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## G = D134order 268 = 22·67

### Dihedral group

Aliases: D134, C2×D67, C134⋊C2, C67⋊C22, sometimes denoted D268 or Dih134 or Dih268, SmallGroup(268,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C67 — D134
 Chief series C1 — C67 — D67 — D134
 Lower central C67 — D134
 Upper central C1 — C2

Generators and relations for D134
G = < a,b | a134=b2=1, bab=a-1 >

Smallest permutation representation of D134
On 134 points
Generators in S134
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134)
(1 134)(2 133)(3 132)(4 131)(5 130)(6 129)(7 128)(8 127)(9 126)(10 125)(11 124)(12 123)(13 122)(14 121)(15 120)(16 119)(17 118)(18 117)(19 116)(20 115)(21 114)(22 113)(23 112)(24 111)(25 110)(26 109)(27 108)(28 107)(29 106)(30 105)(31 104)(32 103)(33 102)(34 101)(35 100)(36 99)(37 98)(38 97)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)```

`G:=sub<Sym(134)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134)], [(1,134),(2,133),(3,132),(4,131),(5,130),(6,129),(7,128),(8,127),(9,126),(10,125),(11,124),(12,123),(13,122),(14,121),(15,120),(16,119),(17,118),(18,117),(19,116),(20,115),(21,114),(22,113),(23,112),(24,111),(25,110),(26,109),(27,108),(28,107),(29,106),(30,105),(31,104),(32,103),(33,102),(34,101),(35,100),(36,99),(37,98),(38,97),(39,96),(40,95),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68)]])`

70 conjugacy classes

 class 1 2A 2B 2C 67A ··· 67AG 134A ··· 134AG order 1 2 2 2 67 ··· 67 134 ··· 134 size 1 1 67 67 2 ··· 2 2 ··· 2

70 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D67 D134 kernel D134 D67 C134 C2 C1 # reps 1 2 1 33 33

Matrix representation of D134 in GL3(𝔽269) generated by

 268 0 0 0 93 261 0 8 8
,
 1 0 0 0 93 261 0 5 176
`G:=sub<GL(3,GF(269))| [268,0,0,0,93,8,0,261,8],[1,0,0,0,93,5,0,261,176] >;`

D134 in GAP, Magma, Sage, TeX

`D_{134}`
`% in TeX`

`G:=Group("D134");`
`// GroupNames label`

`G:=SmallGroup(268,3);`
`// by ID`

`G=gap.SmallGroup(268,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-67,2378]);`
`// Polycyclic`

`G:=Group<a,b|a^134=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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