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## G = SD16×Dic7order 448 = 26·7

### Direct product of SD16 and Dic7

Series: Derived Chief Lower central Upper central

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

Generators and relations for SD16×Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 484 in 122 conjugacy classes, 59 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C4×SD16, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4⋊Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×SD16, C22×Dic7, D4×C14, Q8×C14, C8×Dic7, C8⋊Dic7, D4⋊Dic7, Q8⋊Dic7, D4×Dic7, Q8×Dic7, C14×SD16, SD16×Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, SD16, C22×C4, C2×D4, C4○D4, Dic7, D14, C4×D4, C2×SD16, C4○D8, C2×Dic7, C22×D7, C4×SD16, D4×D7, D42D7, C22×Dic7, D7×SD16, SD163D7, D4×Dic7, SD16×Dic7

Smallest permutation representation of SD16×Dic7
On 224 points
Generators in S224
(1 121 205 98 150 24 63 32)(2 122 206 85 151 25 64 33)(3 123 207 86 152 26 65 34)(4 124 208 87 153 27 66 35)(5 125 209 88 154 28 67 36)(6 126 210 89 141 15 68 37)(7 113 197 90 142 16 69 38)(8 114 198 91 143 17 70 39)(9 115 199 92 144 18 57 40)(10 116 200 93 145 19 58 41)(11 117 201 94 146 20 59 42)(12 118 202 95 147 21 60 29)(13 119 203 96 148 22 61 30)(14 120 204 97 149 23 62 31)(43 111 79 181 221 192 166 133)(44 112 80 182 222 193 167 134)(45 99 81 169 223 194 168 135)(46 100 82 170 224 195 155 136)(47 101 83 171 211 196 156 137)(48 102 84 172 212 183 157 138)(49 103 71 173 213 184 158 139)(50 104 72 174 214 185 159 140)(51 105 73 175 215 186 160 127)(52 106 74 176 216 187 161 128)(53 107 75 177 217 188 162 129)(54 108 76 178 218 189 163 130)(55 109 77 179 219 190 164 131)(56 110 78 180 220 191 165 132)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 29)(43 159)(44 160)(45 161)(46 162)(47 163)(48 164)(49 165)(50 166)(51 167)(52 168)(53 155)(54 156)(55 157)(56 158)(57 206)(58 207)(59 208)(60 209)(61 210)(62 197)(63 198)(64 199)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 220)(72 221)(73 222)(74 223)(75 224)(76 211)(77 212)(78 213)(79 214)(80 215)(81 216)(82 217)(83 218)(84 219)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 113)(98 114)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(127 182)(128 169)(129 170)(130 171)(131 172)(132 173)(133 174)(134 175)(135 176)(136 177)(137 178)(138 179)(139 180)(140 181)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)
(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)
(1 185 8 192)(2 184 9 191)(3 183 10 190)(4 196 11 189)(5 195 12 188)(6 194 13 187)(7 193 14 186)(15 81 22 74)(16 80 23 73)(17 79 24 72)(18 78 25 71)(19 77 26 84)(20 76 27 83)(21 75 28 82)(29 217 36 224)(30 216 37 223)(31 215 38 222)(32 214 39 221)(33 213 40 220)(34 212 41 219)(35 211 42 218)(43 98 50 91)(44 97 51 90)(45 96 52 89)(46 95 53 88)(47 94 54 87)(48 93 55 86)(49 92 56 85)(57 180 64 173)(58 179 65 172)(59 178 66 171)(60 177 67 170)(61 176 68 169)(62 175 69 182)(63 174 70 181)(99 148 106 141)(100 147 107 154)(101 146 108 153)(102 145 109 152)(103 144 110 151)(104 143 111 150)(105 142 112 149)(113 167 120 160)(114 166 121 159)(115 165 122 158)(116 164 123 157)(117 163 124 156)(118 162 125 155)(119 161 126 168)(127 197 134 204)(128 210 135 203)(129 209 136 202)(130 208 137 201)(131 207 138 200)(132 206 139 199)(133 205 140 198)

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

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

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

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 14J ··· 14O 28A ··· 28F 28G ··· 28L 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 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 2 2 4 4 7 7 7 7 14 14 28 28 28 28 2 2 2 2 2 2 2 14 14 14 14 2 ··· 2 8 ··· 8 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 2 4 4 4 4 type + + + + + + + + + + + - + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D7 SD16 C4○D4 D14 Dic7 D14 D14 C4○D8 D4⋊2D7 D4×D7 D7×SD16 SD16⋊3D7 kernel SD16×Dic7 C8×Dic7 C8⋊Dic7 D4⋊Dic7 Q8⋊Dic7 D4×Dic7 Q8×Dic7 C14×SD16 C7×SD16 C2×Dic7 C2×SD16 Dic7 C28 C2×C8 SD16 C2×D4 C2×Q8 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 3 4 2 3 12 3 3 4 3 3 6 6

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

 112 0 0 0 0 112 0 0 0 0 26 87 0 0 13 0
,
 112 0 0 0 0 112 0 0 0 0 112 0 0 0 112 1
,
 80 112 0 0 2 24 0 0 0 0 112 0 0 0 0 112
,
 28 50 0 0 86 85 0 0 0 0 15 0 0 0 0 15
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,26,13,0,0,87,0],[112,0,0,0,0,112,0,0,0,0,112,112,0,0,0,1],[80,2,0,0,112,24,0,0,0,0,112,0,0,0,0,112],[28,86,0,0,50,85,0,0,0,0,15,0,0,0,0,15] >;

SD16×Dic7 in GAP, Magma, Sage, TeX

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

G:=Group("SD16xDic7");
// GroupNames label

G:=SmallGroup(448,695);
// by ID

G=gap.SmallGroup(448,695);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,219,184,851,438,102,18822]);
// Polycyclic

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

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