direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic41, C82⋊2C4, C2.2D82, C22.D41, C82.4C22, C41⋊3(C2×C4), (C2×C82).C2, SmallGroup(328,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C2×Dic41 |
Generators and relations for C2×Dic41
G = < a,b,c | a2=b82=1, c2=b41, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic41
►Regular action on 328 points
Generators in S328
(1 143)(2 144)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 161)(20 162)(21 163)(22 164)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 142)(165 288)(166 289)(167 290)(168 291)(169 292)(170 293)(171 294)(172 295)(173 296)(174 297)(175 298)(176 299)(177 300)(178 301)(179 302)(180 303)(181 304)(182 305)(183 306)(184 307)(185 308)(186 309)(187 310)(188 311)(189 312)(190 313)(191 314)(192 315)(193 316)(194 317)(195 318)(196 319)(197 320)(198 321)(199 322)(200 323)(201 324)(202 325)(203 326)(204 327)(205 328)(206 247)(207 248)(208 249)(209 250)(210 251)(211 252)(212 253)(213 254)(214 255)(215 256)(216 257)(217 258)(218 259)(219 260)(220 261)(221 262)(222 263)(223 264)(224 265)(225 266)(226 267)(227 268)(228 269)(229 270)(230 271)(231 272)(232 273)(233 274)(234 275)(235 276)(236 277)(237 278)(238 279)(239 280)(240 281)(241 282)(242 283)(243 284)(244 285)(245 286)(246 287)
(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)(247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328)
(1 165 42 206)(2 246 43 205)(3 245 44 204)(4 244 45 203)(5 243 46 202)(6 242 47 201)(7 241 48 200)(8 240 49 199)(9 239 50 198)(10 238 51 197)(11 237 52 196)(12 236 53 195)(13 235 54 194)(14 234 55 193)(15 233 56 192)(16 232 57 191)(17 231 58 190)(18 230 59 189)(19 229 60 188)(20 228 61 187)(21 227 62 186)(22 226 63 185)(23 225 64 184)(24 224 65 183)(25 223 66 182)(26 222 67 181)(27 221 68 180)(28 220 69 179)(29 219 70 178)(30 218 71 177)(31 217 72 176)(32 216 73 175)(33 215 74 174)(34 214 75 173)(35 213 76 172)(36 212 77 171)(37 211 78 170)(38 210 79 169)(39 209 80 168)(40 208 81 167)(41 207 82 166)(83 266 124 307)(84 265 125 306)(85 264 126 305)(86 263 127 304)(87 262 128 303)(88 261 129 302)(89 260 130 301)(90 259 131 300)(91 258 132 299)(92 257 133 298)(93 256 134 297)(94 255 135 296)(95 254 136 295)(96 253 137 294)(97 252 138 293)(98 251 139 292)(99 250 140 291)(100 249 141 290)(101 248 142 289)(102 247 143 288)(103 328 144 287)(104 327 145 286)(105 326 146 285)(106 325 147 284)(107 324 148 283)(108 323 149 282)(109 322 150 281)(110 321 151 280)(111 320 152 279)(112 319 153 278)(113 318 154 277)(114 317 155 276)(115 316 156 275)(116 315 157 274)(117 314 158 273)(118 313 159 272)(119 312 160 271)(120 311 161 270)(121 310 162 269)(122 309 163 268)(123 308 164 267)
G:=sub<Sym(328)| (1,143)(2,144)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,161)(20,162)(21,163)(22,164)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(165,288)(166,289)(167,290)(168,291)(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)(176,299)(177,300)(178,301)(179,302)(180,303)(181,304)(182,305)(183,306)(184,307)(185,308)(186,309)(187,310)(188,311)(189,312)(190,313)(191,314)(192,315)(193,316)(194,317)(195,318)(196,319)(197,320)(198,321)(199,322)(200,323)(201,324)(202,325)(203,326)(204,327)(205,328)(206,247)(207,248)(208,249)(209,250)(210,251)(211,252)(212,253)(213,254)(214,255)(215,256)(216,257)(217,258)(218,259)(219,260)(220,261)(221,262)(222,263)(223,264)(224,265)(225,266)(226,267)(227,268)(228,269)(229,270)(230,271)(231,272)(232,273)(233,274)(234,275)(235,276)(236,277)(237,278)(238,279)(239,280)(240,281)(241,282)(242,283)(243,284)(244,285)(245,286)(246,287), (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)(247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,165,42,206)(2,246,43,205)(3,245,44,204)(4,244,45,203)(5,243,46,202)(6,242,47,201)(7,241,48,200)(8,240,49,199)(9,239,50,198)(10,238,51,197)(11,237,52,196)(12,236,53,195)(13,235,54,194)(14,234,55,193)(15,233,56,192)(16,232,57,191)(17,231,58,190)(18,230,59,189)(19,229,60,188)(20,228,61,187)(21,227,62,186)(22,226,63,185)(23,225,64,184)(24,224,65,183)(25,223,66,182)(26,222,67,181)(27,221,68,180)(28,220,69,179)(29,219,70,178)(30,218,71,177)(31,217,72,176)(32,216,73,175)(33,215,74,174)(34,214,75,173)(35,213,76,172)(36,212,77,171)(37,211,78,170)(38,210,79,169)(39,209,80,168)(40,208,81,167)(41,207,82,166)(83,266,124,307)(84,265,125,306)(85,264,126,305)(86,263,127,304)(87,262,128,303)(88,261,129,302)(89,260,130,301)(90,259,131,300)(91,258,132,299)(92,257,133,298)(93,256,134,297)(94,255,135,296)(95,254,136,295)(96,253,137,294)(97,252,138,293)(98,251,139,292)(99,250,140,291)(100,249,141,290)(101,248,142,289)(102,247,143,288)(103,328,144,287)(104,327,145,286)(105,326,146,285)(106,325,147,284)(107,324,148,283)(108,323,149,282)(109,322,150,281)(110,321,151,280)(111,320,152,279)(112,319,153,278)(113,318,154,277)(114,317,155,276)(115,316,156,275)(116,315,157,274)(117,314,158,273)(118,313,159,272)(119,312,160,271)(120,311,161,270)(121,310,162,269)(122,309,163,268)(123,308,164,267)>;
G:=Group( (1,143)(2,144)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,161)(20,162)(21,163)(22,164)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(165,288)(166,289)(167,290)(168,291)(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)(176,299)(177,300)(178,301)(179,302)(180,303)(181,304)(182,305)(183,306)(184,307)(185,308)(186,309)(187,310)(188,311)(189,312)(190,313)(191,314)(192,315)(193,316)(194,317)(195,318)(196,319)(197,320)(198,321)(199,322)(200,323)(201,324)(202,325)(203,326)(204,327)(205,328)(206,247)(207,248)(208,249)(209,250)(210,251)(211,252)(212,253)(213,254)(214,255)(215,256)(216,257)(217,258)(218,259)(219,260)(220,261)(221,262)(222,263)(223,264)(224,265)(225,266)(226,267)(227,268)(228,269)(229,270)(230,271)(231,272)(232,273)(233,274)(234,275)(235,276)(236,277)(237,278)(238,279)(239,280)(240,281)(241,282)(242,283)(243,284)(244,285)(245,286)(246,287), (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)(247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,165,42,206)(2,246,43,205)(3,245,44,204)(4,244,45,203)(5,243,46,202)(6,242,47,201)(7,241,48,200)(8,240,49,199)(9,239,50,198)(10,238,51,197)(11,237,52,196)(12,236,53,195)(13,235,54,194)(14,234,55,193)(15,233,56,192)(16,232,57,191)(17,231,58,190)(18,230,59,189)(19,229,60,188)(20,228,61,187)(21,227,62,186)(22,226,63,185)(23,225,64,184)(24,224,65,183)(25,223,66,182)(26,222,67,181)(27,221,68,180)(28,220,69,179)(29,219,70,178)(30,218,71,177)(31,217,72,176)(32,216,73,175)(33,215,74,174)(34,214,75,173)(35,213,76,172)(36,212,77,171)(37,211,78,170)(38,210,79,169)(39,209,80,168)(40,208,81,167)(41,207,82,166)(83,266,124,307)(84,265,125,306)(85,264,126,305)(86,263,127,304)(87,262,128,303)(88,261,129,302)(89,260,130,301)(90,259,131,300)(91,258,132,299)(92,257,133,298)(93,256,134,297)(94,255,135,296)(95,254,136,295)(96,253,137,294)(97,252,138,293)(98,251,139,292)(99,250,140,291)(100,249,141,290)(101,248,142,289)(102,247,143,288)(103,328,144,287)(104,327,145,286)(105,326,146,285)(106,325,147,284)(107,324,148,283)(108,323,149,282)(109,322,150,281)(110,321,151,280)(111,320,152,279)(112,319,153,278)(113,318,154,277)(114,317,155,276)(115,316,156,275)(116,315,157,274)(117,314,158,273)(118,313,159,272)(119,312,160,271)(120,311,161,270)(121,310,162,269)(122,309,163,268)(123,308,164,267) );
G=PermutationGroup([[(1,143),(2,144),(3,145),(4,146),(5,147),(6,148),(7,149),(8,150),(9,151),(10,152),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,159),(18,160),(19,161),(20,162),(21,163),(22,164),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,127),(68,128),(69,129),(70,130),(71,131),(72,132),(73,133),(74,134),(75,135),(76,136),(77,137),(78,138),(79,139),(80,140),(81,141),(82,142),(165,288),(166,289),(167,290),(168,291),(169,292),(170,293),(171,294),(172,295),(173,296),(174,297),(175,298),(176,299),(177,300),(178,301),(179,302),(180,303),(181,304),(182,305),(183,306),(184,307),(185,308),(186,309),(187,310),(188,311),(189,312),(190,313),(191,314),(192,315),(193,316),(194,317),(195,318),(196,319),(197,320),(198,321),(199,322),(200,323),(201,324),(202,325),(203,326),(204,327),(205,328),(206,247),(207,248),(208,249),(209,250),(210,251),(211,252),(212,253),(213,254),(214,255),(215,256),(216,257),(217,258),(218,259),(219,260),(220,261),(221,262),(222,263),(223,264),(224,265),(225,266),(226,267),(227,268),(228,269),(229,270),(230,271),(231,272),(232,273),(233,274),(234,275),(235,276),(236,277),(237,278),(238,279),(239,280),(240,281),(241,282),(242,283),(243,284),(244,285),(245,286),(246,287)], [(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),(247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328)], [(1,165,42,206),(2,246,43,205),(3,245,44,204),(4,244,45,203),(5,243,46,202),(6,242,47,201),(7,241,48,200),(8,240,49,199),(9,239,50,198),(10,238,51,197),(11,237,52,196),(12,236,53,195),(13,235,54,194),(14,234,55,193),(15,233,56,192),(16,232,57,191),(17,231,58,190),(18,230,59,189),(19,229,60,188),(20,228,61,187),(21,227,62,186),(22,226,63,185),(23,225,64,184),(24,224,65,183),(25,223,66,182),(26,222,67,181),(27,221,68,180),(28,220,69,179),(29,219,70,178),(30,218,71,177),(31,217,72,176),(32,216,73,175),(33,215,74,174),(34,214,75,173),(35,213,76,172),(36,212,77,171),(37,211,78,170),(38,210,79,169),(39,209,80,168),(40,208,81,167),(41,207,82,166),(83,266,124,307),(84,265,125,306),(85,264,126,305),(86,263,127,304),(87,262,128,303),(88,261,129,302),(89,260,130,301),(90,259,131,300),(91,258,132,299),(92,257,133,298),(93,256,134,297),(94,255,135,296),(95,254,136,295),(96,253,137,294),(97,252,138,293),(98,251,139,292),(99,250,140,291),(100,249,141,290),(101,248,142,289),(102,247,143,288),(103,328,144,287),(104,327,145,286),(105,326,146,285),(106,325,147,284),(107,324,148,283),(108,323,149,282),(109,322,150,281),(110,321,151,280),(111,320,152,279),(112,319,153,278),(113,318,154,277),(114,317,155,276),(115,316,156,275),(116,315,157,274),(117,314,158,273),(118,313,159,272),(119,312,160,271),(120,311,161,270),(121,310,162,269),(122,309,163,268),(123,308,164,267)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 41A | ··· | 41T | 82A | ··· | 82BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 41 | ··· | 41 | 82 | ··· | 82 |
| size | 1 | 1 | 1 | 1 | 41 | 41 | 41 | 41 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D41 | Dic41 | D82 |
| kernel | C2×Dic41 | Dic41 | C2×C82 | C82 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 20 | 40 | 20 |
Matrix representation of C2×Dic41 ►in GL3(𝔽821) generated by
| 1 | 0 | 0 |
| 0 | 820 | 0 |
| 0 | 0 | 820 |
| 820 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 820 | 116 |
| 295 | 0 | 0 |
| 0 | 157 | 632 |
| 0 | 782 | 664 |
G:=sub<GL(3,GF(821))| [1,0,0,0,820,0,0,0,820],[820,0,0,0,0,820,0,1,116],[295,0,0,0,157,782,0,632,664] >;
C2×Dic41 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{41} % in TeX
G:=Group("C2xDic41"); // GroupNames label
G:=SmallGroup(328,7);
// by ID
G=gap.SmallGroup(328,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-41,16,5123]);
// Polycyclic
G:=Group<a,b,c|a^2=b^82=1,c^2=b^41,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export