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## G = D207order 414 = 2·32·23

### Dihedral group

Aliases: D207, C9⋊D23, C23⋊D9, C3.D69, C2071C2, C69.1S3, sometimes denoted D414 or Dih207 or Dih414, SmallGroup(414,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C207 — D207
 Chief series C1 — C3 — C69 — C207 — D207
 Lower central C207 — D207
 Upper central C1

Generators and relations for D207
G = < a,b | a207=b2=1, bab=a-1 >

207C2
69S3
9D23
23D9
3D69

Smallest permutation representation of D207
On 207 points
Generators in S207
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207)
(2 207)(3 206)(4 205)(5 204)(6 203)(7 202)(8 201)(9 200)(10 199)(11 198)(12 197)(13 196)(14 195)(15 194)(16 193)(17 192)(18 191)(19 190)(20 189)(21 188)(22 187)(23 186)(24 185)(25 184)(26 183)(27 182)(28 181)(29 180)(30 179)(31 178)(32 177)(33 176)(34 175)(35 174)(36 173)(37 172)(38 171)(39 170)(40 169)(41 168)(42 167)(43 166)(44 165)(45 164)(46 163)(47 162)(48 161)(49 160)(50 159)(51 158)(52 157)(53 156)(54 155)(55 154)(56 153)(57 152)(58 151)(59 150)(60 149)(61 148)(62 147)(63 146)(64 145)(65 144)(66 143)(67 142)(68 141)(69 140)(70 139)(71 138)(72 137)(73 136)(74 135)(75 134)(76 133)(77 132)(78 131)(79 130)(80 129)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 120)(90 119)(91 118)(92 117)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)```

`G:=sub<Sym(207)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207), (2,207)(3,206)(4,205)(5,204)(6,203)(7,202)(8,201)(9,200)(10,199)(11,198)(12,197)(13,196)(14,195)(15,194)(16,193)(17,192)(18,191)(19,190)(20,189)(21,188)(22,187)(23,186)(24,185)(25,184)(26,183)(27,182)(28,181)(29,180)(30,179)(31,178)(32,177)(33,176)(34,175)(35,174)(36,173)(37,172)(38,171)(39,170)(40,169)(41,168)(42,167)(43,166)(44,165)(45,164)(46,163)(47,162)(48,161)(49,160)(50,159)(51,158)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207), (2,207)(3,206)(4,205)(5,204)(6,203)(7,202)(8,201)(9,200)(10,199)(11,198)(12,197)(13,196)(14,195)(15,194)(16,193)(17,192)(18,191)(19,190)(20,189)(21,188)(22,187)(23,186)(24,185)(25,184)(26,183)(27,182)(28,181)(29,180)(30,179)(31,178)(32,177)(33,176)(34,175)(35,174)(36,173)(37,172)(38,171)(39,170)(40,169)(41,168)(42,167)(43,166)(44,165)(45,164)(46,163)(47,162)(48,161)(49,160)(50,159)(51,158)(52,157)(53,156)(54,155)(55,154)(56,153)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,132)(78,131)(79,130)(80,129)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,120)(90,119)(91,118)(92,117)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)], [(2,207),(3,206),(4,205),(5,204),(6,203),(7,202),(8,201),(9,200),(10,199),(11,198),(12,197),(13,196),(14,195),(15,194),(16,193),(17,192),(18,191),(19,190),(20,189),(21,188),(22,187),(23,186),(24,185),(25,184),(26,183),(27,182),(28,181),(29,180),(30,179),(31,178),(32,177),(33,176),(34,175),(35,174),(36,173),(37,172),(38,171),(39,170),(40,169),(41,168),(42,167),(43,166),(44,165),(45,164),(46,163),(47,162),(48,161),(49,160),(50,159),(51,158),(52,157),(53,156),(54,155),(55,154),(56,153),(57,152),(58,151),(59,150),(60,149),(61,148),(62,147),(63,146),(64,145),(65,144),(66,143),(67,142),(68,141),(69,140),(70,139),(71,138),(72,137),(73,136),(74,135),(75,134),(76,133),(77,132),(78,131),(79,130),(80,129),(81,128),(82,127),(83,126),(84,125),(85,124),(86,123),(87,122),(88,121),(89,120),(90,119),(91,118),(92,117),(93,116),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105)]])`

105 conjugacy classes

 class 1 2 3 9A 9B 9C 23A ··· 23K 69A ··· 69V 207A ··· 207BN order 1 2 3 9 9 9 23 ··· 23 69 ··· 69 207 ··· 207 size 1 207 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 2 2 2 2 2 type + + + + + + + image C1 C2 S3 D9 D23 D69 D207 kernel D207 C207 C69 C23 C9 C3 C1 # reps 1 1 1 3 11 22 66

Matrix representation of D207 in GL2(𝔽829) generated by

 315 116 713 431
,
 0 1 1 0
`G:=sub<GL(2,GF(829))| [315,713,116,431],[0,1,1,0] >;`

D207 in GAP, Magma, Sage, TeX

`D_{207}`
`% in TeX`

`G:=Group("D207");`
`// GroupNames label`

`G:=SmallGroup(414,3);`
`// by ID`

`G=gap.SmallGroup(414,3);`
`# by ID`

`G:=PCGroup([4,-2,-3,-23,-3,1137,1109,1586,4419]);`
`// Polycyclic`

`G:=Group<a,b|a^207=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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