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## G = Dic7⋊8SD16order 448 = 26·7

### 3rd semidirect product of Dic7 and SD16 acting through Inn(Dic7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Dic7⋊8SD16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — D28⋊C4 — Dic7⋊8SD16
 Lower central C7 — C14 — C28 — Dic7⋊8SD16
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for Dic78SD16
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c3 >

Subgroups: 652 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×SD16, C56⋊C2, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C14.D8, C14.Q16, C8×Dic7, C7×C4.Q8, Dic73Q8, D28⋊C4, C2×C56⋊C2, Dic78SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, SD16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×SD16, C4○D8, C4×D7, C22×D7, C4×SD16, C2×C4×D7, D4×D7, Q82D7, D28⋊C4, D7×SD16, SD163D7, Dic78SD16

Smallest permutation representation of Dic78SD16
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 21 8 28)(2 20 9 27)(3 19 10 26)(4 18 11 25)(5 17 12 24)(6 16 13 23)(7 15 14 22)(29 176 36 169)(30 175 37 182)(31 174 38 181)(32 173 39 180)(33 172 40 179)(34 171 41 178)(35 170 42 177)(43 118 50 125)(44 117 51 124)(45 116 52 123)(46 115 53 122)(47 114 54 121)(48 113 55 120)(49 126 56 119)(57 203 64 210)(58 202 65 209)(59 201 66 208)(60 200 67 207)(61 199 68 206)(62 198 69 205)(63 197 70 204)(71 111 78 104)(72 110 79 103)(73 109 80 102)(74 108 81 101)(75 107 82 100)(76 106 83 99)(77 105 84 112)(85 127 92 134)(86 140 93 133)(87 139 94 132)(88 138 95 131)(89 137 96 130)(90 136 97 129)(91 135 98 128)(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)(155 195 162 188)(156 194 163 187)(157 193 164 186)(158 192 165 185)(159 191 166 184)(160 190 167 183)(161 189 168 196)
(1 138 220 69 122 111 167 30)(2 139 221 70 123 112 168 31)(3 140 222 57 124 99 155 32)(4 127 223 58 125 100 156 33)(5 128 224 59 126 101 157 34)(6 129 211 60 113 102 158 35)(7 130 212 61 114 103 159 36)(8 131 213 62 115 104 160 37)(9 132 214 63 116 105 161 38)(10 133 215 64 117 106 162 39)(11 134 216 65 118 107 163 40)(12 135 217 66 119 108 164 41)(13 136 218 67 120 109 165 42)(14 137 219 68 121 110 166 29)(15 89 152 199 54 72 191 169)(16 90 153 200 55 73 192 170)(17 91 154 201 56 74 193 171)(18 92 141 202 43 75 194 172)(19 93 142 203 44 76 195 173)(20 94 143 204 45 77 196 174)(21 95 144 205 46 78 183 175)(22 96 145 206 47 79 184 176)(23 97 146 207 48 80 185 177)(24 98 147 208 49 81 186 178)(25 85 148 209 50 82 187 179)(26 86 149 210 51 83 188 180)(27 87 150 197 52 84 189 181)(28 88 151 198 53 71 190 182)
(1 115)(2 114)(3 113)(4 126)(5 125)(6 124)(7 123)(8 122)(9 121)(10 120)(11 119)(12 118)(13 117)(14 116)(15 45)(16 44)(17 43)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 132)(30 131)(31 130)(32 129)(33 128)(34 127)(35 140)(36 139)(37 138)(38 137)(39 136)(40 135)(41 134)(42 133)(57 102)(58 101)(59 100)(60 99)(61 112)(62 111)(63 110)(64 109)(65 108)(66 107)(67 106)(68 105)(69 104)(70 103)(71 205)(72 204)(73 203)(74 202)(75 201)(76 200)(77 199)(78 198)(79 197)(80 210)(81 209)(82 208)(83 207)(84 206)(85 178)(86 177)(87 176)(88 175)(89 174)(90 173)(91 172)(92 171)(93 170)(94 169)(95 182)(96 181)(97 180)(98 179)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 158)(156 157)(159 168)(160 167)(161 166)(162 165)(163 164)(183 190)(184 189)(185 188)(186 187)(191 196)(192 195)(193 194)(211 222)(212 221)(213 220)(214 219)(215 218)(216 217)(223 224)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D7 SD16 C4○D4 D14 D14 C4○D8 C4×D7 Q8⋊2D7 D4×D7 D7×SD16 SD16⋊3D7 kernel Dic7⋊8SD16 C14.D8 C14.Q16 C8×Dic7 C7×C4.Q8 Dic7⋊3Q8 D28⋊C4 C2×C56⋊C2 C56⋊C2 C2×Dic7 C4.Q8 Dic7 C28 C4⋊C4 C2×C8 C14 C8 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 3 4 2 6 3 4 12 3 3 6 6

Matrix representation of Dic78SD16 in GL4(𝔽113) generated by

 0 112 0 0 1 104 0 0 0 0 1 0 0 0 0 1
,
 15 0 0 0 22 98 0 0 0 0 112 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 87 26 0 0 100 0
,
 112 0 0 0 104 1 0 0 0 0 112 0 0 0 112 1
G:=sub<GL(4,GF(113))| [0,1,0,0,112,104,0,0,0,0,1,0,0,0,0,1],[15,22,0,0,0,98,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,87,100,0,0,26,0],[112,104,0,0,0,1,0,0,0,0,112,112,0,0,0,1] >;

Dic78SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("Dic7:8SD16");
// GroupNames label

G:=SmallGroup(448,386);
// by ID

G=gap.SmallGroup(448,386);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,120,135,268,570,297,136,18822]);
// Polycyclic

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

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