metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D235, C47⋊D5, C5⋊D47, C235⋊1C2, sometimes denoted D470 or Dih235 or Dih470, SmallGroup(470,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C235 — D235 |
Generators and relations for D235
G = < a,b | a235=b2=1, bab=a-1 >
Smallest permutation representation of D235
►On 235 points
Generators in S235
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235)
(1 235)(2 234)(3 233)(4 232)(5 231)(6 230)(7 229)(8 228)(9 227)(10 226)(11 225)(12 224)(13 223)(14 222)(15 221)(16 220)(17 219)(18 218)(19 217)(20 216)(21 215)(22 214)(23 213)(24 212)(25 211)(26 210)(27 209)(28 208)(29 207)(30 206)(31 205)(32 204)(33 203)(34 202)(35 201)(36 200)(37 199)(38 198)(39 197)(40 196)(41 195)(42 194)(43 193)(44 192)(45 191)(46 190)(47 189)(48 188)(49 187)(50 186)(51 185)(52 184)(53 183)(54 182)(55 181)(56 180)(57 179)(58 178)(59 177)(60 176)(61 175)(62 174)(63 173)(64 172)(65 171)(66 170)(67 169)(68 168)(69 167)(70 166)(71 165)(72 164)(73 163)(74 162)(75 161)(76 160)(77 159)(78 158)(79 157)(80 156)(81 155)(82 154)(83 153)(84 152)(85 151)(86 150)(87 149)(88 148)(89 147)(90 146)(91 145)(92 144)(93 143)(94 142)(95 141)(96 140)(97 139)(98 138)(99 137)(100 136)(101 135)(102 134)(103 133)(104 132)(105 131)(106 130)(107 129)(108 128)(109 127)(110 126)(111 125)(112 124)(113 123)(114 122)(115 121)(116 120)(117 119)
G:=sub<Sym(235)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235), (1,235)(2,234)(3,233)(4,232)(5,231)(6,230)(7,229)(8,228)(9,227)(10,226)(11,225)(12,224)(13,223)(14,222)(15,221)(16,220)(17,219)(18,218)(19,217)(20,216)(21,215)(22,214)(23,213)(24,212)(25,211)(26,210)(27,209)(28,208)(29,207)(30,206)(31,205)(32,204)(33,203)(34,202)(35,201)(36,200)(37,199)(38,198)(39,197)(40,196)(41,195)(42,194)(43,193)(44,192)(45,191)(46,190)(47,189)(48,188)(49,187)(50,186)(51,185)(52,184)(53,183)(54,182)(55,181)(56,180)(57,179)(58,178)(59,177)(60,176)(61,175)(62,174)(63,173)(64,172)(65,171)(66,170)(67,169)(68,168)(69,167)(70,166)(71,165)(72,164)(73,163)(74,162)(75,161)(76,160)(77,159)(78,158)(79,157)(80,156)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,120)(117,119)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235), (1,235)(2,234)(3,233)(4,232)(5,231)(6,230)(7,229)(8,228)(9,227)(10,226)(11,225)(12,224)(13,223)(14,222)(15,221)(16,220)(17,219)(18,218)(19,217)(20,216)(21,215)(22,214)(23,213)(24,212)(25,211)(26,210)(27,209)(28,208)(29,207)(30,206)(31,205)(32,204)(33,203)(34,202)(35,201)(36,200)(37,199)(38,198)(39,197)(40,196)(41,195)(42,194)(43,193)(44,192)(45,191)(46,190)(47,189)(48,188)(49,187)(50,186)(51,185)(52,184)(53,183)(54,182)(55,181)(56,180)(57,179)(58,178)(59,177)(60,176)(61,175)(62,174)(63,173)(64,172)(65,171)(66,170)(67,169)(68,168)(69,167)(70,166)(71,165)(72,164)(73,163)(74,162)(75,161)(76,160)(77,159)(78,158)(79,157)(80,156)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,120)(117,119) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235)], [(1,235),(2,234),(3,233),(4,232),(5,231),(6,230),(7,229),(8,228),(9,227),(10,226),(11,225),(12,224),(13,223),(14,222),(15,221),(16,220),(17,219),(18,218),(19,217),(20,216),(21,215),(22,214),(23,213),(24,212),(25,211),(26,210),(27,209),(28,208),(29,207),(30,206),(31,205),(32,204),(33,203),(34,202),(35,201),(36,200),(37,199),(38,198),(39,197),(40,196),(41,195),(42,194),(43,193),(44,192),(45,191),(46,190),(47,189),(48,188),(49,187),(50,186),(51,185),(52,184),(53,183),(54,182),(55,181),(56,180),(57,179),(58,178),(59,177),(60,176),(61,175),(62,174),(63,173),(64,172),(65,171),(66,170),(67,169),(68,168),(69,167),(70,166),(71,165),(72,164),(73,163),(74,162),(75,161),(76,160),(77,159),(78,158),(79,157),(80,156),(81,155),(82,154),(83,153),(84,152),(85,151),(86,150),(87,149),(88,148),(89,147),(90,146),(91,145),(92,144),(93,143),(94,142),(95,141),(96,140),(97,139),(98,138),(99,137),(100,136),(101,135),(102,134),(103,133),(104,132),(105,131),(106,130),(107,129),(108,128),(109,127),(110,126),(111,125),(112,124),(113,123),(114,122),(115,121),(116,120),(117,119)]])
119 conjugacy classes
| class | 1 | 2 | 5A | 5B | 47A | ··· | 47W | 235A | ··· | 235CN |
| order | 1 | 2 | 5 | 5 | 47 | ··· | 47 | 235 | ··· | 235 |
| size | 1 | 235 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
119 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D5 | D47 | D235 |
| kernel | D235 | C235 | C47 | C5 | C1 |
| # reps | 1 | 1 | 2 | 23 | 92 |
Matrix representation of D235 ►in GL2(𝔽941) generated by
| 514 | 102 |
| 839 | 399 |
| 521 | 489 |
| 409 | 420 |
G:=sub<GL(2,GF(941))| [514,839,102,399],[521,409,489,420] >;
D235 in GAP, Magma, Sage, TeX
D_{235} % in TeX
G:=Group("D235"); // GroupNames label
G:=SmallGroup(470,3);
// by ID
G=gap.SmallGroup(470,3);
# by ID
G:=PCGroup([3,-2,-5,-47,49,4142]);
// Polycyclic
G:=Group<a,b|a^235=b^2=1,b*a*b=a^-1>;
// generators/relations
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