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## G = D200order 400 = 24·52

### Dihedral group

Aliases: D200, C251D8, C5.D40, C81D25, C2001C2, C50.3D4, C40.2D5, D1001C2, C4.10D50, C10.3D20, C2.5D100, C20.42D10, C100.10C22, sometimes denoted D400 or Dih200 or Dih400, SmallGroup(400,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C100 — D200
 Chief series C1 — C5 — C25 — C50 — C100 — D100 — D200
 Lower central C25 — C50 — C100 — D200
 Upper central C1 — C2 — C4 — C8

Generators and relations for D200
G = < a,b | a200=b2=1, bab=a-1 >

100C2
100C2
50C22
50C22
20D5
20D5
25D4
25D4
10D10
10D10
4D25
4D25
25D8
5D20
5D20
2D50
2D50
5D40

Smallest permutation representation of D200
On 200 points
Generators in S200
```(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)
(1 175)(2 174)(3 173)(4 172)(5 171)(6 170)(7 169)(8 168)(9 167)(10 166)(11 165)(12 164)(13 163)(14 162)(15 161)(16 160)(17 159)(18 158)(19 157)(20 156)(21 155)(22 154)(23 153)(24 152)(25 151)(26 150)(27 149)(28 148)(29 147)(30 146)(31 145)(32 144)(33 143)(34 142)(35 141)(36 140)(37 139)(38 138)(39 137)(40 136)(41 135)(42 134)(43 133)(44 132)(45 131)(46 130)(47 129)(48 128)(49 127)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 120)(57 119)(58 118)(59 117)(60 116)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 103)(74 102)(75 101)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(176 200)(177 199)(178 198)(179 197)(180 196)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)```

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

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

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

103 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 8A 8B 10A 10B 20A 20B 20C 20D 25A ··· 25J 40A ··· 40H 50A ··· 50J 100A ··· 100T 200A ··· 200AN order 1 2 2 2 4 5 5 8 8 10 10 20 20 20 20 25 ··· 25 40 ··· 40 50 ··· 50 100 ··· 100 200 ··· 200 size 1 1 100 100 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

103 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 D4 D5 D8 D10 D20 D25 D40 D50 D100 D200 kernel D200 C200 D100 C50 C40 C25 C20 C10 C8 C5 C4 C2 C1 # reps 1 1 2 1 2 2 2 4 10 8 10 20 40

Matrix representation of D200 in GL2(𝔽401) generated by

 22 265 136 16
,
 190 75 48 211
`G:=sub<GL(2,GF(401))| [22,136,265,16],[190,48,75,211] >;`

D200 in GAP, Magma, Sage, TeX

`D_{200}`
`% in TeX`

`G:=Group("D200");`
`// GroupNames label`

`G:=SmallGroup(400,8);`
`// by ID`

`G=gap.SmallGroup(400,8);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,79,218,50,4324,628,11525]);`
`// Polycyclic`

`G:=Group<a,b|a^200=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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