direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic39, C78⋊3C4, C6⋊Dic13, C2.2D78, C22.D39, C26⋊2Dic3, C26.11D6, C6.11D26, C78.11C22, (C2×C26).S3, (C2×C6).D13, C39⋊10(C2×C4), (C2×C78).1C2, C3⋊2(C2×Dic13), C13⋊3(C2×Dic3), SmallGroup(312,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C2×Dic39 |
Generators and relations for C2×Dic39
G = < a,b,c | a2=b78=1, c2=b39, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic39
►Regular action on 312 points
Generators in S312
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 136)(72 137)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(157 261)(158 262)(159 263)(160 264)(161 265)(162 266)(163 267)(164 268)(165 269)(166 270)(167 271)(168 272)(169 273)(170 274)(171 275)(172 276)(173 277)(174 278)(175 279)(176 280)(177 281)(178 282)(179 283)(180 284)(181 285)(182 286)(183 287)(184 288)(185 289)(186 290)(187 291)(188 292)(189 293)(190 294)(191 295)(192 296)(193 297)(194 298)(195 299)(196 300)(197 301)(198 302)(199 303)(200 304)(201 305)(202 306)(203 307)(204 308)(205 309)(206 310)(207 311)(208 312)(209 235)(210 236)(211 237)(212 238)(213 239)(214 240)(215 241)(216 242)(217 243)(218 244)(219 245)(220 246)(221 247)(222 248)(223 249)(224 250)(225 251)(226 252)(227 253)(228 254)(229 255)(230 256)(231 257)(232 258)(233 259)(234 260)
(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)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312)
(1 248 40 287)(2 247 41 286)(3 246 42 285)(4 245 43 284)(5 244 44 283)(6 243 45 282)(7 242 46 281)(8 241 47 280)(9 240 48 279)(10 239 49 278)(11 238 50 277)(12 237 51 276)(13 236 52 275)(14 235 53 274)(15 312 54 273)(16 311 55 272)(17 310 56 271)(18 309 57 270)(19 308 58 269)(20 307 59 268)(21 306 60 267)(22 305 61 266)(23 304 62 265)(24 303 63 264)(25 302 64 263)(26 301 65 262)(27 300 66 261)(28 299 67 260)(29 298 68 259)(30 297 69 258)(31 296 70 257)(32 295 71 256)(33 294 72 255)(34 293 73 254)(35 292 74 253)(36 291 75 252)(37 290 76 251)(38 289 77 250)(39 288 78 249)(79 209 118 170)(80 208 119 169)(81 207 120 168)(82 206 121 167)(83 205 122 166)(84 204 123 165)(85 203 124 164)(86 202 125 163)(87 201 126 162)(88 200 127 161)(89 199 128 160)(90 198 129 159)(91 197 130 158)(92 196 131 157)(93 195 132 234)(94 194 133 233)(95 193 134 232)(96 192 135 231)(97 191 136 230)(98 190 137 229)(99 189 138 228)(100 188 139 227)(101 187 140 226)(102 186 141 225)(103 185 142 224)(104 184 143 223)(105 183 144 222)(106 182 145 221)(107 181 146 220)(108 180 147 219)(109 179 148 218)(110 178 149 217)(111 177 150 216)(112 176 151 215)(113 175 152 214)(114 174 153 213)(115 173 154 212)(116 172 155 211)(117 171 156 210)
G:=sub<Sym(312)| (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(157,261)(158,262)(159,263)(160,264)(161,265)(162,266)(163,267)(164,268)(165,269)(166,270)(167,271)(168,272)(169,273)(170,274)(171,275)(172,276)(173,277)(174,278)(175,279)(176,280)(177,281)(178,282)(179,283)(180,284)(181,285)(182,286)(183,287)(184,288)(185,289)(186,290)(187,291)(188,292)(189,293)(190,294)(191,295)(192,296)(193,297)(194,298)(195,299)(196,300)(197,301)(198,302)(199,303)(200,304)(201,305)(202,306)(203,307)(204,308)(205,309)(206,310)(207,311)(208,312)(209,235)(210,236)(211,237)(212,238)(213,239)(214,240)(215,241)(216,242)(217,243)(218,244)(219,245)(220,246)(221,247)(222,248)(223,249)(224,250)(225,251)(226,252)(227,253)(228,254)(229,255)(230,256)(231,257)(232,258)(233,259)(234,260), (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)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,248,40,287)(2,247,41,286)(3,246,42,285)(4,245,43,284)(5,244,44,283)(6,243,45,282)(7,242,46,281)(8,241,47,280)(9,240,48,279)(10,239,49,278)(11,238,50,277)(12,237,51,276)(13,236,52,275)(14,235,53,274)(15,312,54,273)(16,311,55,272)(17,310,56,271)(18,309,57,270)(19,308,58,269)(20,307,59,268)(21,306,60,267)(22,305,61,266)(23,304,62,265)(24,303,63,264)(25,302,64,263)(26,301,65,262)(27,300,66,261)(28,299,67,260)(29,298,68,259)(30,297,69,258)(31,296,70,257)(32,295,71,256)(33,294,72,255)(34,293,73,254)(35,292,74,253)(36,291,75,252)(37,290,76,251)(38,289,77,250)(39,288,78,249)(79,209,118,170)(80,208,119,169)(81,207,120,168)(82,206,121,167)(83,205,122,166)(84,204,123,165)(85,203,124,164)(86,202,125,163)(87,201,126,162)(88,200,127,161)(89,199,128,160)(90,198,129,159)(91,197,130,158)(92,196,131,157)(93,195,132,234)(94,194,133,233)(95,193,134,232)(96,192,135,231)(97,191,136,230)(98,190,137,229)(99,189,138,228)(100,188,139,227)(101,187,140,226)(102,186,141,225)(103,185,142,224)(104,184,143,223)(105,183,144,222)(106,182,145,221)(107,181,146,220)(108,180,147,219)(109,179,148,218)(110,178,149,217)(111,177,150,216)(112,176,151,215)(113,175,152,214)(114,174,153,213)(115,173,154,212)(116,172,155,211)(117,171,156,210)>;
G:=Group( (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(157,261)(158,262)(159,263)(160,264)(161,265)(162,266)(163,267)(164,268)(165,269)(166,270)(167,271)(168,272)(169,273)(170,274)(171,275)(172,276)(173,277)(174,278)(175,279)(176,280)(177,281)(178,282)(179,283)(180,284)(181,285)(182,286)(183,287)(184,288)(185,289)(186,290)(187,291)(188,292)(189,293)(190,294)(191,295)(192,296)(193,297)(194,298)(195,299)(196,300)(197,301)(198,302)(199,303)(200,304)(201,305)(202,306)(203,307)(204,308)(205,309)(206,310)(207,311)(208,312)(209,235)(210,236)(211,237)(212,238)(213,239)(214,240)(215,241)(216,242)(217,243)(218,244)(219,245)(220,246)(221,247)(222,248)(223,249)(224,250)(225,251)(226,252)(227,253)(228,254)(229,255)(230,256)(231,257)(232,258)(233,259)(234,260), (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)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,248,40,287)(2,247,41,286)(3,246,42,285)(4,245,43,284)(5,244,44,283)(6,243,45,282)(7,242,46,281)(8,241,47,280)(9,240,48,279)(10,239,49,278)(11,238,50,277)(12,237,51,276)(13,236,52,275)(14,235,53,274)(15,312,54,273)(16,311,55,272)(17,310,56,271)(18,309,57,270)(19,308,58,269)(20,307,59,268)(21,306,60,267)(22,305,61,266)(23,304,62,265)(24,303,63,264)(25,302,64,263)(26,301,65,262)(27,300,66,261)(28,299,67,260)(29,298,68,259)(30,297,69,258)(31,296,70,257)(32,295,71,256)(33,294,72,255)(34,293,73,254)(35,292,74,253)(36,291,75,252)(37,290,76,251)(38,289,77,250)(39,288,78,249)(79,209,118,170)(80,208,119,169)(81,207,120,168)(82,206,121,167)(83,205,122,166)(84,204,123,165)(85,203,124,164)(86,202,125,163)(87,201,126,162)(88,200,127,161)(89,199,128,160)(90,198,129,159)(91,197,130,158)(92,196,131,157)(93,195,132,234)(94,194,133,233)(95,193,134,232)(96,192,135,231)(97,191,136,230)(98,190,137,229)(99,189,138,228)(100,188,139,227)(101,187,140,226)(102,186,141,225)(103,185,142,224)(104,184,143,223)(105,183,144,222)(106,182,145,221)(107,181,146,220)(108,180,147,219)(109,179,148,218)(110,178,149,217)(111,177,150,216)(112,176,151,215)(113,175,152,214)(114,174,153,213)(115,173,154,212)(116,172,155,211)(117,171,156,210) );
G=PermutationGroup([[(1,144),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,151),(9,152),(10,153),(11,154),(12,155),(13,156),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,113),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,136),(72,137),(73,138),(74,139),(75,140),(76,141),(77,142),(78,143),(157,261),(158,262),(159,263),(160,264),(161,265),(162,266),(163,267),(164,268),(165,269),(166,270),(167,271),(168,272),(169,273),(170,274),(171,275),(172,276),(173,277),(174,278),(175,279),(176,280),(177,281),(178,282),(179,283),(180,284),(181,285),(182,286),(183,287),(184,288),(185,289),(186,290),(187,291),(188,292),(189,293),(190,294),(191,295),(192,296),(193,297),(194,298),(195,299),(196,300),(197,301),(198,302),(199,303),(200,304),(201,305),(202,306),(203,307),(204,308),(205,309),(206,310),(207,311),(208,312),(209,235),(210,236),(211,237),(212,238),(213,239),(214,240),(215,241),(216,242),(217,243),(218,244),(219,245),(220,246),(221,247),(222,248),(223,249),(224,250),(225,251),(226,252),(227,253),(228,254),(229,255),(230,256),(231,257),(232,258),(233,259),(234,260)], [(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),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312)], [(1,248,40,287),(2,247,41,286),(3,246,42,285),(4,245,43,284),(5,244,44,283),(6,243,45,282),(7,242,46,281),(8,241,47,280),(9,240,48,279),(10,239,49,278),(11,238,50,277),(12,237,51,276),(13,236,52,275),(14,235,53,274),(15,312,54,273),(16,311,55,272),(17,310,56,271),(18,309,57,270),(19,308,58,269),(20,307,59,268),(21,306,60,267),(22,305,61,266),(23,304,62,265),(24,303,63,264),(25,302,64,263),(26,301,65,262),(27,300,66,261),(28,299,67,260),(29,298,68,259),(30,297,69,258),(31,296,70,257),(32,295,71,256),(33,294,72,255),(34,293,73,254),(35,292,74,253),(36,291,75,252),(37,290,76,251),(38,289,77,250),(39,288,78,249),(79,209,118,170),(80,208,119,169),(81,207,120,168),(82,206,121,167),(83,205,122,166),(84,204,123,165),(85,203,124,164),(86,202,125,163),(87,201,126,162),(88,200,127,161),(89,199,128,160),(90,198,129,159),(91,197,130,158),(92,196,131,157),(93,195,132,234),(94,194,133,233),(95,193,134,232),(96,192,135,231),(97,191,136,230),(98,190,137,229),(99,189,138,228),(100,188,139,227),(101,187,140,226),(102,186,141,225),(103,185,142,224),(104,184,143,223),(105,183,144,222),(106,182,145,221),(107,181,146,220),(108,180,147,219),(109,179,148,218),(110,178,149,217),(111,177,150,216),(112,176,151,215),(113,175,152,214),(114,174,153,213),(115,173,154,212),(116,172,155,211),(117,171,156,210)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 13A | ··· | 13F | 26A | ··· | 26R | 39A | ··· | 39L | 78A | ··· | 78AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 2 | 39 | 39 | 39 | 39 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 | D13 | Dic13 | D26 | D39 | Dic39 | D78 |
| kernel | C2×Dic39 | Dic39 | C2×C78 | C78 | C2×C26 | C26 | C26 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 6 | 12 | 6 | 12 | 24 | 12 |
Matrix representation of C2×Dic39 ►in GL3(𝔽157) generated by
| 156 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 156 | 0 | 0 |
| 0 | 95 | 85 |
| 0 | 72 | 76 |
| 28 | 0 | 0 |
| 0 | 84 | 146 |
| 0 | 142 | 73 |
G:=sub<GL(3,GF(157))| [156,0,0,0,1,0,0,0,1],[156,0,0,0,95,72,0,85,76],[28,0,0,0,84,142,0,146,73] >;
C2×Dic39 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{39} % in TeX
G:=Group("C2xDic39"); // GroupNames label
G:=SmallGroup(312,40);
// by ID
G=gap.SmallGroup(312,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,20,323,7204]);
// Polycyclic
G:=Group<a,b,c|a^2=b^78=1,c^2=b^39,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export