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## G = Dic7.SD16order 448 = 26·7

### 1st non-split extension by Dic7 of SD16 acting via SD16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — Dic7.SD16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C28⋊D4 — Dic7.SD16
 Lower central C7 — C14 — C2×C28 — Dic7.SD16
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic7.SD16
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a7b, dcd=a7c3 >

Subgroups: 756 in 118 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C4×C8, D4⋊C4, D4⋊C4, C41D4, C4⋊Q8, C7⋊C8, C56, Dic14, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C4.4D8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×D28, C2×C7⋊D4, D4×C14, C14.D8, C8×Dic7, C2.D56, D4⋊Dic7, C7×D4⋊C4, C28⋊Q8, C28⋊D4, Dic7.SD16
Quotients: C1, C2, C22, D4, C23, D7, D8, SD16, C2×D4, C4○D4, D14, C4.4D4, C2×D8, C2×SD16, C22×D7, C4.4D8, C4○D28, D4×D7, D42D7, Dic7.D4, D7×D8, D7×SD16, Dic7.SD16

Smallest permutation representation of Dic7.SD16
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)
(1 134 8 127)(2 133 9 140)(3 132 10 139)(4 131 11 138)(5 130 12 137)(6 129 13 136)(7 128 14 135)(15 204 22 197)(16 203 23 210)(17 202 24 209)(18 201 25 208)(19 200 26 207)(20 199 27 206)(21 198 28 205)(29 192 36 185)(30 191 37 184)(31 190 38 183)(32 189 39 196)(33 188 40 195)(34 187 41 194)(35 186 42 193)(43 86 50 93)(44 85 51 92)(45 98 52 91)(46 97 53 90)(47 96 54 89)(48 95 55 88)(49 94 56 87)(57 156 64 163)(58 155 65 162)(59 168 66 161)(60 167 67 160)(61 166 68 159)(62 165 69 158)(63 164 70 157)(71 101 78 108)(72 100 79 107)(73 99 80 106)(74 112 81 105)(75 111 82 104)(76 110 83 103)(77 109 84 102)(113 177 120 170)(114 176 121 169)(115 175 122 182)(116 174 123 181)(117 173 124 180)(118 172 125 179)(119 171 126 178)(141 216 148 223)(142 215 149 222)(143 214 150 221)(144 213 151 220)(145 212 152 219)(146 211 153 218)(147 224 154 217)
(1 18 173 74 211 64 29 49)(2 19 174 75 212 65 30 50)(3 20 175 76 213 66 31 51)(4 21 176 77 214 67 32 52)(5 22 177 78 215 68 33 53)(6 23 178 79 216 69 34 54)(7 24 179 80 217 70 35 55)(8 25 180 81 218 57 36 56)(9 26 181 82 219 58 37 43)(10 27 182 83 220 59 38 44)(11 28 169 84 221 60 39 45)(12 15 170 71 222 61 40 46)(13 16 171 72 223 62 41 47)(14 17 172 73 224 63 42 48)(85 139 206 115 103 144 168 183)(86 140 207 116 104 145 155 184)(87 127 208 117 105 146 156 185)(88 128 209 118 106 147 157 186)(89 129 210 119 107 148 158 187)(90 130 197 120 108 149 159 188)(91 131 198 121 109 150 160 189)(92 132 199 122 110 151 161 190)(93 133 200 123 111 152 162 191)(94 134 201 124 112 153 163 192)(95 135 202 125 99 154 164 193)(96 136 203 126 100 141 165 194)(97 137 204 113 101 142 166 195)(98 138 205 114 102 143 167 196)
(1 117)(2 118)(3 119)(4 120)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 113)(12 114)(13 115)(14 116)(15 160)(16 161)(17 162)(18 163)(19 164)(20 165)(21 166)(22 167)(23 168)(24 155)(25 156)(26 157)(27 158)(28 159)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 141)(39 142)(40 143)(41 144)(42 145)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 85)(55 86)(56 87)(57 208)(58 209)(59 210)(60 197)(61 198)(62 199)(63 200)(64 201)(65 202)(66 203)(67 204)(68 205)(69 206)(70 207)(71 109)(72 110)(73 111)(74 112)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(127 173)(128 174)(129 175)(130 176)(131 177)(132 178)(133 179)(134 180)(135 181)(136 182)(137 169)(138 170)(139 171)(140 172)(183 223)(184 224)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 221)(196 222)

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D7 D8 SD16 C4○D4 D14 D14 D14 C4○D28 D4⋊2D7 D4×D7 D7×D8 D7×SD16 kernel Dic7.SD16 C14.D8 C8×Dic7 C2.D56 D4⋊Dic7 C7×D4⋊C4 C28⋊Q8 C28⋊D4 C2×Dic7 D4⋊C4 Dic7 Dic7 C28 C4⋊C4 C2×C8 C2×D4 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 3 4 4 4 3 3 3 12 3 3 6 6

Matrix representation of Dic7.SD16 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 25 112 0 0 2 9
,
 112 22 0 0 41 1 0 0 0 0 65 54 0 0 18 48
,
 51 4 0 0 28 0 0 0 0 0 15 0 0 0 0 15
,
 112 0 0 0 41 1 0 0 0 0 17 12 0 0 89 96
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,25,2,0,0,112,9],[112,41,0,0,22,1,0,0,0,0,65,18,0,0,54,48],[51,28,0,0,4,0,0,0,0,0,15,0,0,0,0,15],[112,41,0,0,0,1,0,0,0,0,17,89,0,0,12,96] >;

Dic7.SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_7.{\rm SD}_{16}
% in TeX

G:=Group("Dic7.SD16");
// GroupNames label

G:=SmallGroup(448,294);
// by ID

G=gap.SmallGroup(448,294);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,422,135,100,570,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^7*b,d*c*d=a^7*c^3>;
// generators/relations

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