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## G = D222order 444 = 22·3·37

### Dihedral group

Aliases: D222, C2×D111, C74⋊S3, C6⋊D37, C32D74, C372D6, C2221C2, C1112C22, sometimes denoted D444 or Dih222 or Dih444, SmallGroup(444,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C111 — D222
 Chief series C1 — C37 — C111 — D111 — D222
 Lower central C111 — D222
 Upper central C1 — C2

Generators and relations for D222
G = < a,b | a222=b2=1, bab=a-1 >

111C2
111C2
111C22
37S3
37S3
3D37
3D37
37D6
3D74

Smallest permutation representation of D222
On 222 points
Generators in S222
```(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)
(1 222)(2 221)(3 220)(4 219)(5 218)(6 217)(7 216)(8 215)(9 214)(10 213)(11 212)(12 211)(13 210)(14 209)(15 208)(16 207)(17 206)(18 205)(19 204)(20 203)(21 202)(22 201)(23 200)(24 199)(25 198)(26 197)(27 196)(28 195)(29 194)(30 193)(31 192)(32 191)(33 190)(34 189)(35 188)(36 187)(37 186)(38 185)(39 184)(40 183)(41 182)(42 181)(43 180)(44 179)(45 178)(46 177)(47 176)(48 175)(49 174)(50 173)(51 172)(52 171)(53 170)(54 169)(55 168)(56 167)(57 166)(58 165)(59 164)(60 163)(61 162)(62 161)(63 160)(64 159)(65 158)(66 157)(67 156)(68 155)(69 154)(70 153)(71 152)(72 151)(73 150)(74 149)(75 148)(76 147)(77 146)(78 145)(79 144)(80 143)(81 142)(82 141)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)```

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

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

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

114 conjugacy classes

 class 1 2A 2B 2C 3 6 37A ··· 37R 74A ··· 74R 111A ··· 111AJ 222A ··· 222AJ order 1 2 2 2 3 6 37 ··· 37 74 ··· 74 111 ··· 111 222 ··· 222 size 1 1 111 111 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D37 D74 D111 D222 kernel D222 D111 C222 C74 C37 C6 C3 C2 C1 # reps 1 2 1 1 1 18 18 36 36

Matrix representation of D222 in GL3(𝔽223) generated by

 222 0 0 0 98 87 0 136 32
,
 1 0 0 0 98 87 0 128 125
`G:=sub<GL(3,GF(223))| [222,0,0,0,98,136,0,87,32],[1,0,0,0,98,128,0,87,125] >;`

D222 in GAP, Magma, Sage, TeX

`D_{222}`
`% in TeX`

`G:=Group("D222");`
`// GroupNames label`

`G:=SmallGroup(444,17);`
`// by ID`

`G=gap.SmallGroup(444,17);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,-37,98,6915]);`
`// Polycyclic`

`G:=Group<a,b|a^222=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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