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## G = D218order 436 = 22·109

### Dihedral group

Aliases: D218, C2×D109, C218⋊C2, C109⋊C22, sometimes denoted D436 or Dih218 or Dih436, SmallGroup(436,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C109 — D218
 Chief series C1 — C109 — D109 — D218
 Lower central C109 — D218
 Upper central C1 — C2

Generators and relations for D218
G = < a,b | a218=b2=1, bab=a-1 >

109C2
109C2
109C22

Smallest permutation representation of D218
On 218 points
Generators in S218
```(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)
(1 218)(2 217)(3 216)(4 215)(5 214)(6 213)(7 212)(8 211)(9 210)(10 209)(11 208)(12 207)(13 206)(14 205)(15 204)(16 203)(17 202)(18 201)(19 200)(20 199)(21 198)(22 197)(23 196)(24 195)(25 194)(26 193)(27 192)(28 191)(29 190)(30 189)(31 188)(32 187)(33 186)(34 185)(35 184)(36 183)(37 182)(38 181)(39 180)(40 179)(41 178)(42 177)(43 176)(44 175)(45 174)(46 173)(47 172)(48 171)(49 170)(50 169)(51 168)(52 167)(53 166)(54 165)(55 164)(56 163)(57 162)(58 161)(59 160)(60 159)(61 158)(62 157)(63 156)(64 155)(65 154)(66 153)(67 152)(68 151)(69 150)(70 149)(71 148)(72 147)(73 146)(74 145)(75 144)(76 143)(77 142)(78 141)(79 140)(80 139)(81 138)(82 137)(83 136)(84 135)(85 134)(86 133)(87 132)(88 131)(89 130)(90 129)(91 128)(92 127)(93 126)(94 125)(95 124)(96 123)(97 122)(98 121)(99 120)(100 119)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(109 110)```

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 109A ··· 109BB 218A ··· 218BB order 1 2 2 2 109 ··· 109 218 ··· 218 size 1 1 109 109 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D109 D218 kernel D218 D109 C218 C2 C1 # reps 1 2 1 54 54

Matrix representation of D218 in GL2(𝔽1091) generated by

 728 792 138 841
,
 622 853 695 469
`G:=sub<GL(2,GF(1091))| [728,138,792,841],[622,695,853,469] >;`

D218 in GAP, Magma, Sage, TeX

`D_{218}`
`% in TeX`

`G:=Group("D218");`
`// GroupNames label`

`G:=SmallGroup(436,4);`
`// by ID`

`G=gap.SmallGroup(436,4);`
`# by ID`

`G:=PCGroup([3,-2,-2,-109,3890]);`
`// Polycyclic`

`G:=Group<a,b|a^218=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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