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## G = D203order 406 = 2·7·29

### Dihedral group

Aliases: D203, C29⋊D7, C7⋊D29, C2031C2, sometimes denoted D406 or Dih203 or Dih406, SmallGroup(406,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C203 — D203
 Chief series C1 — C29 — C203 — D203
 Lower central C203 — D203
 Upper central C1

Generators and relations for D203
G = < a,b | a203=b2=1, bab=a-1 >

203C2
29D7
7D29

Smallest permutation representation of D203
On 203 points
Generators in S203
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203)
(1 203)(2 202)(3 201)(4 200)(5 199)(6 198)(7 197)(8 196)(9 195)(10 194)(11 193)(12 192)(13 191)(14 190)(15 189)(16 188)(17 187)(18 186)(19 185)(20 184)(21 183)(22 182)(23 181)(24 180)(25 179)(26 178)(27 177)(28 176)(29 175)(30 174)(31 173)(32 172)(33 171)(34 170)(35 169)(36 168)(37 167)(38 166)(39 165)(40 164)(41 163)(42 162)(43 161)(44 160)(45 159)(46 158)(47 157)(48 156)(49 155)(50 154)(51 153)(52 152)(53 151)(54 150)(55 149)(56 148)(57 147)(58 146)(59 145)(60 144)(61 143)(62 142)(63 141)(64 140)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(71 133)(72 132)(73 131)(74 130)(75 129)(76 128)(77 127)(78 126)(79 125)(80 124)(81 123)(82 122)(83 121)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)```

`G:=sub<Sym(203)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203), (1,203)(2,202)(3,201)(4,200)(5,199)(6,198)(7,197)(8,196)(9,195)(10,194)(11,193)(12,192)(13,191)(14,190)(15,189)(16,188)(17,187)(18,186)(19,185)(20,184)(21,183)(22,182)(23,181)(24,180)(25,179)(26,178)(27,177)(28,176)(29,175)(30,174)(31,173)(32,172)(33,171)(34,170)(35,169)(36,168)(37,167)(38,166)(39,165)(40,164)(41,163)(42,162)(43,161)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(61,143)(62,142)(63,141)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,131)(74,130)(75,129)(76,128)(77,127)(78,126)(79,125)(80,124)(81,123)(82,122)(83,121)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203), (1,203)(2,202)(3,201)(4,200)(5,199)(6,198)(7,197)(8,196)(9,195)(10,194)(11,193)(12,192)(13,191)(14,190)(15,189)(16,188)(17,187)(18,186)(19,185)(20,184)(21,183)(22,182)(23,181)(24,180)(25,179)(26,178)(27,177)(28,176)(29,175)(30,174)(31,173)(32,172)(33,171)(34,170)(35,169)(36,168)(37,167)(38,166)(39,165)(40,164)(41,163)(42,162)(43,161)(44,160)(45,159)(46,158)(47,157)(48,156)(49,155)(50,154)(51,153)(52,152)(53,151)(54,150)(55,149)(56,148)(57,147)(58,146)(59,145)(60,144)(61,143)(62,142)(63,141)(64,140)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,131)(74,130)(75,129)(76,128)(77,127)(78,126)(79,125)(80,124)(81,123)(82,122)(83,121)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203)], [(1,203),(2,202),(3,201),(4,200),(5,199),(6,198),(7,197),(8,196),(9,195),(10,194),(11,193),(12,192),(13,191),(14,190),(15,189),(16,188),(17,187),(18,186),(19,185),(20,184),(21,183),(22,182),(23,181),(24,180),(25,179),(26,178),(27,177),(28,176),(29,175),(30,174),(31,173),(32,172),(33,171),(34,170),(35,169),(36,168),(37,167),(38,166),(39,165),(40,164),(41,163),(42,162),(43,161),(44,160),(45,159),(46,158),(47,157),(48,156),(49,155),(50,154),(51,153),(52,152),(53,151),(54,150),(55,149),(56,148),(57,147),(58,146),(59,145),(60,144),(61,143),(62,142),(63,141),(64,140),(65,139),(66,138),(67,137),(68,136),(69,135),(70,134),(71,133),(72,132),(73,131),(74,130),(75,129),(76,128),(77,127),(78,126),(79,125),(80,124),(81,123),(82,122),(83,121),(84,120),(85,119),(86,118),(87,117),(88,116),(89,115),(90,114),(91,113),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103)]])`

103 conjugacy classes

 class 1 2 7A 7B 7C 29A ··· 29N 203A ··· 203CF order 1 2 7 7 7 29 ··· 29 203 ··· 203 size 1 203 2 2 2 2 ··· 2 2 ··· 2

103 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D7 D29 D203 kernel D203 C203 C29 C7 C1 # reps 1 1 3 14 84

Matrix representation of D203 in GL2(𝔽2437) generated by

 757 1833 604 1675
,
 757 1833 513 1680
`G:=sub<GL(2,GF(2437))| [757,604,1833,1675],[757,513,1833,1680] >;`

D203 in GAP, Magma, Sage, TeX

`D_{203}`
`% in TeX`

`G:=Group("D203");`
`// GroupNames label`

`G:=SmallGroup(406,5);`
`// by ID`

`G=gap.SmallGroup(406,5);`
`# by ID`

`G:=PCGroup([3,-2,-7,-29,73,3530]);`
`// Polycyclic`

`G:=Group<a,b|a^203=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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