Copied to
clipboard

## G = D137order 274 = 2·137

### Dihedral group

Aliases: D137, C137⋊C2, sometimes denoted D274 or Dih137 or Dih274, SmallGroup(274,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C137 — D137
 Chief series C1 — C137 — D137
 Lower central C137 — D137
 Upper central C1

Generators and relations for D137
G = < a,b | a137=b2=1, bab=a-1 >

137C2

Smallest permutation representation of D137
On 137 points: primitive
Generators in S137
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137)
(1 137)(2 136)(3 135)(4 134)(5 133)(6 132)(7 131)(8 130)(9 129)(10 128)(11 127)(12 126)(13 125)(14 124)(15 123)(16 122)(17 121)(18 120)(19 119)(20 118)(21 117)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)```

`G:=sub<Sym(137)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,116)(23,115)(24,114)(25,113)(26,112)(27,111)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137)], [(1,137),(2,136),(3,135),(4,134),(5,133),(6,132),(7,131),(8,130),(9,129),(10,128),(11,127),(12,126),(13,125),(14,124),(15,123),(16,122),(17,121),(18,120),(19,119),(20,118),(21,117),(22,116),(23,115),(24,114),(25,113),(26,112),(27,111),(28,110),(29,109),(30,108),(31,107),(32,106),(33,105),(34,104),(35,103),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70)]])`

70 conjugacy classes

 class 1 2 137A ··· 137BP order 1 2 137 ··· 137 size 1 137 2 ··· 2

70 irreducible representations

 dim 1 1 2 type + + + image C1 C2 D137 kernel D137 C137 C1 # reps 1 1 68

Matrix representation of D137 in GL2(𝔽823) generated by

 521 822 1 0
,
 521 822 673 302
`G:=sub<GL(2,GF(823))| [521,1,822,0],[521,673,822,302] >;`

D137 in GAP, Magma, Sage, TeX

`D_{137}`
`% in TeX`

`G:=Group("D137");`
`// GroupNames label`

`G:=SmallGroup(274,1);`
`// by ID`

`G=gap.SmallGroup(274,1);`
`# by ID`

`G:=PCGroup([2,-2,-137,1089]);`
`// Polycyclic`

`G:=Group<a,b|a^137=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

Export

׿
×
𝔽