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## G = C2×C28.17D4order 448 = 26·7

### Direct product of C2 and C28.17D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C28.17D4
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×C4×Dic7 — C2×C28.17D4
 Lower central C7 — C2×C14 — C2×C28.17D4
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C28.17D4
G = < a,b,c,d | a2=b28=c4=1, d2=b14, ab=ba, ac=ca, ad=da, cbc-1=b13, dbd-1=b-1, dcd-1=b14c-1 >

Subgroups: 1172 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C4.4D4, C4×Dic7, C23.D7, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C2×C4×Dic7, C28.17D4, C2×C23.D7, C22×Dic14, D4×C2×C14, C2×C28.17D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C4.4D4, C22×D4, C2×C4○D4, C7⋊D4, C22×D7, C2×C4.4D4, D42D7, C2×C7⋊D4, C23×D7, C28.17D4, C2×D42D7, C22×C7⋊D4, C2×C28.17D4

Smallest permutation representation of C2×C28.17D4
On 224 points
Generators in S224
(1 174)(2 175)(3 176)(4 177)(5 178)(6 179)(7 180)(8 181)(9 182)(10 183)(11 184)(12 185)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 193)(21 194)(22 195)(23 196)(24 169)(25 170)(26 171)(27 172)(28 173)(29 201)(30 202)(31 203)(32 204)(33 205)(34 206)(35 207)(36 208)(37 209)(38 210)(39 211)(40 212)(41 213)(42 214)(43 215)(44 216)(45 217)(46 218)(47 219)(48 220)(49 221)(50 222)(51 223)(52 224)(53 197)(54 198)(55 199)(56 200)(57 159)(58 160)(59 161)(60 162)(61 163)(62 164)(63 165)(64 166)(65 167)(66 168)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 155)(82 156)(83 157)(84 158)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 113)(111 114)(112 115)
(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 68 48 137)(2 81 49 122)(3 66 50 135)(4 79 51 120)(5 64 52 133)(6 77 53 118)(7 62 54 131)(8 75 55 116)(9 60 56 129)(10 73 29 114)(11 58 30 127)(12 71 31 140)(13 84 32 125)(14 69 33 138)(15 82 34 123)(16 67 35 136)(17 80 36 121)(18 65 37 134)(19 78 38 119)(20 63 39 132)(21 76 40 117)(22 61 41 130)(23 74 42 115)(24 59 43 128)(25 72 44 113)(26 57 45 126)(27 70 46 139)(28 83 47 124)(85 181 149 199)(86 194 150 212)(87 179 151 197)(88 192 152 210)(89 177 153 223)(90 190 154 208)(91 175 155 221)(92 188 156 206)(93 173 157 219)(94 186 158 204)(95 171 159 217)(96 184 160 202)(97 169 161 215)(98 182 162 200)(99 195 163 213)(100 180 164 198)(101 193 165 211)(102 178 166 224)(103 191 167 209)(104 176 168 222)(105 189 141 207)(106 174 142 220)(107 187 143 205)(108 172 144 218)(109 185 145 203)(110 170 146 216)(111 183 147 201)(112 196 148 214)
(1 75 15 61)(2 74 16 60)(3 73 17 59)(4 72 18 58)(5 71 19 57)(6 70 20 84)(7 69 21 83)(8 68 22 82)(9 67 23 81)(10 66 24 80)(11 65 25 79)(12 64 26 78)(13 63 27 77)(14 62 28 76)(29 135 43 121)(30 134 44 120)(31 133 45 119)(32 132 46 118)(33 131 47 117)(34 130 48 116)(35 129 49 115)(36 128 50 114)(37 127 51 113)(38 126 52 140)(39 125 53 139)(40 124 54 138)(41 123 55 137)(42 122 56 136)(85 206 99 220)(86 205 100 219)(87 204 101 218)(88 203 102 217)(89 202 103 216)(90 201 104 215)(91 200 105 214)(92 199 106 213)(93 198 107 212)(94 197 108 211)(95 224 109 210)(96 223 110 209)(97 222 111 208)(98 221 112 207)(141 196 155 182)(142 195 156 181)(143 194 157 180)(144 193 158 179)(145 192 159 178)(146 191 160 177)(147 190 161 176)(148 189 162 175)(149 188 163 174)(150 187 164 173)(151 186 165 172)(152 185 166 171)(153 184 167 170)(154 183 168 169)

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 7A 7B 7C 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 4 4 4 4 2 2 2 2 14 ··· 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 D14 C7⋊D4 D4⋊2D7 kernel C2×C28.17D4 C2×C4×Dic7 C28.17D4 C2×C23.D7 C22×Dic14 D4×C2×C14 C2×C28 C22×D4 C2×C14 C22×C4 C2×D4 C24 C2×C4 C22 # reps 1 1 8 4 1 1 4 3 8 3 12 6 24 12

Matrix representation of C2×C28.17D4 in GL5(𝔽29)

 28 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 24 27 0 0 0 0 23 0 0 0 0 0 1 1 0 0 0 27 28
,
 1 0 0 0 0 0 28 4 0 0 0 14 1 0 0 0 0 0 17 0 0 0 0 0 17
,
 28 0 0 0 0 0 28 0 0 0 0 14 1 0 0 0 0 0 12 0 0 0 0 5 17

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,24,0,0,0,0,27,23,0,0,0,0,0,1,27,0,0,0,1,28],[1,0,0,0,0,0,28,14,0,0,0,4,1,0,0,0,0,0,17,0,0,0,0,0,17],[28,0,0,0,0,0,28,14,0,0,0,0,1,0,0,0,0,0,12,5,0,0,0,0,17] >;

C2×C28.17D4 in GAP, Magma, Sage, TeX

C_2\times C_{28}._{17}D_4
% in TeX

G:=Group("C2xC28.17D4");
// GroupNames label

G:=SmallGroup(448,1250);
// by ID

G=gap.SmallGroup(448,1250);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,100,1571,185,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^13,d*b*d^-1=b^-1,d*c*d^-1=b^14*c^-1>;
// generators/relations

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