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G = C22×C82order 328 = 23·41

Abelian group of type [2,2,82]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C82, SmallGroup(328,15)

Series: Derived Chief Lower central Upper central

C1 — C22×C82
C1C41C82C2×C82 — 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
(1 224)(2 225)(3 226)(4 227)(5 228)(6 229)(7 230)(8 231)(9 232)(10 233)(11 234)(12 235)(13 236)(14 237)(15 238)(16 239)(17 240)(18 241)(19 242)(20 243)(21 244)(22 245)(23 246)(24 165)(25 166)(26 167)(27 168)(28 169)(29 170)(30 171)(31 172)(32 173)(33 174)(34 175)(35 176)(36 177)(37 178)(38 179)(39 180)(40 181)(41 182)(42 183)(43 184)(44 185)(45 186)(46 187)(47 188)(48 189)(49 190)(50 191)(51 192)(52 193)(53 194)(54 195)(55 196)(56 197)(57 198)(58 199)(59 200)(60 201)(61 202)(62 203)(63 204)(64 205)(65 206)(66 207)(67 208)(68 209)(69 210)(70 211)(71 212)(72 213)(73 214)(74 215)(75 216)(76 217)(77 218)(78 219)(79 220)(80 221)(81 222)(82 223)(83 310)(84 311)(85 312)(86 313)(87 314)(88 315)(89 316)(90 317)(91 318)(92 319)(93 320)(94 321)(95 322)(96 323)(97 324)(98 325)(99 326)(100 327)(101 328)(102 247)(103 248)(104 249)(105 250)(106 251)(107 252)(108 253)(109 254)(110 255)(111 256)(112 257)(113 258)(114 259)(115 260)(116 261)(117 262)(118 263)(119 264)(120 265)(121 266)(122 267)(123 268)(124 269)(125 270)(126 271)(127 272)(128 273)(129 274)(130 275)(131 276)(132 277)(133 278)(134 279)(135 280)(136 281)(137 282)(138 283)(139 284)(140 285)(141 286)(142 287)(143 288)(144 289)(145 290)(146 291)(147 292)(148 293)(149 294)(150 295)(151 296)(152 297)(153 298)(154 299)(155 300)(156 301)(157 302)(158 303)(159 304)(160 305)(161 306)(162 307)(163 308)(164 309)
(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)(165 322)(166 323)(167 324)(168 325)(169 326)(170 327)(171 328)(172 247)(173 248)(174 249)(175 250)(176 251)(177 252)(178 253)(179 254)(180 255)(181 256)(182 257)(183 258)(184 259)(185 260)(186 261)(187 262)(188 263)(189 264)(190 265)(191 266)(192 267)(193 268)(194 269)(195 270)(196 271)(197 272)(198 273)(199 274)(200 275)(201 276)(202 277)(203 278)(204 279)(205 280)(206 281)(207 282)(208 283)(209 284)(210 285)(211 286)(212 287)(213 288)(214 289)(215 290)(216 291)(217 292)(218 293)(219 294)(220 295)(221 296)(222 297)(223 298)(224 299)(225 300)(226 301)(227 302)(228 303)(229 304)(230 305)(231 306)(232 307)(233 308)(234 309)(235 310)(236 311)(237 312)(238 313)(239 314)(240 315)(241 316)(242 317)(243 318)(244 319)(245 320)(246 321)
(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)| (1,224)(2,225)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,241)(19,242)(20,243)(21,244)(22,245)(23,246)(24,165)(25,166)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,176)(36,177)(37,178)(38,179)(39,180)(40,181)(41,182)(42,183)(43,184)(44,185)(45,186)(46,187)(47,188)(48,189)(49,190)(50,191)(51,192)(52,193)(53,194)(54,195)(55,196)(56,197)(57,198)(58,199)(59,200)(60,201)(61,202)(62,203)(63,204)(64,205)(65,206)(66,207)(67,208)(68,209)(69,210)(70,211)(71,212)(72,213)(73,214)(74,215)(75,216)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,310)(84,311)(85,312)(86,313)(87,314)(88,315)(89,316)(90,317)(91,318)(92,319)(93,320)(94,321)(95,322)(96,323)(97,324)(98,325)(99,326)(100,327)(101,328)(102,247)(103,248)(104,249)(105,250)(106,251)(107,252)(108,253)(109,254)(110,255)(111,256)(112,257)(113,258)(114,259)(115,260)(116,261)(117,262)(118,263)(119,264)(120,265)(121,266)(122,267)(123,268)(124,269)(125,270)(126,271)(127,272)(128,273)(129,274)(130,275)(131,276)(132,277)(133,278)(134,279)(135,280)(136,281)(137,282)(138,283)(139,284)(140,285)(141,286)(142,287)(143,288)(144,289)(145,290)(146,291)(147,292)(148,293)(149,294)(150,295)(151,296)(152,297)(153,298)(154,299)(155,300)(156,301)(157,302)(158,303)(159,304)(160,305)(161,306)(162,307)(163,308)(164,309), (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)(165,322)(166,323)(167,324)(168,325)(169,326)(170,327)(171,328)(172,247)(173,248)(174,249)(175,250)(176,251)(177,252)(178,253)(179,254)(180,255)(181,256)(182,257)(183,258)(184,259)(185,260)(186,261)(187,262)(188,263)(189,264)(190,265)(191,266)(192,267)(193,268)(194,269)(195,270)(196,271)(197,272)(198,273)(199,274)(200,275)(201,276)(202,277)(203,278)(204,279)(205,280)(206,281)(207,282)(208,283)(209,284)(210,285)(211,286)(212,287)(213,288)(214,289)(215,290)(216,291)(217,292)(218,293)(219,294)(220,295)(221,296)(222,297)(223,298)(224,299)(225,300)(226,301)(227,302)(228,303)(229,304)(230,305)(231,306)(232,307)(233,308)(234,309)(235,310)(236,311)(237,312)(238,313)(239,314)(240,315)(241,316)(242,317)(243,318)(244,319)(245,320)(246,321), (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( (1,224)(2,225)(3,226)(4,227)(5,228)(6,229)(7,230)(8,231)(9,232)(10,233)(11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,241)(19,242)(20,243)(21,244)(22,245)(23,246)(24,165)(25,166)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,176)(36,177)(37,178)(38,179)(39,180)(40,181)(41,182)(42,183)(43,184)(44,185)(45,186)(46,187)(47,188)(48,189)(49,190)(50,191)(51,192)(52,193)(53,194)(54,195)(55,196)(56,197)(57,198)(58,199)(59,200)(60,201)(61,202)(62,203)(63,204)(64,205)(65,206)(66,207)(67,208)(68,209)(69,210)(70,211)(71,212)(72,213)(73,214)(74,215)(75,216)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,310)(84,311)(85,312)(86,313)(87,314)(88,315)(89,316)(90,317)(91,318)(92,319)(93,320)(94,321)(95,322)(96,323)(97,324)(98,325)(99,326)(100,327)(101,328)(102,247)(103,248)(104,249)(105,250)(106,251)(107,252)(108,253)(109,254)(110,255)(111,256)(112,257)(113,258)(114,259)(115,260)(116,261)(117,262)(118,263)(119,264)(120,265)(121,266)(122,267)(123,268)(124,269)(125,270)(126,271)(127,272)(128,273)(129,274)(130,275)(131,276)(132,277)(133,278)(134,279)(135,280)(136,281)(137,282)(138,283)(139,284)(140,285)(141,286)(142,287)(143,288)(144,289)(145,290)(146,291)(147,292)(148,293)(149,294)(150,295)(151,296)(152,297)(153,298)(154,299)(155,300)(156,301)(157,302)(158,303)(159,304)(160,305)(161,306)(162,307)(163,308)(164,309), (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)(165,322)(166,323)(167,324)(168,325)(169,326)(170,327)(171,328)(172,247)(173,248)(174,249)(175,250)(176,251)(177,252)(178,253)(179,254)(180,255)(181,256)(182,257)(183,258)(184,259)(185,260)(186,261)(187,262)(188,263)(189,264)(190,265)(191,266)(192,267)(193,268)(194,269)(195,270)(196,271)(197,272)(198,273)(199,274)(200,275)(201,276)(202,277)(203,278)(204,279)(205,280)(206,281)(207,282)(208,283)(209,284)(210,285)(211,286)(212,287)(213,288)(214,289)(215,290)(216,291)(217,292)(218,293)(219,294)(220,295)(221,296)(222,297)(223,298)(224,299)(225,300)(226,301)(227,302)(228,303)(229,304)(230,305)(231,306)(232,307)(233,308)(234,309)(235,310)(236,311)(237,312)(238,313)(239,314)(240,315)(241,316)(242,317)(243,318)(244,319)(245,320)(246,321), (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([(1,224),(2,225),(3,226),(4,227),(5,228),(6,229),(7,230),(8,231),(9,232),(10,233),(11,234),(12,235),(13,236),(14,237),(15,238),(16,239),(17,240),(18,241),(19,242),(20,243),(21,244),(22,245),(23,246),(24,165),(25,166),(26,167),(27,168),(28,169),(29,170),(30,171),(31,172),(32,173),(33,174),(34,175),(35,176),(36,177),(37,178),(38,179),(39,180),(40,181),(41,182),(42,183),(43,184),(44,185),(45,186),(46,187),(47,188),(48,189),(49,190),(50,191),(51,192),(52,193),(53,194),(54,195),(55,196),(56,197),(57,198),(58,199),(59,200),(60,201),(61,202),(62,203),(63,204),(64,205),(65,206),(66,207),(67,208),(68,209),(69,210),(70,211),(71,212),(72,213),(73,214),(74,215),(75,216),(76,217),(77,218),(78,219),(79,220),(80,221),(81,222),(82,223),(83,310),(84,311),(85,312),(86,313),(87,314),(88,315),(89,316),(90,317),(91,318),(92,319),(93,320),(94,321),(95,322),(96,323),(97,324),(98,325),(99,326),(100,327),(101,328),(102,247),(103,248),(104,249),(105,250),(106,251),(107,252),(108,253),(109,254),(110,255),(111,256),(112,257),(113,258),(114,259),(115,260),(116,261),(117,262),(118,263),(119,264),(120,265),(121,266),(122,267),(123,268),(124,269),(125,270),(126,271),(127,272),(128,273),(129,274),(130,275),(131,276),(132,277),(133,278),(134,279),(135,280),(136,281),(137,282),(138,283),(139,284),(140,285),(141,286),(142,287),(143,288),(144,289),(145,290),(146,291),(147,292),(148,293),(149,294),(150,295),(151,296),(152,297),(153,298),(154,299),(155,300),(156,301),(157,302),(158,303),(159,304),(160,305),(161,306),(162,307),(163,308),(164,309)], [(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),(165,322),(166,323),(167,324),(168,325),(169,326),(170,327),(171,328),(172,247),(173,248),(174,249),(175,250),(176,251),(177,252),(178,253),(179,254),(180,255),(181,256),(182,257),(183,258),(184,259),(185,260),(186,261),(187,262),(188,263),(189,264),(190,265),(191,266),(192,267),(193,268),(194,269),(195,270),(196,271),(197,272),(198,273),(199,274),(200,275),(201,276),(202,277),(203,278),(204,279),(205,280),(206,281),(207,282),(208,283),(209,284),(210,285),(211,286),(212,287),(213,288),(214,289),(215,290),(216,291),(217,292),(218,293),(219,294),(220,295),(221,296),(222,297),(223,298),(224,299),(225,300),(226,301),(227,302),(228,303),(229,304),(230,305),(231,306),(232,307),(233,308),(234,309),(235,310),(236,311),(237,312),(238,313),(239,314),(240,315),(241,316),(242,317),(243,318),(244,319),(245,320),(246,321)], [(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···2G41A···41AN82A···82JT
order12···241···4182···82
size11···11···11···1

328 irreducible representations

dim1111
type++
imageC1C2C41C82
kernelC22×C82C2×C82C23C22
# reps1740280

Matrix representation of C22×C82 in GL3(𝔽83) generated by

100
0820
0082
,
100
010
0082
,
3900
0390
0054
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

Subgroup lattice of C22×C82 in TeX

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