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## G = D227order 454 = 2·227

### Dihedral group

Aliases: D227, C227⋊C2, sometimes denoted D454 or Dih227 or Dih454, SmallGroup(454,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C227 — D227
 Chief series C1 — C227 — D227
 Lower central C227 — D227
 Upper central C1

Generators and relations for D227
G = < a,b | a227=b2=1, bab=a-1 >

227C2

Smallest permutation representation of D227
On 227 points: primitive
Generators in S227
```(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227)
(1 227)(2 226)(3 225)(4 224)(5 223)(6 222)(7 221)(8 220)(9 219)(10 218)(11 217)(12 216)(13 215)(14 214)(15 213)(16 212)(17 211)(18 210)(19 209)(20 208)(21 207)(22 206)(23 205)(24 204)(25 203)(26 202)(27 201)(28 200)(29 199)(30 198)(31 197)(32 196)(33 195)(34 194)(35 193)(36 192)(37 191)(38 190)(39 189)(40 188)(41 187)(42 186)(43 185)(44 184)(45 183)(46 182)(47 181)(48 180)(49 179)(50 178)(51 177)(52 176)(53 175)(54 174)(55 173)(56 172)(57 171)(58 170)(59 169)(60 168)(61 167)(62 166)(63 165)(64 164)(65 163)(66 162)(67 161)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(81 147)(82 146)(83 145)(84 144)(85 143)(86 142)(87 141)(88 140)(89 139)(90 138)(91 137)(92 136)(93 135)(94 134)(95 133)(96 132)(97 131)(98 130)(99 129)(100 128)(101 127)(102 126)(103 125)(104 124)(105 123)(106 122)(107 121)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)```

`G:=sub<Sym(227)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227), (1,227)(2,226)(3,225)(4,224)(5,223)(6,222)(7,221)(8,220)(9,219)(10,218)(11,217)(12,216)(13,215)(14,214)(15,213)(16,212)(17,211)(18,210)(19,209)(20,208)(21,207)(22,206)(23,205)(24,204)(25,203)(26,202)(27,201)(28,200)(29,199)(30,198)(31,197)(32,196)(33,195)(34,194)(35,193)(36,192)(37,191)(38,190)(39,189)(40,188)(41,187)(42,186)(43,185)(44,184)(45,183)(46,182)(47,181)(48,180)(49,179)(50,178)(51,177)(52,176)(53,175)(54,174)(55,173)(56,172)(57,171)(58,170)(59,169)(60,168)(61,167)(62,166)(63,165)(64,164)(65,163)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,147)(82,146)(83,145)(84,144)(85,143)(86,142)(87,141)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227), (1,227)(2,226)(3,225)(4,224)(5,223)(6,222)(7,221)(8,220)(9,219)(10,218)(11,217)(12,216)(13,215)(14,214)(15,213)(16,212)(17,211)(18,210)(19,209)(20,208)(21,207)(22,206)(23,205)(24,204)(25,203)(26,202)(27,201)(28,200)(29,199)(30,198)(31,197)(32,196)(33,195)(34,194)(35,193)(36,192)(37,191)(38,190)(39,189)(40,188)(41,187)(42,186)(43,185)(44,184)(45,183)(46,182)(47,181)(48,180)(49,179)(50,178)(51,177)(52,176)(53,175)(54,174)(55,173)(56,172)(57,171)(58,170)(59,169)(60,168)(61,167)(62,166)(63,165)(64,164)(65,163)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,147)(82,146)(83,145)(84,144)(85,143)(86,142)(87,141)(88,140)(89,139)(90,138)(91,137)(92,136)(93,135)(94,134)(95,133)(96,132)(97,131)(98,130)(99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227)], [(1,227),(2,226),(3,225),(4,224),(5,223),(6,222),(7,221),(8,220),(9,219),(10,218),(11,217),(12,216),(13,215),(14,214),(15,213),(16,212),(17,211),(18,210),(19,209),(20,208),(21,207),(22,206),(23,205),(24,204),(25,203),(26,202),(27,201),(28,200),(29,199),(30,198),(31,197),(32,196),(33,195),(34,194),(35,193),(36,192),(37,191),(38,190),(39,189),(40,188),(41,187),(42,186),(43,185),(44,184),(45,183),(46,182),(47,181),(48,180),(49,179),(50,178),(51,177),(52,176),(53,175),(54,174),(55,173),(56,172),(57,171),(58,170),(59,169),(60,168),(61,167),(62,166),(63,165),(64,164),(65,163),(66,162),(67,161),(68,160),(69,159),(70,158),(71,157),(72,156),(73,155),(74,154),(75,153),(76,152),(77,151),(78,150),(79,149),(80,148),(81,147),(82,146),(83,145),(84,144),(85,143),(86,142),(87,141),(88,140),(89,139),(90,138),(91,137),(92,136),(93,135),(94,134),(95,133),(96,132),(97,131),(98,130),(99,129),(100,128),(101,127),(102,126),(103,125),(104,124),(105,123),(106,122),(107,121),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115)]])`

115 conjugacy classes

 class 1 2 227A ··· 227DI order 1 2 227 ··· 227 size 1 227 2 ··· 2

115 irreducible representations

 dim 1 1 2 type + + + image C1 C2 D227 kernel D227 C227 C1 # reps 1 1 113

Matrix representation of D227 in GL2(𝔽5449) generated by

 3647 5448 1 0
,
 3647 5448 5048 1802
`G:=sub<GL(2,GF(5449))| [3647,1,5448,0],[3647,5048,5448,1802] >;`

D227 in GAP, Magma, Sage, TeX

`D_{227}`
`% in TeX`

`G:=Group("D227");`
`// GroupNames label`

`G:=SmallGroup(454,1);`
`// by ID`

`G=gap.SmallGroup(454,1);`
`# by ID`

`G:=PCGroup([2,-2,-227,1809]);`
`// Polycyclic`

`G:=Group<a,b|a^227=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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