direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D83, C83⋊C6, C249⋊2C2, SmallGroup(498,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C83 — C3×D83 |
Generators and relations for C3×D83
G = < a,b,c | a3=b83=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D83
►On 249 points
Generators in S249
(1 197 93)(2 198 94)(3 199 95)(4 200 96)(5 201 97)(6 202 98)(7 203 99)(8 204 100)(9 205 101)(10 206 102)(11 207 103)(12 208 104)(13 209 105)(14 210 106)(15 211 107)(16 212 108)(17 213 109)(18 214 110)(19 215 111)(20 216 112)(21 217 113)(22 218 114)(23 219 115)(24 220 116)(25 221 117)(26 222 118)(27 223 119)(28 224 120)(29 225 121)(30 226 122)(31 227 123)(32 228 124)(33 229 125)(34 230 126)(35 231 127)(36 232 128)(37 233 129)(38 234 130)(39 235 131)(40 236 132)(41 237 133)(42 238 134)(43 239 135)(44 240 136)(45 241 137)(46 242 138)(47 243 139)(48 244 140)(49 245 141)(50 246 142)(51 247 143)(52 248 144)(53 249 145)(54 167 146)(55 168 147)(56 169 148)(57 170 149)(58 171 150)(59 172 151)(60 173 152)(61 174 153)(62 175 154)(63 176 155)(64 177 156)(65 178 157)(66 179 158)(67 180 159)(68 181 160)(69 182 161)(70 183 162)(71 184 163)(72 185 164)(73 186 165)(74 187 166)(75 188 84)(76 189 85)(77 190 86)(78 191 87)(79 192 88)(80 193 89)(81 194 90)(82 195 91)(83 196 92)
(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)(167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249)
(1 83)(2 82)(3 81)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(102 166)(103 165)(104 164)(105 163)(106 162)(107 161)(108 160)(109 159)(110 158)(111 157)(112 156)(113 155)(114 154)(115 153)(116 152)(117 151)(118 150)(119 149)(120 148)(121 147)(122 146)(123 145)(124 144)(125 143)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(167 226)(168 225)(169 224)(170 223)(171 222)(172 221)(173 220)(174 219)(175 218)(176 217)(177 216)(178 215)(179 214)(180 213)(181 212)(182 211)(183 210)(184 209)(185 208)(186 207)(187 206)(188 205)(189 204)(190 203)(191 202)(192 201)(193 200)(194 199)(195 198)(196 197)(227 249)(228 248)(229 247)(230 246)(231 245)(232 244)(233 243)(234 242)(235 241)(236 240)(237 239)
G:=sub<Sym(249)| (1,197,93)(2,198,94)(3,199,95)(4,200,96)(5,201,97)(6,202,98)(7,203,99)(8,204,100)(9,205,101)(10,206,102)(11,207,103)(12,208,104)(13,209,105)(14,210,106)(15,211,107)(16,212,108)(17,213,109)(18,214,110)(19,215,111)(20,216,112)(21,217,113)(22,218,114)(23,219,115)(24,220,116)(25,221,117)(26,222,118)(27,223,119)(28,224,120)(29,225,121)(30,226,122)(31,227,123)(32,228,124)(33,229,125)(34,230,126)(35,231,127)(36,232,128)(37,233,129)(38,234,130)(39,235,131)(40,236,132)(41,237,133)(42,238,134)(43,239,135)(44,240,136)(45,241,137)(46,242,138)(47,243,139)(48,244,140)(49,245,141)(50,246,142)(51,247,143)(52,248,144)(53,249,145)(54,167,146)(55,168,147)(56,169,148)(57,170,149)(58,171,150)(59,172,151)(60,173,152)(61,174,153)(62,175,154)(63,176,155)(64,177,156)(65,178,157)(66,179,158)(67,180,159)(68,181,160)(69,182,161)(70,183,162)(71,184,163)(72,185,164)(73,186,165)(74,187,166)(75,188,84)(76,189,85)(77,190,86)(78,191,87)(79,192,88)(80,193,89)(81,194,90)(82,195,91)(83,196,92), (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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,83)(2,82)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(102,166)(103,165)(104,164)(105,163)(106,162)(107,161)(108,160)(109,159)(110,158)(111,157)(112,156)(113,155)(114,154)(115,153)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(167,226)(168,225)(169,224)(170,223)(171,222)(172,221)(173,220)(174,219)(175,218)(176,217)(177,216)(178,215)(179,214)(180,213)(181,212)(182,211)(183,210)(184,209)(185,208)(186,207)(187,206)(188,205)(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197)(227,249)(228,248)(229,247)(230,246)(231,245)(232,244)(233,243)(234,242)(235,241)(236,240)(237,239)>;
G:=Group( (1,197,93)(2,198,94)(3,199,95)(4,200,96)(5,201,97)(6,202,98)(7,203,99)(8,204,100)(9,205,101)(10,206,102)(11,207,103)(12,208,104)(13,209,105)(14,210,106)(15,211,107)(16,212,108)(17,213,109)(18,214,110)(19,215,111)(20,216,112)(21,217,113)(22,218,114)(23,219,115)(24,220,116)(25,221,117)(26,222,118)(27,223,119)(28,224,120)(29,225,121)(30,226,122)(31,227,123)(32,228,124)(33,229,125)(34,230,126)(35,231,127)(36,232,128)(37,233,129)(38,234,130)(39,235,131)(40,236,132)(41,237,133)(42,238,134)(43,239,135)(44,240,136)(45,241,137)(46,242,138)(47,243,139)(48,244,140)(49,245,141)(50,246,142)(51,247,143)(52,248,144)(53,249,145)(54,167,146)(55,168,147)(56,169,148)(57,170,149)(58,171,150)(59,172,151)(60,173,152)(61,174,153)(62,175,154)(63,176,155)(64,177,156)(65,178,157)(66,179,158)(67,180,159)(68,181,160)(69,182,161)(70,183,162)(71,184,163)(72,185,164)(73,186,165)(74,187,166)(75,188,84)(76,189,85)(77,190,86)(78,191,87)(79,192,88)(80,193,89)(81,194,90)(82,195,91)(83,196,92), (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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,83)(2,82)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(102,166)(103,165)(104,164)(105,163)(106,162)(107,161)(108,160)(109,159)(110,158)(111,157)(112,156)(113,155)(114,154)(115,153)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(167,226)(168,225)(169,224)(170,223)(171,222)(172,221)(173,220)(174,219)(175,218)(176,217)(177,216)(178,215)(179,214)(180,213)(181,212)(182,211)(183,210)(184,209)(185,208)(186,207)(187,206)(188,205)(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197)(227,249)(228,248)(229,247)(230,246)(231,245)(232,244)(233,243)(234,242)(235,241)(236,240)(237,239) );
G=PermutationGroup([[(1,197,93),(2,198,94),(3,199,95),(4,200,96),(5,201,97),(6,202,98),(7,203,99),(8,204,100),(9,205,101),(10,206,102),(11,207,103),(12,208,104),(13,209,105),(14,210,106),(15,211,107),(16,212,108),(17,213,109),(18,214,110),(19,215,111),(20,216,112),(21,217,113),(22,218,114),(23,219,115),(24,220,116),(25,221,117),(26,222,118),(27,223,119),(28,224,120),(29,225,121),(30,226,122),(31,227,123),(32,228,124),(33,229,125),(34,230,126),(35,231,127),(36,232,128),(37,233,129),(38,234,130),(39,235,131),(40,236,132),(41,237,133),(42,238,134),(43,239,135),(44,240,136),(45,241,137),(46,242,138),(47,243,139),(48,244,140),(49,245,141),(50,246,142),(51,247,143),(52,248,144),(53,249,145),(54,167,146),(55,168,147),(56,169,148),(57,170,149),(58,171,150),(59,172,151),(60,173,152),(61,174,153),(62,175,154),(63,176,155),(64,177,156),(65,178,157),(66,179,158),(67,180,159),(68,181,160),(69,182,161),(70,183,162),(71,184,163),(72,185,164),(73,186,165),(74,187,166),(75,188,84),(76,189,85),(77,190,86),(78,191,87),(79,192,88),(80,193,89),(81,194,90),(82,195,91),(83,196,92)], [(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),(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249)], [(1,83),(2,82),(3,81),(4,80),(5,79),(6,78),(7,77),(8,76),(9,75),(10,74),(11,73),(12,72),(13,71),(14,70),(15,69),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(84,101),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(102,166),(103,165),(104,164),(105,163),(106,162),(107,161),(108,160),(109,159),(110,158),(111,157),(112,156),(113,155),(114,154),(115,153),(116,152),(117,151),(118,150),(119,149),(120,148),(121,147),(122,146),(123,145),(124,144),(125,143),(126,142),(127,141),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(167,226),(168,225),(169,224),(170,223),(171,222),(172,221),(173,220),(174,219),(175,218),(176,217),(177,216),(178,215),(179,214),(180,213),(181,212),(182,211),(183,210),(184,209),(185,208),(186,207),(187,206),(188,205),(189,204),(190,203),(191,202),(192,201),(193,200),(194,199),(195,198),(196,197),(227,249),(228,248),(229,247),(230,246),(231,245),(232,244),(233,243),(234,242),(235,241),(236,240),(237,239)]])
129 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 83A | ··· | 83AO | 249A | ··· | 249CD |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 83 | ··· | 83 | 249 | ··· | 249 |
| size | 1 | 83 | 1 | 1 | 83 | 83 | 2 | ··· | 2 | 2 | ··· | 2 |
129 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D83 | C3×D83 |
| kernel | C3×D83 | C249 | D83 | C83 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 41 | 82 |
Matrix representation of C3×D83 ►in GL3(𝔽499) generated by
| 359 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 313 | 1 |
| 0 | 50 | 378 |
| 498 | 0 | 0 |
| 0 | 378 | 498 |
| 0 | 169 | 121 |
G:=sub<GL(3,GF(499))| [359,0,0,0,1,0,0,0,1],[1,0,0,0,313,50,0,1,378],[498,0,0,0,378,169,0,498,121] >;
C3×D83 in GAP, Magma, Sage, TeX
C_3\times D_{83} % in TeX
G:=Group("C3xD83"); // GroupNames label
G:=SmallGroup(498,2);
// by ID
G=gap.SmallGroup(498,2);
# by ID
G:=PCGroup([3,-2,-3,-83,4430]);
// Polycyclic
G:=Group<a,b,c|a^3=b^83=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export