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## G = D102order 204 = 22·3·17

### Dihedral group

Aliases: D102, C2×D51, C34⋊S3, C6⋊D17, C172D6, C32D34, C1021C2, C512C22, sometimes denoted D204 or Dih102 or Dih204, SmallGroup(204,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C51 — D102
 Chief series C1 — C17 — C51 — D51 — D102
 Lower central C51 — D102
 Upper central C1 — C2

Generators and relations for D102
G = < a,b | a102=b2=1, bab=a-1 >

51C2
51C2
51C22
17S3
17S3
3D17
3D17
17D6
3D34

Smallest permutation representation of D102
On 102 points
Generators in S102
```(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)
(1 102)(2 101)(3 100)(4 99)(5 98)(6 97)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 90)(14 89)(15 88)(16 87)(17 86)(18 85)(19 84)(20 83)(21 82)(22 81)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)```

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

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

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

D102 is a maximal subgroup of   D512C4  C3⋊D68  C17⋊D12  D204  C517D4  C2×S3×D17
D102 is a maximal quotient of   Dic102  D204  C517D4

54 conjugacy classes

 class 1 2A 2B 2C 3 6 17A ··· 17H 34A ··· 34H 51A ··· 51P 102A ··· 102P order 1 2 2 2 3 6 17 ··· 17 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 51 51 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

54 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D17 D34 D51 D102 kernel D102 D51 C102 C34 C17 C6 C3 C2 C1 # reps 1 2 1 1 1 8 8 16 16

Matrix representation of D102 in GL2(𝔽103) generated by

 41 85 9 84
,
 10 4 1 93
`G:=sub<GL(2,GF(103))| [41,9,85,84],[10,1,4,93] >;`

D102 in GAP, Magma, Sage, TeX

`D_{102}`
`% in TeX`

`G:=Group("D102");`
`// GroupNames label`

`G:=SmallGroup(204,11);`
`// by ID`

`G=gap.SmallGroup(204,11);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,-17,98,3075]);`
`// Polycyclic`

`G:=Group<a,b|a^102=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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