direct product, abelian, monomial, 2-elementary
Aliases: C22×C82, SmallGroup(328,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C82 |
| C1 — C22×C82 |
| C1 — C22×C82 |
Generators and relations for C22×C82
G = < a,b,c | a2=b2=c82=1, ab=ba, ac=ca, bc=cb >
Smallest permutation representation of C22×C82
►Regular action on 328 points
Generators in S328
(1 229)(2 230)(3 231)(4 232)(5 233)(6 234)(7 235)(8 236)(9 237)(10 238)(11 239)(12 240)(13 241)(14 242)(15 243)(16 244)(17 245)(18 246)(19 165)(20 166)(21 167)(22 168)(23 169)(24 170)(25 171)(26 172)(27 173)(28 174)(29 175)(30 176)(31 177)(32 178)(33 179)(34 180)(35 181)(36 182)(37 183)(38 184)(39 185)(40 186)(41 187)(42 188)(43 189)(44 190)(45 191)(46 192)(47 193)(48 194)(49 195)(50 196)(51 197)(52 198)(53 199)(54 200)(55 201)(56 202)(57 203)(58 204)(59 205)(60 206)(61 207)(62 208)(63 209)(64 210)(65 211)(66 212)(67 213)(68 214)(69 215)(70 216)(71 217)(72 218)(73 219)(74 220)(75 221)(76 222)(77 223)(78 224)(79 225)(80 226)(81 227)(82 228)(83 327)(84 328)(85 247)(86 248)(87 249)(88 250)(89 251)(90 252)(91 253)(92 254)(93 255)(94 256)(95 257)(96 258)(97 259)(98 260)(99 261)(100 262)(101 263)(102 264)(103 265)(104 266)(105 267)(106 268)(107 269)(108 270)(109 271)(110 272)(111 273)(112 274)(113 275)(114 276)(115 277)(116 278)(117 279)(118 280)(119 281)(120 282)(121 283)(122 284)(123 285)(124 286)(125 287)(126 288)(127 289)(128 290)(129 291)(130 292)(131 293)(132 294)(133 295)(134 296)(135 297)(136 298)(137 299)(138 300)(139 301)(140 302)(141 303)(142 304)(143 305)(144 306)(145 307)(146 308)(147 309)(148 310)(149 311)(150 312)(151 313)(152 314)(153 315)(154 316)(155 317)(156 318)(157 319)(158 320)(159 321)(160 322)(161 323)(162 324)(163 325)(164 326)
(1 154)(2 155)(3 156)(4 157)(5 158)(6 159)(7 160)(8 161)(9 162)(10 163)(11 164)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 121)(51 122)(52 123)(53 124)(54 125)(55 126)(56 127)(57 128)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(165 252)(166 253)(167 254)(168 255)(169 256)(170 257)(171 258)(172 259)(173 260)(174 261)(175 262)(176 263)(177 264)(178 265)(179 266)(180 267)(181 268)(182 269)(183 270)(184 271)(185 272)(186 273)(187 274)(188 275)(189 276)(190 277)(191 278)(192 279)(193 280)(194 281)(195 282)(196 283)(197 284)(198 285)(199 286)(200 287)(201 288)(202 289)(203 290)(204 291)(205 292)(206 293)(207 294)(208 295)(209 296)(210 297)(211 298)(212 299)(213 300)(214 301)(215 302)(216 303)(217 304)(218 305)(219 306)(220 307)(221 308)(222 309)(223 310)(224 311)(225 312)(226 313)(227 314)(228 315)(229 316)(230 317)(231 318)(232 319)(233 320)(234 321)(235 322)(236 323)(237 324)(238 325)(239 326)(240 327)(241 328)(242 247)(243 248)(244 249)(245 250)(246 251)
(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)
G:=sub<Sym(328)| (1,229)(2,230)(3,231)(4,232)(5,233)(6,234)(7,235)(8,236)(9,237)(10,238)(11,239)(12,240)(13,241)(14,242)(15,243)(16,244)(17,245)(18,246)(19,165)(20,166)(21,167)(22,168)(23,169)(24,170)(25,171)(26,172)(27,173)(28,174)(29,175)(30,176)(31,177)(32,178)(33,179)(34,180)(35,181)(36,182)(37,183)(38,184)(39,185)(40,186)(41,187)(42,188)(43,189)(44,190)(45,191)(46,192)(47,193)(48,194)(49,195)(50,196)(51,197)(52,198)(53,199)(54,200)(55,201)(56,202)(57,203)(58,204)(59,205)(60,206)(61,207)(62,208)(63,209)(64,210)(65,211)(66,212)(67,213)(68,214)(69,215)(70,216)(71,217)(72,218)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,327)(84,328)(85,247)(86,248)(87,249)(88,250)(89,251)(90,252)(91,253)(92,254)(93,255)(94,256)(95,257)(96,258)(97,259)(98,260)(99,261)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324)(163,325)(164,326), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(165,252)(166,253)(167,254)(168,255)(169,256)(170,257)(171,258)(172,259)(173,260)(174,261)(175,262)(176,263)(177,264)(178,265)(179,266)(180,267)(181,268)(182,269)(183,270)(184,271)(185,272)(186,273)(187,274)(188,275)(189,276)(190,277)(191,278)(192,279)(193,280)(194,281)(195,282)(196,283)(197,284)(198,285)(199,286)(200,287)(201,288)(202,289)(203,290)(204,291)(205,292)(206,293)(207,294)(208,295)(209,296)(210,297)(211,298)(212,299)(213,300)(214,301)(215,302)(216,303)(217,304)(218,305)(219,306)(220,307)(221,308)(222,309)(223,310)(224,311)(225,312)(226,313)(227,314)(228,315)(229,316)(230,317)(231,318)(232,319)(233,320)(234,321)(235,322)(236,323)(237,324)(238,325)(239,326)(240,327)(241,328)(242,247)(243,248)(244,249)(245,250)(246,251), (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)>;
G:=Group( (1,229)(2,230)(3,231)(4,232)(5,233)(6,234)(7,235)(8,236)(9,237)(10,238)(11,239)(12,240)(13,241)(14,242)(15,243)(16,244)(17,245)(18,246)(19,165)(20,166)(21,167)(22,168)(23,169)(24,170)(25,171)(26,172)(27,173)(28,174)(29,175)(30,176)(31,177)(32,178)(33,179)(34,180)(35,181)(36,182)(37,183)(38,184)(39,185)(40,186)(41,187)(42,188)(43,189)(44,190)(45,191)(46,192)(47,193)(48,194)(49,195)(50,196)(51,197)(52,198)(53,199)(54,200)(55,201)(56,202)(57,203)(58,204)(59,205)(60,206)(61,207)(62,208)(63,209)(64,210)(65,211)(66,212)(67,213)(68,214)(69,215)(70,216)(71,217)(72,218)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,327)(84,328)(85,247)(86,248)(87,249)(88,250)(89,251)(90,252)(91,253)(92,254)(93,255)(94,256)(95,257)(96,258)(97,259)(98,260)(99,261)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324)(163,325)(164,326), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,127)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(165,252)(166,253)(167,254)(168,255)(169,256)(170,257)(171,258)(172,259)(173,260)(174,261)(175,262)(176,263)(177,264)(178,265)(179,266)(180,267)(181,268)(182,269)(183,270)(184,271)(185,272)(186,273)(187,274)(188,275)(189,276)(190,277)(191,278)(192,279)(193,280)(194,281)(195,282)(196,283)(197,284)(198,285)(199,286)(200,287)(201,288)(202,289)(203,290)(204,291)(205,292)(206,293)(207,294)(208,295)(209,296)(210,297)(211,298)(212,299)(213,300)(214,301)(215,302)(216,303)(217,304)(218,305)(219,306)(220,307)(221,308)(222,309)(223,310)(224,311)(225,312)(226,313)(227,314)(228,315)(229,316)(230,317)(231,318)(232,319)(233,320)(234,321)(235,322)(236,323)(237,324)(238,325)(239,326)(240,327)(241,328)(242,247)(243,248)(244,249)(245,250)(246,251), (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) );
G=PermutationGroup([[(1,229),(2,230),(3,231),(4,232),(5,233),(6,234),(7,235),(8,236),(9,237),(10,238),(11,239),(12,240),(13,241),(14,242),(15,243),(16,244),(17,245),(18,246),(19,165),(20,166),(21,167),(22,168),(23,169),(24,170),(25,171),(26,172),(27,173),(28,174),(29,175),(30,176),(31,177),(32,178),(33,179),(34,180),(35,181),(36,182),(37,183),(38,184),(39,185),(40,186),(41,187),(42,188),(43,189),(44,190),(45,191),(46,192),(47,193),(48,194),(49,195),(50,196),(51,197),(52,198),(53,199),(54,200),(55,201),(56,202),(57,203),(58,204),(59,205),(60,206),(61,207),(62,208),(63,209),(64,210),(65,211),(66,212),(67,213),(68,214),(69,215),(70,216),(71,217),(72,218),(73,219),(74,220),(75,221),(76,222),(77,223),(78,224),(79,225),(80,226),(81,227),(82,228),(83,327),(84,328),(85,247),(86,248),(87,249),(88,250),(89,251),(90,252),(91,253),(92,254),(93,255),(94,256),(95,257),(96,258),(97,259),(98,260),(99,261),(100,262),(101,263),(102,264),(103,265),(104,266),(105,267),(106,268),(107,269),(108,270),(109,271),(110,272),(111,273),(112,274),(113,275),(114,276),(115,277),(116,278),(117,279),(118,280),(119,281),(120,282),(121,283),(122,284),(123,285),(124,286),(125,287),(126,288),(127,289),(128,290),(129,291),(130,292),(131,293),(132,294),(133,295),(134,296),(135,297),(136,298),(137,299),(138,300),(139,301),(140,302),(141,303),(142,304),(143,305),(144,306),(145,307),(146,308),(147,309),(148,310),(149,311),(150,312),(151,313),(152,314),(153,315),(154,316),(155,317),(156,318),(157,319),(158,320),(159,321),(160,322),(161,323),(162,324),(163,325),(164,326)], [(1,154),(2,155),(3,156),(4,157),(5,158),(6,159),(7,160),(8,161),(9,162),(10,163),(11,164),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,120),(50,121),(51,122),(52,123),(53,124),(54,125),(55,126),(56,127),(57,128),(58,129),(59,130),(60,131),(61,132),(62,133),(63,134),(64,135),(65,136),(66,137),(67,138),(68,139),(69,140),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(165,252),(166,253),(167,254),(168,255),(169,256),(170,257),(171,258),(172,259),(173,260),(174,261),(175,262),(176,263),(177,264),(178,265),(179,266),(180,267),(181,268),(182,269),(183,270),(184,271),(185,272),(186,273),(187,274),(188,275),(189,276),(190,277),(191,278),(192,279),(193,280),(194,281),(195,282),(196,283),(197,284),(198,285),(199,286),(200,287),(201,288),(202,289),(203,290),(204,291),(205,292),(206,293),(207,294),(208,295),(209,296),(210,297),(211,298),(212,299),(213,300),(214,301),(215,302),(216,303),(217,304),(218,305),(219,306),(220,307),(221,308),(222,309),(223,310),(224,311),(225,312),(226,313),(227,314),(228,315),(229,316),(230,317),(231,318),(232,319),(233,320),(234,321),(235,322),(236,323),(237,324),(238,325),(239,326),(240,327),(241,328),(242,247),(243,248),(244,249),(245,250),(246,251)], [(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)]])
328 conjugacy classes
| class | 1 | 2A | ··· | 2G | 41A | ··· | 41AN | 82A | ··· | 82JT |
| order | 1 | 2 | ··· | 2 | 41 | ··· | 41 | 82 | ··· | 82 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
328 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C41 | C82 |
| kernel | C22×C82 | C2×C82 | C23 | C22 |
| # reps | 1 | 7 | 40 | 280 |
Matrix representation of C22×C82 ►in GL3(𝔽83) generated by
| 1 | 0 | 0 |
| 0 | 82 | 0 |
| 0 | 0 | 82 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 82 |
| 39 | 0 | 0 |
| 0 | 39 | 0 |
| 0 | 0 | 54 |
G:=sub<GL(3,GF(83))| [1,0,0,0,82,0,0,0,82],[1,0,0,0,1,0,0,0,82],[39,0,0,0,39,0,0,0,54] >;
C22×C82 in GAP, Magma, Sage, TeX
C_2^2\times C_{82} % in TeX
G:=Group("C2^2xC82"); // GroupNames label
G:=SmallGroup(328,15);
// by ID
G=gap.SmallGroup(328,15);
# by ID
G:=PCGroup([4,-2,-2,-2,-41]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^82=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
Export