metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D233, C233⋊C2, sometimes denoted D466 or Dih233 or Dih466, SmallGroup(466,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C233 — D233 |
Generators and relations for D233
G = < a,b | a233=b2=1, bab=a-1 >
Smallest permutation representation of D233
►On 233 points: primitive
Generators in S233
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233)
(1 233)(2 232)(3 231)(4 230)(5 229)(6 228)(7 227)(8 226)(9 225)(10 224)(11 223)(12 222)(13 221)(14 220)(15 219)(16 218)(17 217)(18 216)(19 215)(20 214)(21 213)(22 212)(23 211)(24 210)(25 209)(26 208)(27 207)(28 206)(29 205)(30 204)(31 203)(32 202)(33 201)(34 200)(35 199)(36 198)(37 197)(38 196)(39 195)(40 194)(41 193)(42 192)(43 191)(44 190)(45 189)(46 188)(47 187)(48 186)(49 185)(50 184)(51 183)(52 182)(53 181)(54 180)(55 179)(56 178)(57 177)(58 176)(59 175)(60 174)(61 173)(62 172)(63 171)(64 170)(65 169)(66 168)(67 167)(68 166)(69 165)(70 164)(71 163)(72 162)(73 161)(74 160)(75 159)(76 158)(77 157)(78 156)(79 155)(80 154)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)(88 146)(89 145)(90 144)(91 143)(92 142)(93 141)(94 140)(95 139)(96 138)(97 137)(98 136)(99 135)(100 134)(101 133)(102 132)(103 131)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)
G:=sub<Sym(233)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233), (1,233)(2,232)(3,231)(4,230)(5,229)(6,228)(7,227)(8,226)(9,225)(10,224)(11,223)(12,222)(13,221)(14,220)(15,219)(16,218)(17,217)(18,216)(19,215)(20,214)(21,213)(22,212)(23,211)(24,210)(25,209)(26,208)(27,207)(28,206)(29,205)(30,204)(31,203)(32,202)(33,201)(34,200)(35,199)(36,198)(37,197)(38,196)(39,195)(40,194)(41,193)(42,192)(43,191)(44,190)(45,189)(46,188)(47,187)(48,186)(49,185)(50,184)(51,183)(52,182)(53,181)(54,180)(55,179)(56,178)(57,177)(58,176)(59,175)(60,174)(61,173)(62,172)(63,171)(64,170)(65,169)(66,168)(67,167)(68,166)(69,165)(70,164)(71,163)(72,162)(73,161)(74,160)(75,159)(76,158)(77,157)(78,156)(79,155)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,140)(95,139)(96,138)(97,137)(98,136)(99,135)(100,134)(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233), (1,233)(2,232)(3,231)(4,230)(5,229)(6,228)(7,227)(8,226)(9,225)(10,224)(11,223)(12,222)(13,221)(14,220)(15,219)(16,218)(17,217)(18,216)(19,215)(20,214)(21,213)(22,212)(23,211)(24,210)(25,209)(26,208)(27,207)(28,206)(29,205)(30,204)(31,203)(32,202)(33,201)(34,200)(35,199)(36,198)(37,197)(38,196)(39,195)(40,194)(41,193)(42,192)(43,191)(44,190)(45,189)(46,188)(47,187)(48,186)(49,185)(50,184)(51,183)(52,182)(53,181)(54,180)(55,179)(56,178)(57,177)(58,176)(59,175)(60,174)(61,173)(62,172)(63,171)(64,170)(65,169)(66,168)(67,167)(68,166)(69,165)(70,164)(71,163)(72,162)(73,161)(74,160)(75,159)(76,158)(77,157)(78,156)(79,155)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,140)(95,139)(96,138)(97,137)(98,136)(99,135)(100,134)(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233)], [(1,233),(2,232),(3,231),(4,230),(5,229),(6,228),(7,227),(8,226),(9,225),(10,224),(11,223),(12,222),(13,221),(14,220),(15,219),(16,218),(17,217),(18,216),(19,215),(20,214),(21,213),(22,212),(23,211),(24,210),(25,209),(26,208),(27,207),(28,206),(29,205),(30,204),(31,203),(32,202),(33,201),(34,200),(35,199),(36,198),(37,197),(38,196),(39,195),(40,194),(41,193),(42,192),(43,191),(44,190),(45,189),(46,188),(47,187),(48,186),(49,185),(50,184),(51,183),(52,182),(53,181),(54,180),(55,179),(56,178),(57,177),(58,176),(59,175),(60,174),(61,173),(62,172),(63,171),(64,170),(65,169),(66,168),(67,167),(68,166),(69,165),(70,164),(71,163),(72,162),(73,161),(74,160),(75,159),(76,158),(77,157),(78,156),(79,155),(80,154),(81,153),(82,152),(83,151),(84,150),(85,149),(86,148),(87,147),(88,146),(89,145),(90,144),(91,143),(92,142),(93,141),(94,140),(95,139),(96,138),(97,137),(98,136),(99,135),(100,134),(101,133),(102,132),(103,131),(104,130),(105,129),(106,128),(107,127),(108,126),(109,125),(110,124),(111,123),(112,122),(113,121),(114,120),(115,119),(116,118)]])
118 conjugacy classes
| class | 1 | 2 | 233A | ··· | 233DL |
| order | 1 | 2 | 233 | ··· | 233 |
| size | 1 | 233 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D233 |
| kernel | D233 | C233 | C1 |
| # reps | 1 | 1 | 116 |
Matrix representation of D233 ►in GL2(𝔽467) generated by
| 258 | 466 |
| 2 | 324 |
| 242 | 319 |
| 77 | 225 |
G:=sub<GL(2,GF(467))| [258,2,466,324],[242,77,319,225] >;
D233 in GAP, Magma, Sage, TeX
D_{233} % in TeX
G:=Group("D233"); // GroupNames label
G:=SmallGroup(466,1);
// by ID
G=gap.SmallGroup(466,1);
# by ID
G:=PCGroup([2,-2,-233,1857]);
// Polycyclic
G:=Group<a,b|a^233=b^2=1,b*a*b=a^-1>;
// generators/relations
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