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## G = D225order 450 = 2·32·52

### Dihedral group

Aliases: D225, C25⋊D9, C9⋊D25, C3.D75, C5.D45, C2251C2, C75.1S3, C45.1D5, C15.1D15, sometimes denoted D450 or Dih225 or Dih450, SmallGroup(450,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C225 — D225
 Chief series C1 — C5 — C15 — C75 — C225 — D225
 Lower central C225 — D225
 Upper central C1

Generators and relations for D225
G = < a,b | a225=b2=1, bab=a-1 >

225C2
75S3
45D5
25D9
15D15
9D25
5D45
3D75

Smallest permutation representation of D225
On 225 points
Generators in S225
```(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)
(2 225)(3 224)(4 223)(5 222)(6 221)(7 220)(8 219)(9 218)(10 217)(11 216)(12 215)(13 214)(14 213)(15 212)(16 211)(17 210)(18 209)(19 208)(20 207)(21 206)(22 205)(23 204)(24 203)(25 202)(26 201)(27 200)(28 199)(29 198)(30 197)(31 196)(32 195)(33 194)(34 193)(35 192)(36 191)(37 190)(38 189)(39 188)(40 187)(41 186)(42 185)(43 184)(44 183)(45 182)(46 181)(47 180)(48 179)(49 178)(50 177)(51 176)(52 175)(53 174)(54 173)(55 172)(56 171)(57 170)(58 169)(59 168)(60 167)(61 166)(62 165)(63 164)(64 163)(65 162)(66 161)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)(81 146)(82 145)(83 144)(84 143)(85 142)(86 141)(87 140)(88 139)(89 138)(90 137)(91 136)(92 135)(93 134)(94 133)(95 132)(96 131)(97 130)(98 129)(99 128)(100 127)(101 126)(102 125)(103 124)(104 123)(105 122)(106 121)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)```

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

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

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

114 conjugacy classes

 class 1 2 3 5A 5B 9A 9B 9C 15A 15B 15C 15D 25A ··· 25J 45A ··· 45L 75A ··· 75T 225A ··· 225BH order 1 2 3 5 5 9 9 9 15 15 15 15 25 ··· 25 45 ··· 45 75 ··· 75 225 ··· 225 size 1 225 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 S3 D5 D9 D15 D25 D45 D75 D225 kernel D225 C225 C75 C45 C25 C15 C9 C5 C3 C1 # reps 1 1 1 2 3 4 10 12 20 60

Matrix representation of D225 in GL2(𝔽1801) generated by

 462 1331 470 1241
,
 871 1145 234 930
`G:=sub<GL(2,GF(1801))| [462,470,1331,1241],[871,234,1145,930] >;`

D225 in GAP, Magma, Sage, TeX

`D_{225}`
`% in TeX`

`G:=Group("D225");`
`// GroupNames label`

`G:=SmallGroup(450,3);`
`// by ID`

`G=gap.SmallGroup(450,3);`
`# by ID`

`G:=PCGroup([5,-2,-3,-5,-3,-5,341,306,1712,912,1203,9004]);`
`// Polycyclic`

`G:=Group<a,b|a^225=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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