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## G = D205order 410 = 2·5·41

### Dihedral group

Aliases: D205, C41⋊D5, C5⋊D41, C2051C2, sometimes denoted D410 or Dih205 or Dih410, SmallGroup(410,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C205 — D205
 Chief series C1 — C41 — C205 — D205
 Lower central C205 — D205
 Upper central C1

Generators and relations for D205
G = < a,b | a205=b2=1, bab=a-1 >

205C2
41D5
5D41

Smallest permutation representation of D205
On 205 points
Generators in S205
```(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)
(1 205)(2 204)(3 203)(4 202)(5 201)(6 200)(7 199)(8 198)(9 197)(10 196)(11 195)(12 194)(13 193)(14 192)(15 191)(16 190)(17 189)(18 188)(19 187)(20 186)(21 185)(22 184)(23 183)(24 182)(25 181)(26 180)(27 179)(28 178)(29 177)(30 176)(31 175)(32 174)(33 173)(34 172)(35 171)(36 170)(37 169)(38 168)(39 167)(40 166)(41 165)(42 164)(43 163)(44 162)(45 161)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(61 145)(62 144)(63 143)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 135)(72 134)(73 133)(74 132)(75 131)(76 130)(77 129)(78 128)(79 127)(80 126)(81 125)(82 124)(83 123)(84 122)(85 121)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)```

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

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

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

104 conjugacy classes

 class 1 2 5A 5B 41A ··· 41T 205A ··· 205CB order 1 2 5 5 41 ··· 41 205 ··· 205 size 1 205 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D5 D41 D205 kernel D205 C205 C41 C5 C1 # reps 1 1 2 20 80

Matrix representation of D205 in GL2(𝔽821) generated by

 600 12 809 387
,
 600 12 35 221
`G:=sub<GL(2,GF(821))| [600,809,12,387],[600,35,12,221] >;`

D205 in GAP, Magma, Sage, TeX

`D_{205}`
`% in TeX`

`G:=Group("D205");`
`// GroupNames label`

`G:=SmallGroup(410,5);`
`// by ID`

`G=gap.SmallGroup(410,5);`
`# by ID`

`G:=PCGroup([3,-2,-5,-41,49,3602]);`
`// Polycyclic`

`G:=Group<a,b|a^205=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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