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## G = D214order 428 = 22·107

### Dihedral group

Aliases: D214, C2×D107, C214⋊C2, C107⋊C22, sometimes denoted D428 or Dih214 or Dih428, SmallGroup(428,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C107 — D214
 Chief series C1 — C107 — D107 — D214
 Lower central C107 — D214
 Upper central C1 — C2

Generators and relations for D214
G = < a,b | a214=b2=1, bab=a-1 >

107C2
107C2
107C22

Smallest permutation representation of D214
On 214 points
Generators in S214
```(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)
(1 214)(2 213)(3 212)(4 211)(5 210)(6 209)(7 208)(8 207)(9 206)(10 205)(11 204)(12 203)(13 202)(14 201)(15 200)(16 199)(17 198)(18 197)(19 196)(20 195)(21 194)(22 193)(23 192)(24 191)(25 190)(26 189)(27 188)(28 187)(29 186)(30 185)(31 184)(32 183)(33 182)(34 181)(35 180)(36 179)(37 178)(38 177)(39 176)(40 175)(41 174)(42 173)(43 172)(44 171)(45 170)(46 169)(47 168)(48 167)(49 166)(50 165)(51 164)(52 163)(53 162)(54 161)(55 160)(56 159)(57 158)(58 157)(59 156)(60 155)(61 154)(62 153)(63 152)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 144)(72 143)(73 142)(74 141)(75 140)(76 139)(77 138)(78 137)(79 136)(80 135)(81 134)(82 133)(83 132)(84 131)(85 130)(86 129)(87 128)(88 127)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)```

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 107A ··· 107BA 214A ··· 214BA order 1 2 2 2 107 ··· 107 214 ··· 214 size 1 1 107 107 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D107 D214 kernel D214 D107 C214 C2 C1 # reps 1 2 1 53 53

Matrix representation of D214 in GL3(𝔽643) generated by

 642 0 0 0 19 67 0 576 576
,
 1 0 0 0 19 67 0 81 624
`G:=sub<GL(3,GF(643))| [642,0,0,0,19,576,0,67,576],[1,0,0,0,19,81,0,67,624] >;`

D214 in GAP, Magma, Sage, TeX

`D_{214}`
`% in TeX`

`G:=Group("D214");`
`// GroupNames label`

`G:=SmallGroup(428,3);`
`// by ID`

`G=gap.SmallGroup(428,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-107,3818]);`
`// Polycyclic`

`G:=Group<a,b|a^214=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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