metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D237, C79⋊S3, C3⋊D79, C237⋊1C2, sometimes denoted D474 or Dih237 or Dih474, SmallGroup(474,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C237 — D237 |
Generators and relations for D237
G = < a,b | a237=b2=1, bab=a-1 >
Smallest permutation representation of D237
►On 237 points
Generators in S237
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237)
(1 237)(2 236)(3 235)(4 234)(5 233)(6 232)(7 231)(8 230)(9 229)(10 228)(11 227)(12 226)(13 225)(14 224)(15 223)(16 222)(17 221)(18 220)(19 219)(20 218)(21 217)(22 216)(23 215)(24 214)(25 213)(26 212)(27 211)(28 210)(29 209)(30 208)(31 207)(32 206)(33 205)(34 204)(35 203)(36 202)(37 201)(38 200)(39 199)(40 198)(41 197)(42 196)(43 195)(44 194)(45 193)(46 192)(47 191)(48 190)(49 189)(50 188)(51 187)(52 186)(53 185)(54 184)(55 183)(56 182)(57 181)(58 180)(59 179)(60 178)(61 177)(62 176)(63 175)(64 174)(65 173)(66 172)(67 171)(68 170)(69 169)(70 168)(71 167)(72 166)(73 165)(74 164)(75 163)(76 162)(77 161)(78 160)(79 159)(80 158)(81 157)(82 156)(83 155)(84 154)(85 153)(86 152)(87 151)(88 150)(89 149)(90 148)(91 147)(92 146)(93 145)(94 144)(95 143)(96 142)(97 141)(98 140)(99 139)(100 138)(101 137)(102 136)(103 135)(104 134)(105 133)(106 132)(107 131)(108 130)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)
G:=sub<Sym(237)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,237)(2,236)(3,235)(4,234)(5,233)(6,232)(7,231)(8,230)(9,229)(10,228)(11,227)(12,226)(13,225)(14,224)(15,223)(16,222)(17,221)(18,220)(19,219)(20,218)(21,217)(22,216)(23,215)(24,214)(25,213)(26,212)(27,211)(28,210)(29,209)(30,208)(31,207)(32,206)(33,205)(34,204)(35,203)(36,202)(37,201)(38,200)(39,199)(40,198)(41,197)(42,196)(43,195)(44,194)(45,193)(46,192)(47,191)(48,190)(49,189)(50,188)(51,187)(52,186)(53,185)(54,184)(55,183)(56,182)(57,181)(58,180)(59,179)(60,178)(61,177)(62,176)(63,175)(64,174)(65,173)(66,172)(67,171)(68,170)(69,169)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,160)(79,159)(80,158)(81,157)(82,156)(83,155)(84,154)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237), (1,237)(2,236)(3,235)(4,234)(5,233)(6,232)(7,231)(8,230)(9,229)(10,228)(11,227)(12,226)(13,225)(14,224)(15,223)(16,222)(17,221)(18,220)(19,219)(20,218)(21,217)(22,216)(23,215)(24,214)(25,213)(26,212)(27,211)(28,210)(29,209)(30,208)(31,207)(32,206)(33,205)(34,204)(35,203)(36,202)(37,201)(38,200)(39,199)(40,198)(41,197)(42,196)(43,195)(44,194)(45,193)(46,192)(47,191)(48,190)(49,189)(50,188)(51,187)(52,186)(53,185)(54,184)(55,183)(56,182)(57,181)(58,180)(59,179)(60,178)(61,177)(62,176)(63,175)(64,174)(65,173)(66,172)(67,171)(68,170)(69,169)(70,168)(71,167)(72,166)(73,165)(74,164)(75,163)(76,162)(77,161)(78,160)(79,159)(80,158)(81,157)(82,156)(83,155)(84,154)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237)], [(1,237),(2,236),(3,235),(4,234),(5,233),(6,232),(7,231),(8,230),(9,229),(10,228),(11,227),(12,226),(13,225),(14,224),(15,223),(16,222),(17,221),(18,220),(19,219),(20,218),(21,217),(22,216),(23,215),(24,214),(25,213),(26,212),(27,211),(28,210),(29,209),(30,208),(31,207),(32,206),(33,205),(34,204),(35,203),(36,202),(37,201),(38,200),(39,199),(40,198),(41,197),(42,196),(43,195),(44,194),(45,193),(46,192),(47,191),(48,190),(49,189),(50,188),(51,187),(52,186),(53,185),(54,184),(55,183),(56,182),(57,181),(58,180),(59,179),(60,178),(61,177),(62,176),(63,175),(64,174),(65,173),(66,172),(67,171),(68,170),(69,169),(70,168),(71,167),(72,166),(73,165),(74,164),(75,163),(76,162),(77,161),(78,160),(79,159),(80,158),(81,157),(82,156),(83,155),(84,154),(85,153),(86,152),(87,151),(88,150),(89,149),(90,148),(91,147),(92,146),(93,145),(94,144),(95,143),(96,142),(97,141),(98,140),(99,139),(100,138),(101,137),(102,136),(103,135),(104,134),(105,133),(106,132),(107,131),(108,130),(109,129),(110,128),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120)]])
120 conjugacy classes
| class | 1 | 2 | 3 | 79A | ··· | 79AM | 237A | ··· | 237BZ |
| order | 1 | 2 | 3 | 79 | ··· | 79 | 237 | ··· | 237 |
| size | 1 | 237 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D79 | D237 |
| kernel | D237 | C237 | C79 | C3 | C1 |
| # reps | 1 | 1 | 1 | 39 | 78 |
Matrix representation of D237 ►in GL2(𝔽1423) generated by
| 162 | 953 |
| 470 | 402 |
| 162 | 953 |
| 840 | 1261 |
G:=sub<GL(2,GF(1423))| [162,470,953,402],[162,840,953,1261] >;
D237 in GAP, Magma, Sage, TeX
D_{237} % in TeX
G:=Group("D237"); // GroupNames label
G:=SmallGroup(474,5);
// by ID
G=gap.SmallGroup(474,5);
# by ID
G:=PCGroup([3,-2,-3,-79,25,4214]);
// Polycyclic
G:=Group<a,b|a^237=b^2=1,b*a*b=a^-1>;
// generators/relations
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