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G = D87order 174 = 2·3·29

Dihedral group

Aliases: D87, C29⋊S3, C3⋊D29, C871C2, sometimes denoted D174 or Dih87 or Dih174, SmallGroup(174,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C87 — D87
 Chief series C1 — C29 — C87 — D87
 Lower central C87 — D87
 Upper central C1

Generators and relations for D87
G = < a,b | a87=b2=1, bab=a-1 >

87C2
29S3
3D29

Smallest permutation representation of D87
On 87 points
Generators in S87
```(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)
(1 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)```

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

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

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

D87 is a maximal subgroup of   S3×D29
D87 is a maximal quotient of   Dic87

45 conjugacy classes

 class 1 2 3 29A ··· 29N 87A ··· 87AB order 1 2 3 29 ··· 29 87 ··· 87 size 1 87 2 2 ··· 2 2 ··· 2

45 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 S3 D29 D87 kernel D87 C87 C29 C3 C1 # reps 1 1 1 14 28

Matrix representation of D87 in GL2(𝔽349) generated by

 46 103 104 286
,
 72 100 182 277
`G:=sub<GL(2,GF(349))| [46,104,103,286],[72,182,100,277] >;`

D87 in GAP, Magma, Sage, TeX

`D_{87}`
`% in TeX`

`G:=Group("D87");`
`// GroupNames label`

`G:=SmallGroup(174,3);`
`// by ID`

`G=gap.SmallGroup(174,3);`
`# by ID`

`G:=PCGroup([3,-2,-3,-29,25,1514]);`
`// Polycyclic`

`G:=Group<a,b|a^87=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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