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## G = Dic14⋊14D4order 448 = 26·7

### 2nd semidirect product of Dic14 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — Dic14⋊14D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×Dic14 — C22×Dic14 — Dic14⋊14D4
 Lower central C7 — C14 — C2×C28 — Dic14⋊14D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for Dic1414D4
G = < a,b,c,d | a28=c4=d2=1, b2=a14, bab-1=cac-1=a-1, ad=da, cbc-1=a21b, bd=db, dcd=c-1 >

Subgroups: 956 in 158 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C56, Dic14, Dic14, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, Q8⋊D4, C56⋊C2, C4⋊Dic7, D14⋊C4, C2×C56, C2×Dic14, C2×Dic14, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, C28.44D4, C7×C22⋊C8, C2×C56⋊C2, C287D4, C22×Dic14, Dic1414D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8.C22, D28, C22×D7, Q8⋊D4, C56⋊C2, C2×D28, D4×D7, C22⋊D28, C2×C56⋊C2, C8.D14, Dic1414D4

Smallest permutation representation of Dic1414D4
On 224 points
Generators in S224
(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)
(1 58 15 72)(2 57 16 71)(3 84 17 70)(4 83 18 69)(5 82 19 68)(6 81 20 67)(7 80 21 66)(8 79 22 65)(9 78 23 64)(10 77 24 63)(11 76 25 62)(12 75 26 61)(13 74 27 60)(14 73 28 59)(29 170 43 184)(30 169 44 183)(31 196 45 182)(32 195 46 181)(33 194 47 180)(34 193 48 179)(35 192 49 178)(36 191 50 177)(37 190 51 176)(38 189 52 175)(39 188 53 174)(40 187 54 173)(41 186 55 172)(42 185 56 171)(85 152 99 166)(86 151 100 165)(87 150 101 164)(88 149 102 163)(89 148 103 162)(90 147 104 161)(91 146 105 160)(92 145 106 159)(93 144 107 158)(94 143 108 157)(95 142 109 156)(96 141 110 155)(97 168 111 154)(98 167 112 153)(113 224 127 210)(114 223 128 209)(115 222 129 208)(116 221 130 207)(117 220 131 206)(118 219 132 205)(119 218 133 204)(120 217 134 203)(121 216 135 202)(122 215 136 201)(123 214 137 200)(124 213 138 199)(125 212 139 198)(126 211 140 197)
(1 138 100 39)(2 137 101 38)(3 136 102 37)(4 135 103 36)(5 134 104 35)(6 133 105 34)(7 132 106 33)(8 131 107 32)(9 130 108 31)(10 129 109 30)(11 128 110 29)(12 127 111 56)(13 126 112 55)(14 125 85 54)(15 124 86 53)(16 123 87 52)(17 122 88 51)(18 121 89 50)(19 120 90 49)(20 119 91 48)(21 118 92 47)(22 117 93 46)(23 116 94 45)(24 115 95 44)(25 114 96 43)(26 113 97 42)(27 140 98 41)(28 139 99 40)(57 207 164 196)(58 206 165 195)(59 205 166 194)(60 204 167 193)(61 203 168 192)(62 202 141 191)(63 201 142 190)(64 200 143 189)(65 199 144 188)(66 198 145 187)(67 197 146 186)(68 224 147 185)(69 223 148 184)(70 222 149 183)(71 221 150 182)(72 220 151 181)(73 219 152 180)(74 218 153 179)(75 217 154 178)(76 216 155 177)(77 215 156 176)(78 214 157 175)(79 213 158 174)(80 212 159 173)(81 211 160 172)(82 210 161 171)(83 209 162 170)(84 208 163 169)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 113)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 222)(170 223)(171 224)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)(181 206)(182 207)(183 208)(184 209)(185 210)(186 211)(187 212)(188 213)(189 214)(190 215)(191 216)(192 217)(193 218)(194 219)(195 220)(196 221)

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

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

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

79 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 56 2 2 4 28 28 28 28 56 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 SD16 D14 D14 D28 D28 C56⋊C2 C8.C22 D4×D7 C8.D14 kernel Dic14⋊14D4 C28.44D4 C7×C22⋊C8 C2×C56⋊C2 C28⋊7D4 C22×Dic14 Dic14 C2×C28 C22×C14 C22⋊C8 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 C14 C4 C2 # reps 1 2 1 2 1 1 4 1 1 3 4 6 3 6 6 24 1 6 6

Matrix representation of Dic1414D4 in GL4(𝔽113) generated by

 67 9 0 0 104 100 0 0 0 0 1 0 0 0 0 1
,
 9 47 0 0 80 104 0 0 0 0 1 0 0 0 0 1
,
 1 34 0 0 0 112 0 0 0 0 0 112 0 0 1 0
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
G:=sub<GL(4,GF(113))| [67,104,0,0,9,100,0,0,0,0,1,0,0,0,0,1],[9,80,0,0,47,104,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,112,0,0,0,0,0,1,0,0,112,0],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112] >;

Dic1414D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{14}\rtimes_{14}D_4
% in TeX

G:=Group("Dic14:14D4");
// GroupNames label

G:=SmallGroup(448,272);
// by ID

G=gap.SmallGroup(448,272);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,254,219,58,1123,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^28=c^4=d^2=1,b^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^21*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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