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G = C20.5Q16order 320 = 26·5

5th non-split extension by C20 of Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.5Q16, C20.9SD16, C42.10D10, C4⋊Q8.1D5, C4.8(Q8⋊D5), (C2×C20).107D4, (Q8×C10).12C4, (C2×Q8).2Dic5, C203C8.12C2, C4.6(C5⋊Q16), C53(C4.6Q16), (C4×C20).48C22, C2.5(C20.D4), C2.4(Q8⋊Dic5), C10.19(Q8⋊C4), C10.17(C4.D4), C22.42(C23.D5), (C5×C4⋊Q8).1C2, (C2×C20).345(C2×C4), (C2×C4).12(C2×Dic5), (C2×C4).177(C5⋊D4), (C2×C10).168(C22⋊C4), SmallGroup(320,104)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.5Q16
C1C5C10C2×C10C2×C20C4×C20C203C8 — C20.5Q16
C5C2×C10C2×C20 — C20.5Q16
C1C22C42C4⋊Q8

Generators and relations for C20.5Q16
 G = < a,b,c | a20=b8=1, c2=a10b4, bab-1=a-1, cac-1=a11, cbc-1=a15b-1 >

Subgroups: 174 in 64 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C4 [×4], C4 [×3], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], Q8 [×2], C10, C10 [×2], C42, C4⋊C4 [×2], C2×C8 [×2], C2×Q8 [×2], C20 [×4], C20 [×3], C2×C10, C4⋊C8 [×2], C4⋊Q8, C52C8 [×2], C2×C20, C2×C20 [×2], C2×C20 [×2], C5×Q8 [×2], C4.6Q16, C2×C52C8 [×2], C4×C20, C5×C4⋊C4 [×2], Q8×C10 [×2], C203C8 [×2], C5×C4⋊Q8, C20.5Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, SD16 [×2], Q16 [×2], Dic5 [×2], D10, C4.D4, Q8⋊C4 [×2], C2×Dic5, C5⋊D4 [×2], C4.6Q16, Q8⋊D5 [×2], C5⋊Q16 [×2], C23.D5, C20.D4, Q8⋊Dic5 [×2], C20.5Q16

Smallest permutation representation of C20.5Q16
Regular action on 320 points
Generators in S320
(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)
(1 233 46 297 178 318 272 114)(2 232 47 296 179 317 273 113)(3 231 48 295 180 316 274 112)(4 230 49 294 161 315 275 111)(5 229 50 293 162 314 276 110)(6 228 51 292 163 313 277 109)(7 227 52 291 164 312 278 108)(8 226 53 290 165 311 279 107)(9 225 54 289 166 310 280 106)(10 224 55 288 167 309 261 105)(11 223 56 287 168 308 262 104)(12 222 57 286 169 307 263 103)(13 221 58 285 170 306 264 102)(14 240 59 284 171 305 265 101)(15 239 60 283 172 304 266 120)(16 238 41 282 173 303 267 119)(17 237 42 281 174 302 268 118)(18 236 43 300 175 301 269 117)(19 235 44 299 176 320 270 116)(20 234 45 298 177 319 271 115)(21 140 153 66 83 245 186 201)(22 139 154 65 84 244 187 220)(23 138 155 64 85 243 188 219)(24 137 156 63 86 242 189 218)(25 136 157 62 87 241 190 217)(26 135 158 61 88 260 191 216)(27 134 159 80 89 259 192 215)(28 133 160 79 90 258 193 214)(29 132 141 78 91 257 194 213)(30 131 142 77 92 256 195 212)(31 130 143 76 93 255 196 211)(32 129 144 75 94 254 197 210)(33 128 145 74 95 253 198 209)(34 127 146 73 96 252 199 208)(35 126 147 72 97 251 200 207)(36 125 148 71 98 250 181 206)(37 124 149 70 99 249 182 205)(38 123 150 69 100 248 183 204)(39 122 151 68 81 247 184 203)(40 121 152 67 82 246 185 202)
(1 28 168 100)(2 39 169 91)(3 30 170 82)(4 21 171 93)(5 32 172 84)(6 23 173 95)(7 34 174 86)(8 25 175 97)(9 36 176 88)(10 27 177 99)(11 38 178 90)(12 29 179 81)(13 40 180 92)(14 31 161 83)(15 22 162 94)(16 33 163 85)(17 24 164 96)(18 35 165 87)(19 26 166 98)(20 37 167 89)(41 198 277 155)(42 189 278 146)(43 200 279 157)(44 191 280 148)(45 182 261 159)(46 193 262 150)(47 184 263 141)(48 195 264 152)(49 186 265 143)(50 197 266 154)(51 188 267 145)(52 199 268 156)(53 190 269 147)(54 181 270 158)(55 192 271 149)(56 183 272 160)(57 194 273 151)(58 185 274 142)(59 196 275 153)(60 187 276 144)(61 230 206 305)(62 221 207 316)(63 232 208 307)(64 223 209 318)(65 234 210 309)(66 225 211 320)(67 236 212 311)(68 227 213 302)(69 238 214 313)(70 229 215 304)(71 240 216 315)(72 231 217 306)(73 222 218 317)(74 233 219 308)(75 224 220 319)(76 235 201 310)(77 226 202 301)(78 237 203 312)(79 228 204 303)(80 239 205 314)(101 135 294 250)(102 126 295 241)(103 137 296 252)(104 128 297 243)(105 139 298 254)(106 130 299 245)(107 121 300 256)(108 132 281 247)(109 123 282 258)(110 134 283 249)(111 125 284 260)(112 136 285 251)(113 127 286 242)(114 138 287 253)(115 129 288 244)(116 140 289 255)(117 131 290 246)(118 122 291 257)(119 133 292 248)(120 124 293 259)

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G5A5B8A···8H10A···10F20A···20L20M···20T
order12224444444558···810···1020···2020···20
size111122224882220···202···24···48···8

47 irreducible representations

dim111122222224444
type+++++-+-++-
imageC1C2C2C4D4D5SD16Q16D10Dic5C5⋊D4C4.D4Q8⋊D5C5⋊Q16C20.D4
kernelC20.5Q16C203C8C5×C4⋊Q8Q8×C10C2×C20C4⋊Q8C20C20C42C2×Q8C2×C4C10C4C4C2
# reps121422442481444

Matrix representation of C20.5Q16 in GL6(𝔽41)

4000000
0400000
0004000
0013400
000001
0000400
,
30220000
2800000
00113200
00273000
0000639
00003935
,
31370000
15100000
0040000
0004000
00002734
00003414

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,34,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[30,28,0,0,0,0,22,0,0,0,0,0,0,0,11,27,0,0,0,0,32,30,0,0,0,0,0,0,6,39,0,0,0,0,39,35],[31,15,0,0,0,0,37,10,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,27,34,0,0,0,0,34,14] >;

C20.5Q16 in GAP, Magma, Sage, TeX

C_{20}._5Q_{16}
% in TeX

G:=Group("C20.5Q16");
// GroupNames label

G:=SmallGroup(320,104);
// by ID

G=gap.SmallGroup(320,104);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,100,1571,570,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^10*b^4,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b^-1>;
// generators/relations

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