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G = C14×D4⋊C4order 448 = 26·7

Direct product of C14 and D4⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for C14×D4⋊C4
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 402 in 202 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C14, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C28, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C2×D4⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C22×C28, C22×C28, D4×C14, D4×C14, C23×C14, C7×D4⋊C4, C14×C4⋊C4, C22×C56, D4×C2×C14, C14×D4⋊C4
Quotients:

Smallest permutation representation of C14×D4⋊C4
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 74 89 189)(2 75 90 190)(3 76 91 191)(4 77 92 192)(5 78 93 193)(6 79 94 194)(7 80 95 195)(8 81 96 196)(9 82 97 183)(10 83 98 184)(11 84 85 185)(12 71 86 186)(13 72 87 187)(14 73 88 188)(15 111 135 58)(16 112 136 59)(17 99 137 60)(18 100 138 61)(19 101 139 62)(20 102 140 63)(21 103 127 64)(22 104 128 65)(23 105 129 66)(24 106 130 67)(25 107 131 68)(26 108 132 69)(27 109 133 70)(28 110 134 57)(29 50 146 199)(30 51 147 200)(31 52 148 201)(32 53 149 202)(33 54 150 203)(34 55 151 204)(35 56 152 205)(36 43 153 206)(37 44 154 207)(38 45 141 208)(39 46 142 209)(40 47 143 210)(41 48 144 197)(42 49 145 198)(113 167 216 180)(114 168 217 181)(115 155 218 182)(116 156 219 169)(117 157 220 170)(118 158 221 171)(119 159 222 172)(120 160 223 173)(121 161 224 174)(122 162 211 175)(123 163 212 176)(124 164 213 177)(125 165 214 178)(126 166 215 179)
(1 81)(2 82)(3 83)(4 84)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 205)(50 206)(51 207)(52 208)(53 209)(54 210)(55 197)(56 198)(85 192)(86 193)(87 194)(88 195)(89 196)(90 183)(91 184)(92 185)(93 186)(94 187)(95 188)(96 189)(97 190)(98 191)(99 130)(100 131)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 127)(111 128)(112 129)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(155 175)(156 176)(157 177)(158 178)(159 179)(160 180)(161 181)(162 182)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(1 212 58 151)(2 213 59 152)(3 214 60 153)(4 215 61 154)(5 216 62 141)(6 217 63 142)(7 218 64 143)(8 219 65 144)(9 220 66 145)(10 221 67 146)(11 222 68 147)(12 223 69 148)(13 224 70 149)(14 211 57 150)(15 55 74 163)(16 56 75 164)(17 43 76 165)(18 44 77 166)(19 45 78 167)(20 46 79 168)(21 47 80 155)(22 48 81 156)(23 49 82 157)(24 50 83 158)(25 51 84 159)(26 52 71 160)(27 53 72 161)(28 54 73 162)(29 98 118 106)(30 85 119 107)(31 86 120 108)(32 87 121 109)(33 88 122 110)(34 89 123 111)(35 90 124 112)(36 91 125 99)(37 92 126 100)(38 93 113 101)(39 94 114 102)(40 95 115 103)(41 96 116 104)(42 97 117 105)(127 210 195 182)(128 197 196 169)(129 198 183 170)(130 199 184 171)(131 200 185 172)(132 201 186 173)(133 202 187 174)(134 203 188 175)(135 204 189 176)(136 205 190 177)(137 206 191 178)(138 207 192 179)(139 208 193 180)(140 209 194 181)

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

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

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

196 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 8A ··· 8H 14A ··· 14AP 14AQ ··· 14BN 28A ··· 28X 28Y ··· 28AV 56A ··· 56AV order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 ··· 1 4 4 4 4 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C7 C14 C14 C14 C14 C28 D4 D4 D8 SD16 C7×D4 C7×D4 C7×D8 C7×SD16 kernel C14×D4⋊C4 C7×D4⋊C4 C14×C4⋊C4 C22×C56 D4×C2×C14 D4×C14 C2×D4⋊C4 D4⋊C4 C2×C4⋊C4 C22×C8 C22×D4 C2×D4 C2×C28 C22×C14 C2×C14 C2×C14 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 6 24 6 6 6 48 3 1 4 4 18 6 24 24

Matrix representation of C14×D4⋊C4 in GL5(𝔽113)

 112 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 109 0 0 0 0 0 109
,
 1 0 0 0 0 0 112 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 112 0
,
 1 0 0 0 0 0 112 87 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 112 0
,
 112 0 0 0 0 0 87 57 0 0 0 2 26 0 0 0 0 0 100 13 0 0 0 13 13

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,109,0,0,0,0,0,109],[1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,1,0],[1,0,0,0,0,0,112,0,0,0,0,87,1,0,0,0,0,0,0,112,0,0,0,112,0],[112,0,0,0,0,0,87,2,0,0,0,57,26,0,0,0,0,0,100,13,0,0,0,13,13] >;

C14×D4⋊C4 in GAP, Magma, Sage, TeX

C_{14}\times D_4\rtimes C_4
% in TeX

G:=Group("C14xD4:C4");
// GroupNames label

G:=SmallGroup(448,822);
// by ID

G=gap.SmallGroup(448,822);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,9804,4911,172]);
// Polycyclic

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

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