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## G = D155order 310 = 2·5·31

### Dihedral group

Aliases: D155, C5⋊D31, C31⋊D5, C1551C2, sometimes denoted D310 or Dih155 or Dih310, SmallGroup(310,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C155 — D155
 Chief series C1 — C31 — C155 — D155
 Lower central C155 — D155
 Upper central C1

Generators and relations for D155
G = < a,b | a155=b2=1, bab=a-1 >

155C2
31D5
5D31

Smallest permutation representation of D155
On 155 points
Generators in S155
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155)
(1 155)(2 154)(3 153)(4 152)(5 151)(6 150)(7 149)(8 148)(9 147)(10 146)(11 145)(12 144)(13 143)(14 142)(15 141)(16 140)(17 139)(18 138)(19 137)(20 136)(21 135)(22 134)(23 133)(24 132)(25 131)(26 130)(27 129)(28 128)(29 127)(30 126)(31 125)(32 124)(33 123)(34 122)(35 121)(36 120)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)```

`G:=sub<Sym(155)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155), (1,155)(2,154)(3,153)(4,152)(5,151)(6,150)(7,149)(8,148)(9,147)(10,146)(11,145)(12,144)(13,143)(14,142)(15,141)(16,140)(17,139)(18,138)(19,137)(20,136)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,120)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155), (1,155)(2,154)(3,153)(4,152)(5,151)(6,150)(7,149)(8,148)(9,147)(10,146)(11,145)(12,144)(13,143)(14,142)(15,141)(16,140)(17,139)(18,138)(19,137)(20,136)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,120)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155)], [(1,155),(2,154),(3,153),(4,152),(5,151),(6,150),(7,149),(8,148),(9,147),(10,146),(11,145),(12,144),(13,143),(14,142),(15,141),(16,140),(17,139),(18,138),(19,137),(20,136),(21,135),(22,134),(23,133),(24,132),(25,131),(26,130),(27,129),(28,128),(29,127),(30,126),(31,125),(32,124),(33,123),(34,122),(35,121),(36,120),(37,119),(38,118),(39,117),(40,116),(41,115),(42,114),(43,113),(44,112),(45,111),(46,110),(47,109),(48,108),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79)]])`

79 conjugacy classes

 class 1 2 5A 5B 31A ··· 31O 155A ··· 155BH order 1 2 5 5 31 ··· 31 155 ··· 155 size 1 155 2 2 2 ··· 2 2 ··· 2

79 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D5 D31 D155 kernel D155 C155 C31 C5 C1 # reps 1 1 2 15 60

Matrix representation of D155 in GL2(𝔽311) generated by

 143 33 104 111
,
 77 128 12 234
`G:=sub<GL(2,GF(311))| [143,104,33,111],[77,12,128,234] >;`

D155 in GAP, Magma, Sage, TeX

`D_{155}`
`% in TeX`

`G:=Group("D155");`
`// GroupNames label`

`G:=SmallGroup(310,5);`
`// by ID`

`G=gap.SmallGroup(310,5);`
`# by ID`

`G:=PCGroup([3,-2,-5,-31,49,2702]);`
`// Polycyclic`

`G:=Group<a,b|a^155=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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