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## G = D221order 442 = 2·13·17

### Dihedral group

Aliases: D221, C17⋊D13, C13⋊D17, C2211C2, sometimes denoted D442 or Dih221 or Dih442, SmallGroup(442,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C221 — D221
 Chief series C1 — C17 — C221 — D221
 Lower central C221 — D221
 Upper central C1

Generators and relations for D221
G = < a,b | a221=b2=1, bab=a-1 >

221C2
17D13
13D17

Smallest permutation representation of D221
On 221 points
Generators in S221
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221)
(1 221)(2 220)(3 219)(4 218)(5 217)(6 216)(7 215)(8 214)(9 213)(10 212)(11 211)(12 210)(13 209)(14 208)(15 207)(16 206)(17 205)(18 204)(19 203)(20 202)(21 201)(22 200)(23 199)(24 198)(25 197)(26 196)(27 195)(28 194)(29 193)(30 192)(31 191)(32 190)(33 189)(34 188)(35 187)(36 186)(37 185)(38 184)(39 183)(40 182)(41 181)(42 180)(43 179)(44 178)(45 177)(46 176)(47 175)(48 174)(49 173)(50 172)(51 171)(52 170)(53 169)(54 168)(55 167)(56 166)(57 165)(58 164)(59 163)(60 162)(61 161)(62 160)(63 159)(64 158)(65 157)(66 156)(67 155)(68 154)(69 153)(70 152)(71 151)(72 150)(73 149)(74 148)(75 147)(76 146)(77 145)(78 144)(79 143)(80 142)(81 141)(82 140)(83 139)(84 138)(85 137)(86 136)(87 135)(88 134)(89 133)(90 132)(91 131)(92 130)(93 129)(94 128)(95 127)(96 126)(97 125)(98 124)(99 123)(100 122)(101 121)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)```

`G:=sub<Sym(221)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221), (1,221)(2,220)(3,219)(4,218)(5,217)(6,216)(7,215)(8,214)(9,213)(10,212)(11,211)(12,210)(13,209)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,201)(22,200)(23,199)(24,198)(25,197)(26,196)(27,195)(28,194)(29,193)(30,192)(31,191)(32,190)(33,189)(34,188)(35,187)(36,186)(37,185)(38,184)(39,183)(40,182)(41,181)(42,180)(43,179)(44,178)(45,177)(46,176)(47,175)(48,174)(49,173)(50,172)(51,171)(52,170)(53,169)(54,168)(55,167)(56,166)(57,165)(58,164)(59,163)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141)(82,140)(83,139)(84,138)(85,137)(86,136)(87,135)(88,134)(89,133)(90,132)(91,131)(92,130)(93,129)(94,128)(95,127)(96,126)(97,125)(98,124)(99,123)(100,122)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221), (1,221)(2,220)(3,219)(4,218)(5,217)(6,216)(7,215)(8,214)(9,213)(10,212)(11,211)(12,210)(13,209)(14,208)(15,207)(16,206)(17,205)(18,204)(19,203)(20,202)(21,201)(22,200)(23,199)(24,198)(25,197)(26,196)(27,195)(28,194)(29,193)(30,192)(31,191)(32,190)(33,189)(34,188)(35,187)(36,186)(37,185)(38,184)(39,183)(40,182)(41,181)(42,180)(43,179)(44,178)(45,177)(46,176)(47,175)(48,174)(49,173)(50,172)(51,171)(52,170)(53,169)(54,168)(55,167)(56,166)(57,165)(58,164)(59,163)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141)(82,140)(83,139)(84,138)(85,137)(86,136)(87,135)(88,134)(89,133)(90,132)(91,131)(92,130)(93,129)(94,128)(95,127)(96,126)(97,125)(98,124)(99,123)(100,122)(101,121)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221)], [(1,221),(2,220),(3,219),(4,218),(5,217),(6,216),(7,215),(8,214),(9,213),(10,212),(11,211),(12,210),(13,209),(14,208),(15,207),(16,206),(17,205),(18,204),(19,203),(20,202),(21,201),(22,200),(23,199),(24,198),(25,197),(26,196),(27,195),(28,194),(29,193),(30,192),(31,191),(32,190),(33,189),(34,188),(35,187),(36,186),(37,185),(38,184),(39,183),(40,182),(41,181),(42,180),(43,179),(44,178),(45,177),(46,176),(47,175),(48,174),(49,173),(50,172),(51,171),(52,170),(53,169),(54,168),(55,167),(56,166),(57,165),(58,164),(59,163),(60,162),(61,161),(62,160),(63,159),(64,158),(65,157),(66,156),(67,155),(68,154),(69,153),(70,152),(71,151),(72,150),(73,149),(74,148),(75,147),(76,146),(77,145),(78,144),(79,143),(80,142),(81,141),(82,140),(83,139),(84,138),(85,137),(86,136),(87,135),(88,134),(89,133),(90,132),(91,131),(92,130),(93,129),(94,128),(95,127),(96,126),(97,125),(98,124),(99,123),(100,122),(101,121),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112)])`

112 conjugacy classes

 class 1 2 13A ··· 13F 17A ··· 17H 221A ··· 221CR order 1 2 13 ··· 13 17 ··· 17 221 ··· 221 size 1 221 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D13 D17 D221 kernel D221 C221 C17 C13 C1 # reps 1 1 6 8 96

Matrix representation of D221 in GL2(𝔽443) generated by

 117 321 122 259
,
 117 321 439 326
`G:=sub<GL(2,GF(443))| [117,122,321,259],[117,439,321,326] >;`

D221 in GAP, Magma, Sage, TeX

`D_{221}`
`% in TeX`

`G:=Group("D221");`
`// GroupNames label`

`G:=SmallGroup(442,3);`
`// by ID`

`G=gap.SmallGroup(442,3);`
`# by ID`

`G:=PCGroup([3,-2,-13,-17,145,3746]);`
`// Polycyclic`

`G:=Group<a,b|a^221=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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