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G = C8.8Dic10order 320 = 26·5

5th non-split extension by C8 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.8Q8, C8.8Dic10, C4.22(Q8×D5), C52C8.3Q8, C4⋊C4.35D10, C4.Q8.7D5, C20.58(C2×Q8), C52(C8.5Q8), (C2×C8).257D10, C2.10(C20⋊Q8), C10.15(C4⋊Q8), (C8×Dic5).7C2, C406C4.14C2, C10.53(C4○D8), C4.22(C2×Dic10), C10.D8.6C2, C22.213(D4×D5), C4.Dic10.7C2, (C2×C40).158C22, (C2×C20).274C23, (C2×Dic5).142D4, C4⋊Dic5.106C22, C2.21(SD163D5), (C4×Dic5).261C22, (C5×C4.Q8).5C2, (C2×C10).279(C2×D4), (C5×C4⋊C4).67C22, (C2×C4).377(C22×D5), (C2×C52C8).233C22, SmallGroup(320,485)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8.8Dic10
C1C5C10C2×C10C2×C20C4×Dic5C8×Dic5 — C8.8Dic10
C5C10C2×C20 — C8.8Dic10
C1C22C2×C4C4.Q8

Generators and relations for C8.8Dic10
 G = < a,b,c | a8=b20=1, c2=b10, bab-1=a3, ac=ca, cbc-1=a4b-1 >

Subgroups: 286 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×6], C22, C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×6], C10, C10 [×2], C42, C4⋊C4 [×2], C4⋊C4 [×6], C2×C8, C2×C8, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C4×C8, C4.Q8, C4.Q8, C2.D8 [×2], C42.C2 [×2], C52C8 [×2], C40 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C8.5Q8, C2×C52C8, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C2×C40, C10.D8 [×2], C8×Dic5, C406C4, C5×C4.Q8, C4.Dic10 [×2], C8.8Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×4], C23, D5, C2×D4, C2×Q8 [×2], D10 [×3], C4⋊Q8, C4○D8 [×2], Dic10 [×2], C22×D5, C8.5Q8, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8, SD163D5 [×2], C8.8Dic10

Smallest permutation representation of C8.8Dic10
Regular action on 320 points
Generators in S320
(1 318 244 279 75 95 112 289)(2 280 113 319 76 290 245 96)(3 320 246 261 77 97 114 291)(4 262 115 301 78 292 247 98)(5 302 248 263 79 99 116 293)(6 264 117 303 80 294 249 100)(7 304 250 265 61 81 118 295)(8 266 119 305 62 296 251 82)(9 306 252 267 63 83 120 297)(10 268 101 307 64 298 253 84)(11 308 254 269 65 85 102 299)(12 270 103 309 66 300 255 86)(13 310 256 271 67 87 104 281)(14 272 105 311 68 282 257 88)(15 312 258 273 69 89 106 283)(16 274 107 313 70 284 259 90)(17 314 260 275 71 91 108 285)(18 276 109 315 72 286 241 92)(19 316 242 277 73 93 110 287)(20 278 111 317 74 288 243 94)(21 130 187 172 219 146 234 57)(22 173 235 131 220 58 188 147)(23 132 189 174 201 148 236 59)(24 175 237 133 202 60 190 149)(25 134 191 176 203 150 238 41)(26 177 239 135 204 42 192 151)(27 136 193 178 205 152 240 43)(28 179 221 137 206 44 194 153)(29 138 195 180 207 154 222 45)(30 161 223 139 208 46 196 155)(31 140 197 162 209 156 224 47)(32 163 225 121 210 48 198 157)(33 122 199 164 211 158 226 49)(34 165 227 123 212 50 200 159)(35 124 181 166 213 160 228 51)(36 167 229 125 214 52 182 141)(37 126 183 168 215 142 230 53)(38 169 231 127 216 54 184 143)(39 128 185 170 217 144 232 55)(40 171 233 129 218 56 186 145)
(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 41 11 51)(2 175 12 165)(3 59 13 49)(4 173 14 163)(5 57 15 47)(6 171 16 161)(7 55 17 45)(8 169 18 179)(9 53 19 43)(10 167 20 177)(21 312 31 302)(22 88 32 98)(23 310 33 320)(24 86 34 96)(25 308 35 318)(26 84 36 94)(27 306 37 316)(28 82 38 92)(29 304 39 314)(30 100 40 90)(42 64 52 74)(44 62 54 72)(46 80 56 70)(48 78 58 68)(50 76 60 66)(61 170 71 180)(63 168 73 178)(65 166 75 176)(67 164 77 174)(69 162 79 172)(81 217 91 207)(83 215 93 205)(85 213 95 203)(87 211 97 201)(89 209 99 219)(101 125 111 135)(102 160 112 150)(103 123 113 133)(104 158 114 148)(105 121 115 131)(106 156 116 146)(107 139 117 129)(108 154 118 144)(109 137 119 127)(110 152 120 142)(122 246 132 256)(124 244 134 254)(126 242 136 252)(128 260 138 250)(130 258 140 248)(141 243 151 253)(143 241 153 251)(145 259 155 249)(147 257 157 247)(149 255 159 245)(181 279 191 269)(182 288 192 298)(183 277 193 267)(184 286 194 296)(185 275 195 265)(186 284 196 294)(187 273 197 263)(188 282 198 292)(189 271 199 261)(190 300 200 290)(202 309 212 319)(204 307 214 317)(206 305 216 315)(208 303 218 313)(210 301 220 311)(221 266 231 276)(222 295 232 285)(223 264 233 274)(224 293 234 283)(225 262 235 272)(226 291 236 281)(227 280 237 270)(228 289 238 299)(229 278 239 268)(230 287 240 297)

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

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

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

50 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444558888888810···102020202020···2040···40
size11112288101010104040222222101010102···244448···84···4

50 irreducible representations

dim11111122222222444
type++++++--++++--+
imageC1C2C2C2C2C2Q8Q8D4D5D10D10C4○D8Dic10Q8×D5D4×D5SD163D5
kernelC8.8Dic10C10.D8C8×Dic5C406C4C5×C4.Q8C4.Dic10C52C8C40C2×Dic5C4.Q8C4⋊C4C2×C8C10C8C4C22C2
# reps12111222224288228

Matrix representation of C8.8Dic10 in GL4(𝔽41) generated by

152600
151500
00400
00040
,
141600
162700
001639
00228
,
0900
32000
00186
002123
G:=sub<GL(4,GF(41))| [15,15,0,0,26,15,0,0,0,0,40,0,0,0,0,40],[14,16,0,0,16,27,0,0,0,0,16,2,0,0,39,28],[0,32,0,0,9,0,0,0,0,0,18,21,0,0,6,23] >;

C8.8Dic10 in GAP, Magma, Sage, TeX

C_8._8{\rm Dic}_{10}
% in TeX

G:=Group("C8.8Dic10");
// GroupNames label

G:=SmallGroup(320,485);
// by ID

G=gap.SmallGroup(320,485);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,926,555,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=b^10,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^4*b^-1>;
// generators/relations

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