2 — C4 — C8

Generators and relations for D104
G = < a,b | a104=b2=1, bab=a-1 >

52C2
52C2
26C22
26C22
4D13
4D13
13D4
13D4
2D26
2D26
13D8

Smallest permutation representation of D104
On 104 points
Generators in S104
```(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 99 100 101 102 103 104)
(1 104)(2 103)(3 102)(4 101)(5 100)(6 99)(7 98)(8 97)(9 96)(10 95)(11 94)(12 93)(13 92)(14 91)(15 90)(16 89)(17 88)(18 87)(19 86)(20 85)(21 84)(22 83)(23 82)(24 81)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)```

`G:=sub<Sym(104)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,104)(2,103)(3,102)(4,101)(5,100)(6,99)(7,98)(8,97)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,104)(2,103)(3,102)(4,101)(5,100)(6,99)(7,98)(8,97)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,104),(2,103),(3,102),(4,101),(5,100),(6,99),(7,98),(8,97),(9,96),(10,95),(11,94),(12,93),(13,92),(14,91),(15,90),(16,89),(17,88),(18,87),(19,86),(20,85),(21,84),(22,83),(23,82),(24,81),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53)]])`

D104 is a maximal subgroup of
D208  C16⋊D13  C13⋊D16  C8.6D26  D1047C2  C8⋊D26  D8×D13  Q8⋊D26  D104⋊C2
D104 is a maximal quotient of
D208  C16⋊D13  Dic104  C1045C4  D525C4

55 conjugacy classes

 class 1 2A 2B 2C 4 8A 8B 13A ··· 13F 26A ··· 26F 52A ··· 52L 104A ··· 104X order 1 2 2 2 4 8 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 52 52 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

55 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 D4 D8 D13 D26 D52 D104 kernel D104 C104 D52 C26 C13 C8 C4 C2 C1 # reps 1 1 2 1 2 6 6 12 24

Matrix representation of D104 in GL2(𝔽313) generated by

 21 309 196 231
,
 285 3 52 28
`G:=sub<GL(2,GF(313))| [21,196,309,231],[285,52,3,28] >;`

D104 in GAP, Magma, Sage, TeX

`D_{104}`
`% in TeX`

`G:=Group("D104");`
`// GroupNames label`

`G:=SmallGroup(208,7);`
`// by ID`

`G=gap.SmallGroup(208,7);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-13,61,66,182,42,4804]);`
`// Polycyclic`

`G:=Group<a,b|a^104=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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