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## G = C2×C20⋊3C8order 320 = 26·5

### Direct product of C2 and C20⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C20⋊3C8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×C20⋊3C8
 Lower central C5 — C10 — C2×C20⋊3C8
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C203C8
G = < a,b,c | a2=b20=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 270 in 138 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C20, C20, C20, C2×C10, C2×C10, C4⋊C8, C2×C42, C22×C8, C52C8, C2×C20, C2×C20, C2×C20, C22×C10, C2×C4⋊C8, C2×C52C8, C2×C52C8, C4×C20, C22×C20, C203C8, C22×C52C8, C2×C4×C20, C2×C203C8
Quotients:

Smallest permutation representation of C2×C203C8
Regular action on 320 points
Generators in S320
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 161)(13 162)(14 163)(15 164)(16 165)(17 166)(18 167)(19 168)(20 169)(21 284)(22 285)(23 286)(24 287)(25 288)(26 289)(27 290)(28 291)(29 292)(30 293)(31 294)(32 295)(33 296)(34 297)(35 298)(36 299)(37 300)(38 281)(39 282)(40 283)(41 226)(42 227)(43 228)(44 229)(45 230)(46 231)(47 232)(48 233)(49 234)(50 235)(51 236)(52 237)(53 238)(54 239)(55 240)(56 221)(57 222)(58 223)(59 224)(60 225)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)(121 199)(122 200)(123 181)(124 182)(125 183)(126 184)(127 185)(128 186)(129 187)(130 188)(131 189)(132 190)(133 191)(134 192)(135 193)(136 194)(137 195)(138 196)(139 197)(140 198)(201 274)(202 275)(203 276)(204 277)(205 278)(206 279)(207 280)(208 261)(209 262)(210 263)(211 264)(212 265)(213 266)(214 267)(215 268)(216 269)(217 270)(218 271)(219 272)(220 273)(241 306)(242 307)(243 308)(244 309)(245 310)(246 311)(247 312)(248 313)(249 314)(250 315)(251 316)(252 317)(253 318)(254 319)(255 320)(256 301)(257 302)(258 303)(259 304)(260 305)
(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)
(1 203 74 134 85 288 226 306)(2 202 75 133 86 287 227 305)(3 201 76 132 87 286 228 304)(4 220 77 131 88 285 229 303)(5 219 78 130 89 284 230 302)(6 218 79 129 90 283 231 301)(7 217 80 128 91 282 232 320)(8 216 61 127 92 281 233 319)(9 215 62 126 93 300 234 318)(10 214 63 125 94 299 235 317)(11 213 64 124 95 298 236 316)(12 212 65 123 96 297 237 315)(13 211 66 122 97 296 238 314)(14 210 67 121 98 295 239 313)(15 209 68 140 99 294 240 312)(16 208 69 139 100 293 221 311)(17 207 70 138 81 292 222 310)(18 206 71 137 82 291 223 309)(19 205 72 136 83 290 224 308)(20 204 73 135 84 289 225 307)(21 45 257 174 272 145 188 102)(22 44 258 173 273 144 189 101)(23 43 259 172 274 143 190 120)(24 42 260 171 275 142 191 119)(25 41 241 170 276 141 192 118)(26 60 242 169 277 160 193 117)(27 59 243 168 278 159 194 116)(28 58 244 167 279 158 195 115)(29 57 245 166 280 157 196 114)(30 56 246 165 261 156 197 113)(31 55 247 164 262 155 198 112)(32 54 248 163 263 154 199 111)(33 53 249 162 264 153 200 110)(34 52 250 161 265 152 181 109)(35 51 251 180 266 151 182 108)(36 50 252 179 267 150 183 107)(37 49 253 178 268 149 184 106)(38 48 254 177 269 148 185 105)(39 47 255 176 270 147 186 104)(40 46 256 175 271 146 187 103)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 5A 5B 8A ··· 8P 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 ··· 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + - + - + image C1 C2 C2 C2 C4 C4 C8 D4 Q8 D5 M4(2) Dic5 D10 Dic5 D10 C5⋊2C8 Dic10 D20 C4.Dic5 kernel C2×C20⋊3C8 C20⋊3C8 C22×C5⋊2C8 C2×C4×C20 C4×C20 C22×C20 C2×C20 C2×C20 C2×C20 C2×C42 C2×C10 C42 C42 C22×C4 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 4 2 1 4 4 16 2 2 2 4 4 4 4 2 16 8 8 16

Matrix representation of C2×C203C8 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 11 2 0 0 0 25 27
,
 40 0 0 0 0 0 36 5 0 0 0 5 5 0 0 0 0 0 32 9 0 0 0 0 9

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,11,25,0,0,0,2,27],[40,0,0,0,0,0,36,5,0,0,0,5,5,0,0,0,0,0,32,0,0,0,0,9,9] >;

C2×C203C8 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_3C_8
% in TeX

G:=Group("C2xC20:3C8");
// GroupNames label

G:=SmallGroup(320,550);
// by ID

G=gap.SmallGroup(320,550);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^20=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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