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G = Dic1512D4order 480 = 25·3·5

2nd semidirect product of Dic15 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1512D4, C23.23D30, (C2×C30)⋊3D4, (C6×D4)⋊15D5, (C2×D4)⋊5D15, (D4×C30)⋊27C2, (D4×C10)⋊15S3, C2.27(D4×D15), (C2×C4).18D30, C6.121(D4×D5), C1535(C4⋊D4), (C2×C20).251D6, C10.123(S3×D4), C30.384(C2×D4), D303C437C2, (C2×C12).250D10, C57(C23.14D6), C37(Dic5⋊D4), C222(C157D4), C30.4Q837C2, (C22×C10).81D6, (C22×C6).66D10, C30.226(C4○D4), C30.38D412C2, (C2×C30).310C23, (C2×C60).434C22, (C22×Dic15)⋊6C2, C6.105(D42D5), C2.18(D42D15), (C22×C30).22C22, C10.105(D42S3), C22.61(C22×D15), (C22×D15).12C22, (C2×Dic15).172C22, (C2×C157D4)⋊6C2, (C2×C6)⋊6(C5⋊D4), C6.110(C2×C5⋊D4), C2.15(C2×C157D4), (C2×C10)⋊10(C3⋊D4), C10.110(C2×C3⋊D4), (C2×C6).306(C22×D5), (C2×C10).305(C22×S3), SmallGroup(480,904)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic1512D4
C1C5C15C30C2×C30C22×D15C2×C157D4 — Dic1512D4
C15C2×C30 — Dic1512D4
C1C22C2×D4

Generators and relations for Dic1512D4
 G = < a,b,c,d | a30=c4=d2=1, b2=a15, bab-1=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd=c-1 >

Subgroups: 1076 in 188 conjugacy classes, 57 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], C5, S3, C6 [×3], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, D5, C10 [×3], C10 [×3], Dic3 [×4], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], Dic5 [×4], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3 [×5], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], D15, C30 [×3], C30 [×3], C4⋊D4, C2×Dic5 [×5], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, Dic15 [×2], Dic15 [×2], C60, D30 [×3], C2×C30, C2×C30 [×2], C2×C30 [×5], C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4 [×2], D4×C10, C23.14D6, C2×Dic15 [×3], C2×Dic15 [×2], C157D4 [×4], C2×C60, D4×C15 [×2], C22×D15, C22×C30 [×2], Dic5⋊D4, C30.4Q8, D303C4, C30.38D4, C22×Dic15, C2×C157D4 [×2], D4×C30, Dic1512D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C4⋊D4, C5⋊D4 [×2], C22×D5, S3×D4, D42S3, C2×C3⋊D4, D30 [×3], D4×D5, D42D5, C2×C5⋊D4, C23.14D6, C157D4 [×2], C22×D15, Dic5⋊D4, D4×D15, D42D15, C2×C157D4, Dic1512D4

Smallest permutation representation of Dic1512D4
On 240 points
Generators in S240
(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)
(1 186 16 201)(2 185 17 200)(3 184 18 199)(4 183 19 198)(5 182 20 197)(6 181 21 196)(7 210 22 195)(8 209 23 194)(9 208 24 193)(10 207 25 192)(11 206 26 191)(12 205 27 190)(13 204 28 189)(14 203 29 188)(15 202 30 187)(31 168 46 153)(32 167 47 152)(33 166 48 151)(34 165 49 180)(35 164 50 179)(36 163 51 178)(37 162 52 177)(38 161 53 176)(39 160 54 175)(40 159 55 174)(41 158 56 173)(42 157 57 172)(43 156 58 171)(44 155 59 170)(45 154 60 169)(61 235 76 220)(62 234 77 219)(63 233 78 218)(64 232 79 217)(65 231 80 216)(66 230 81 215)(67 229 82 214)(68 228 83 213)(69 227 84 212)(70 226 85 211)(71 225 86 240)(72 224 87 239)(73 223 88 238)(74 222 89 237)(75 221 90 236)(91 127 106 142)(92 126 107 141)(93 125 108 140)(94 124 109 139)(95 123 110 138)(96 122 111 137)(97 121 112 136)(98 150 113 135)(99 149 114 134)(100 148 115 133)(101 147 116 132)(102 146 117 131)(103 145 118 130)(104 144 119 129)(105 143 120 128)
(1 35 85 123)(2 36 86 124)(3 37 87 125)(4 38 88 126)(5 39 89 127)(6 40 90 128)(7 41 61 129)(8 42 62 130)(9 43 63 131)(10 44 64 132)(11 45 65 133)(12 46 66 134)(13 47 67 135)(14 48 68 136)(15 49 69 137)(16 50 70 138)(17 51 71 139)(18 52 72 140)(19 53 73 141)(20 54 74 142)(21 55 75 143)(22 56 76 144)(23 57 77 145)(24 58 78 146)(25 59 79 147)(26 60 80 148)(27 31 81 149)(28 32 82 150)(29 33 83 121)(30 34 84 122)(91 182 175 237)(92 183 176 238)(93 184 177 239)(94 185 178 240)(95 186 179 211)(96 187 180 212)(97 188 151 213)(98 189 152 214)(99 190 153 215)(100 191 154 216)(101 192 155 217)(102 193 156 218)(103 194 157 219)(104 195 158 220)(105 196 159 221)(106 197 160 222)(107 198 161 223)(108 199 162 224)(109 200 163 225)(110 201 164 226)(111 202 165 227)(112 203 166 228)(113 204 167 229)(114 205 168 230)(115 206 169 231)(116 207 170 232)(117 208 171 233)(118 209 172 234)(119 210 173 235)(120 181 174 236)
(31 149)(32 150)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(91 175)(92 176)(93 177)(94 178)(95 179)(96 180)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)

G:=sub<Sym(240)| (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), (1,186,16,201)(2,185,17,200)(3,184,18,199)(4,183,19,198)(5,182,20,197)(6,181,21,196)(7,210,22,195)(8,209,23,194)(9,208,24,193)(10,207,25,192)(11,206,26,191)(12,205,27,190)(13,204,28,189)(14,203,29,188)(15,202,30,187)(31,168,46,153)(32,167,47,152)(33,166,48,151)(34,165,49,180)(35,164,50,179)(36,163,51,178)(37,162,52,177)(38,161,53,176)(39,160,54,175)(40,159,55,174)(41,158,56,173)(42,157,57,172)(43,156,58,171)(44,155,59,170)(45,154,60,169)(61,235,76,220)(62,234,77,219)(63,233,78,218)(64,232,79,217)(65,231,80,216)(66,230,81,215)(67,229,82,214)(68,228,83,213)(69,227,84,212)(70,226,85,211)(71,225,86,240)(72,224,87,239)(73,223,88,238)(74,222,89,237)(75,221,90,236)(91,127,106,142)(92,126,107,141)(93,125,108,140)(94,124,109,139)(95,123,110,138)(96,122,111,137)(97,121,112,136)(98,150,113,135)(99,149,114,134)(100,148,115,133)(101,147,116,132)(102,146,117,131)(103,145,118,130)(104,144,119,129)(105,143,120,128), (1,35,85,123)(2,36,86,124)(3,37,87,125)(4,38,88,126)(5,39,89,127)(6,40,90,128)(7,41,61,129)(8,42,62,130)(9,43,63,131)(10,44,64,132)(11,45,65,133)(12,46,66,134)(13,47,67,135)(14,48,68,136)(15,49,69,137)(16,50,70,138)(17,51,71,139)(18,52,72,140)(19,53,73,141)(20,54,74,142)(21,55,75,143)(22,56,76,144)(23,57,77,145)(24,58,78,146)(25,59,79,147)(26,60,80,148)(27,31,81,149)(28,32,82,150)(29,33,83,121)(30,34,84,122)(91,182,175,237)(92,183,176,238)(93,184,177,239)(94,185,178,240)(95,186,179,211)(96,187,180,212)(97,188,151,213)(98,189,152,214)(99,190,153,215)(100,191,154,216)(101,192,155,217)(102,193,156,218)(103,194,157,219)(104,195,158,220)(105,196,159,221)(106,197,160,222)(107,198,161,223)(108,199,162,224)(109,200,163,225)(110,201,164,226)(111,202,165,227)(112,203,166,228)(113,204,167,229)(114,205,168,230)(115,206,169,231)(116,207,170,232)(117,208,171,233)(118,209,172,234)(119,210,173,235)(120,181,174,236), (31,149)(32,150)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)>;

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), (1,186,16,201)(2,185,17,200)(3,184,18,199)(4,183,19,198)(5,182,20,197)(6,181,21,196)(7,210,22,195)(8,209,23,194)(9,208,24,193)(10,207,25,192)(11,206,26,191)(12,205,27,190)(13,204,28,189)(14,203,29,188)(15,202,30,187)(31,168,46,153)(32,167,47,152)(33,166,48,151)(34,165,49,180)(35,164,50,179)(36,163,51,178)(37,162,52,177)(38,161,53,176)(39,160,54,175)(40,159,55,174)(41,158,56,173)(42,157,57,172)(43,156,58,171)(44,155,59,170)(45,154,60,169)(61,235,76,220)(62,234,77,219)(63,233,78,218)(64,232,79,217)(65,231,80,216)(66,230,81,215)(67,229,82,214)(68,228,83,213)(69,227,84,212)(70,226,85,211)(71,225,86,240)(72,224,87,239)(73,223,88,238)(74,222,89,237)(75,221,90,236)(91,127,106,142)(92,126,107,141)(93,125,108,140)(94,124,109,139)(95,123,110,138)(96,122,111,137)(97,121,112,136)(98,150,113,135)(99,149,114,134)(100,148,115,133)(101,147,116,132)(102,146,117,131)(103,145,118,130)(104,144,119,129)(105,143,120,128), (1,35,85,123)(2,36,86,124)(3,37,87,125)(4,38,88,126)(5,39,89,127)(6,40,90,128)(7,41,61,129)(8,42,62,130)(9,43,63,131)(10,44,64,132)(11,45,65,133)(12,46,66,134)(13,47,67,135)(14,48,68,136)(15,49,69,137)(16,50,70,138)(17,51,71,139)(18,52,72,140)(19,53,73,141)(20,54,74,142)(21,55,75,143)(22,56,76,144)(23,57,77,145)(24,58,78,146)(25,59,79,147)(26,60,80,148)(27,31,81,149)(28,32,82,150)(29,33,83,121)(30,34,84,122)(91,182,175,237)(92,183,176,238)(93,184,177,239)(94,185,178,240)(95,186,179,211)(96,187,180,212)(97,188,151,213)(98,189,152,214)(99,190,153,215)(100,191,154,216)(101,192,155,217)(102,193,156,218)(103,194,157,219)(104,195,158,220)(105,196,159,221)(106,197,160,222)(107,198,161,223)(108,199,162,224)(109,200,163,225)(110,201,164,226)(111,202,165,227)(112,203,166,228)(113,204,167,229)(114,205,168,230)(115,206,169,231)(116,207,170,232)(117,208,171,233)(118,209,172,234)(119,210,173,235)(120,181,174,236), (31,149)(32,150)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174) );

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)], [(1,186,16,201),(2,185,17,200),(3,184,18,199),(4,183,19,198),(5,182,20,197),(6,181,21,196),(7,210,22,195),(8,209,23,194),(9,208,24,193),(10,207,25,192),(11,206,26,191),(12,205,27,190),(13,204,28,189),(14,203,29,188),(15,202,30,187),(31,168,46,153),(32,167,47,152),(33,166,48,151),(34,165,49,180),(35,164,50,179),(36,163,51,178),(37,162,52,177),(38,161,53,176),(39,160,54,175),(40,159,55,174),(41,158,56,173),(42,157,57,172),(43,156,58,171),(44,155,59,170),(45,154,60,169),(61,235,76,220),(62,234,77,219),(63,233,78,218),(64,232,79,217),(65,231,80,216),(66,230,81,215),(67,229,82,214),(68,228,83,213),(69,227,84,212),(70,226,85,211),(71,225,86,240),(72,224,87,239),(73,223,88,238),(74,222,89,237),(75,221,90,236),(91,127,106,142),(92,126,107,141),(93,125,108,140),(94,124,109,139),(95,123,110,138),(96,122,111,137),(97,121,112,136),(98,150,113,135),(99,149,114,134),(100,148,115,133),(101,147,116,132),(102,146,117,131),(103,145,118,130),(104,144,119,129),(105,143,120,128)], [(1,35,85,123),(2,36,86,124),(3,37,87,125),(4,38,88,126),(5,39,89,127),(6,40,90,128),(7,41,61,129),(8,42,62,130),(9,43,63,131),(10,44,64,132),(11,45,65,133),(12,46,66,134),(13,47,67,135),(14,48,68,136),(15,49,69,137),(16,50,70,138),(17,51,71,139),(18,52,72,140),(19,53,73,141),(20,54,74,142),(21,55,75,143),(22,56,76,144),(23,57,77,145),(24,58,78,146),(25,59,79,147),(26,60,80,148),(27,31,81,149),(28,32,82,150),(29,33,83,121),(30,34,84,122),(91,182,175,237),(92,183,176,238),(93,184,177,239),(94,185,178,240),(95,186,179,211),(96,187,180,212),(97,188,151,213),(98,189,152,214),(99,190,153,215),(100,191,154,216),(101,192,155,217),(102,193,156,218),(103,194,157,219),(104,195,158,220),(105,196,159,221),(106,197,160,222),(107,198,161,223),(108,199,162,224),(109,200,163,225),(110,201,164,226),(111,202,165,227),(112,203,166,228),(113,204,167,229),(114,205,168,230),(115,206,169,231),(116,207,170,232),(117,208,171,233),(118,209,172,234),(119,210,173,235),(120,181,174,236)], [(31,149),(32,150),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(91,175),(92,176),(93,177),(94,178),(95,179),(96,180),(97,151),(98,152),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(112,166),(113,167),(114,168),(115,169),(116,170),(117,171),(118,172),(119,173),(120,174)])

84 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222222344444455666666610···1010···101212151515152020202030···3030···3060···60
size1111224602430303030602222244442···24···444222244442···24···44···4

84 irreducible representations

dim1111111222222222222222444444
type+++++++++++++++++++-+-+-
imageC1C2C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10C3⋊D4D15C5⋊D4D30D30C157D4S3×D4D42S3D4×D5D42D5D4×D15D42D15
kernelDic1512D4C30.4Q8D303C4C30.38D4C22×Dic15C2×C157D4D4×C30D4×C10Dic15C2×C30C6×D4C2×C20C22×C10C30C2×C12C22×C6C2×C10C2×D4C2×C6C2×C4C23C22C10C10C6C6C2C2
# reps11111211222122244484816112244

Matrix representation of Dic1512D4 in GL4(𝔽61) generated by

283700
244700
0010
0001
,
115700
05000
00600
00060
,
144500
164700
00121
005860
,
1000
0100
0010
005860
G:=sub<GL(4,GF(61))| [28,24,0,0,37,47,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,57,50,0,0,0,0,60,0,0,0,0,60],[14,16,0,0,45,47,0,0,0,0,1,58,0,0,21,60],[1,0,0,0,0,1,0,0,0,0,1,58,0,0,0,60] >;

Dic1512D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_{12}D_4
% in TeX

G:=Group("Dic15:12D4");
// GroupNames label

G:=SmallGroup(480,904);
// by ID

G=gap.SmallGroup(480,904);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,219,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=d^2=1,b^2=a^15,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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