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G = C2×Dic40order 320 = 26·5

Direct product of C2 and Dic40

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C2×Dic40
 Chief series C1 — C5 — C10 — C20 — C40 — Dic20 — C2×Dic20 — C2×Dic40
 Lower central C5 — C10 — C20 — C40 — C2×Dic40
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2×Dic40
G = < a,b,c | a2=b80=1, c2=b40, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 382 in 82 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C10, C16, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C2×C16, Q32, C2×Q16, C40, Dic10, C2×Dic5, C2×C20, C2×Q32, C80, Dic20, Dic20, C2×C40, C2×Dic10, Dic40, C2×C80, C2×Dic20, C2×Dic40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, Q32, C2×D8, D20, C22×D5, C2×Q32, D40, C2×D20, Dic40, C2×D40, C2×Dic40

Smallest permutation representation of C2×Dic40
Regular action on 320 points
Generators in S320
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 81)(11 82)(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)(161 295)(162 296)(163 297)(164 298)(165 299)(166 300)(167 301)(168 302)(169 303)(170 304)(171 305)(172 306)(173 307)(174 308)(175 309)(176 310)(177 311)(178 312)(179 313)(180 314)(181 315)(182 316)(183 317)(184 318)(185 319)(186 320)(187 241)(188 242)(189 243)(190 244)(191 245)(192 246)(193 247)(194 248)(195 249)(196 250)(197 251)(198 252)(199 253)(200 254)(201 255)(202 256)(203 257)(204 258)(205 259)(206 260)(207 261)(208 262)(209 263)(210 264)(211 265)(212 266)(213 267)(214 268)(215 269)(216 270)(217 271)(218 272)(219 273)(220 274)(221 275)(222 276)(223 277)(224 278)(225 279)(226 280)(227 281)(228 282)(229 283)(230 284)(231 285)(232 286)(233 287)(234 288)(235 289)(236 290)(237 291)(238 292)(239 293)(240 294)
(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)
(1 310 41 270)(2 309 42 269)(3 308 43 268)(4 307 44 267)(5 306 45 266)(6 305 46 265)(7 304 47 264)(8 303 48 263)(9 302 49 262)(10 301 50 261)(11 300 51 260)(12 299 52 259)(13 298 53 258)(14 297 54 257)(15 296 55 256)(16 295 56 255)(17 294 57 254)(18 293 58 253)(19 292 59 252)(20 291 60 251)(21 290 61 250)(22 289 62 249)(23 288 63 248)(24 287 64 247)(25 286 65 246)(26 285 66 245)(27 284 67 244)(28 283 68 243)(29 282 69 242)(30 281 70 241)(31 280 71 320)(32 279 72 319)(33 278 73 318)(34 277 74 317)(35 276 75 316)(36 275 76 315)(37 274 77 314)(38 273 78 313)(39 272 79 312)(40 271 80 311)(81 167 121 207)(82 166 122 206)(83 165 123 205)(84 164 124 204)(85 163 125 203)(86 162 126 202)(87 161 127 201)(88 240 128 200)(89 239 129 199)(90 238 130 198)(91 237 131 197)(92 236 132 196)(93 235 133 195)(94 234 134 194)(95 233 135 193)(96 232 136 192)(97 231 137 191)(98 230 138 190)(99 229 139 189)(100 228 140 188)(101 227 141 187)(102 226 142 186)(103 225 143 185)(104 224 144 184)(105 223 145 183)(106 222 146 182)(107 221 147 181)(108 220 148 180)(109 219 149 179)(110 218 150 178)(111 217 151 177)(112 216 152 176)(113 215 153 175)(114 214 154 174)(115 213 155 173)(116 212 156 172)(117 211 157 171)(118 210 158 170)(119 209 159 169)(120 208 160 168)

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

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

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

86 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 16A ··· 16H 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 2 2 40 40 40 40 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

86 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + - + + + + - image C1 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 Q32 D20 D20 D40 D40 Dic40 kernel C2×Dic40 Dic40 C2×C80 C2×Dic20 C40 C2×C20 C2×C16 C20 C2×C10 C16 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 1 1 2 2 2 4 2 8 4 4 8 8 32

Matrix representation of C2×Dic40 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 27 85 0 0 156 27 0 0 0 0 98 138 0 0 152 187
,
 41 138 0 0 138 200 0 0 0 0 47 70 0 0 175 194
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[27,156,0,0,85,27,0,0,0,0,98,152,0,0,138,187],[41,138,0,0,138,200,0,0,0,0,47,175,0,0,70,194] >;

C2×Dic40 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{40}
% in TeX

G:=Group("C2xDic40");
// GroupNames label

G:=SmallGroup(320,532);
// by ID

G=gap.SmallGroup(320,532);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,142,675,192,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^80=1,c^2=b^40,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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