direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C11⋊D11, C22⋊D11, C11⋊2D22, C112⋊3C22, (C11×C22)⋊2C2, SmallGroup(484,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C112 — C2×C11⋊D11 |
Generators and relations for C2×C11⋊D11
G = < a,b,c,d | a2=b11=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 790 in 70 conjugacy classes, 31 normal (5 characteristic)
C1, C2, C2, C22, C11, D11, C22, D22, C112, C11⋊D11, C11×C22, C2×C11⋊D11
Quotients: C1, C2, C22, D11, D22, C11⋊D11, C2×C11⋊D11
►On 242 points
Generators in S242
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 78)(9 79)(10 80)(11 81)(12 132)(13 122)(14 123)(15 124)(16 125)(17 126)(18 127)(19 128)(20 129)(21 130)(22 131)(23 143)(24 133)(25 134)(26 135)(27 136)(28 137)(29 138)(30 139)(31 140)(32 141)(33 142)(34 154)(35 144)(36 145)(37 146)(38 147)(39 148)(40 149)(41 150)(42 151)(43 152)(44 153)(45 165)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 161)(53 162)(54 163)(55 164)(56 176)(57 166)(58 167)(59 168)(60 169)(61 170)(62 171)(63 172)(64 173)(65 174)(66 175)(67 187)(68 177)(69 178)(70 179)(71 180)(72 181)(73 182)(74 183)(75 184)(76 185)(77 186)(89 209)(90 199)(91 200)(92 201)(93 202)(94 203)(95 204)(96 205)(97 206)(98 207)(99 208)(100 220)(101 210)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 217)(109 218)(110 219)(111 231)(112 221)(113 222)(114 223)(115 224)(116 225)(117 226)(118 227)(119 228)(120 229)(121 230)(188 232)(189 233)(190 234)(191 235)(192 236)(193 237)(194 238)(195 239)(196 240)(197 241)(198 242)
(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)(221 222 223 224 225 226 227 228 229 230 231)(232 233 234 235 236 237 238 239 240 241 242)
(1 198 145 138 199 171 164 225 218 179 130)(2 188 146 139 200 172 165 226 219 180 131)(3 189 147 140 201 173 155 227 220 181 132)(4 190 148 141 202 174 156 228 210 182 122)(5 191 149 142 203 175 157 229 211 183 123)(6 192 150 143 204 176 158 230 212 184 124)(7 193 151 133 205 166 159 231 213 185 125)(8 194 152 134 206 167 160 221 214 186 126)(9 195 153 135 207 168 161 222 215 187 127)(10 196 154 136 208 169 162 223 216 177 128)(11 197 144 137 209 170 163 224 217 178 129)(12 84 233 38 31 92 64 46 118 100 72)(13 85 234 39 32 93 65 47 119 101 73)(14 86 235 40 33 94 66 48 120 102 74)(15 87 236 41 23 95 56 49 121 103 75)(16 88 237 42 24 96 57 50 111 104 76)(17 78 238 43 25 97 58 51 112 105 77)(18 79 239 44 26 98 59 52 113 106 67)(19 80 240 34 27 99 60 53 114 107 68)(20 81 241 35 28 89 61 54 115 108 69)(21 82 242 36 29 90 62 55 116 109 70)(22 83 232 37 30 91 63 45 117 110 71)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(11 22)(23 231)(24 230)(25 229)(26 228)(27 227)(28 226)(29 225)(30 224)(31 223)(32 222)(33 221)(34 220)(35 219)(36 218)(37 217)(38 216)(39 215)(40 214)(41 213)(42 212)(43 211)(44 210)(45 209)(46 208)(47 207)(48 206)(49 205)(50 204)(51 203)(52 202)(53 201)(54 200)(55 199)(56 166)(57 176)(58 175)(59 174)(60 173)(61 172)(62 171)(63 170)(64 169)(65 168)(66 167)(67 190)(68 189)(69 188)(70 198)(71 197)(72 196)(73 195)(74 194)(75 193)(76 192)(77 191)(78 123)(79 122)(80 132)(81 131)(82 130)(83 129)(84 128)(85 127)(86 126)(87 125)(88 124)(89 165)(90 164)(91 163)(92 162)(93 161)(94 160)(95 159)(96 158)(97 157)(98 156)(99 155)(100 154)(101 153)(102 152)(103 151)(104 150)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 133)(177 233)(178 232)(179 242)(180 241)(181 240)(182 239)(183 238)(184 237)(185 236)(186 235)(187 234)
G:=sub<Sym(242)| (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,78)(9,79)(10,80)(11,81)(12,132)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,143)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,140)(32,141)(33,142)(34,154)(35,144)(36,145)(37,146)(38,147)(39,148)(40,149)(41,150)(42,151)(43,152)(44,153)(45,165)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,176)(57,166)(58,167)(59,168)(60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,187)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,183)(75,184)(76,185)(77,186)(89,209)(90,199)(91,200)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,220)(101,210)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,217)(109,218)(110,219)(111,231)(112,221)(113,222)(114,223)(115,224)(116,225)(117,226)(118,227)(119,228)(120,229)(121,230)(188,232)(189,233)(190,234)(191,235)(192,236)(193,237)(194,238)(195,239)(196,240)(197,241)(198,242), (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)(221,222,223,224,225,226,227,228,229,230,231)(232,233,234,235,236,237,238,239,240,241,242), (1,198,145,138,199,171,164,225,218,179,130)(2,188,146,139,200,172,165,226,219,180,131)(3,189,147,140,201,173,155,227,220,181,132)(4,190,148,141,202,174,156,228,210,182,122)(5,191,149,142,203,175,157,229,211,183,123)(6,192,150,143,204,176,158,230,212,184,124)(7,193,151,133,205,166,159,231,213,185,125)(8,194,152,134,206,167,160,221,214,186,126)(9,195,153,135,207,168,161,222,215,187,127)(10,196,154,136,208,169,162,223,216,177,128)(11,197,144,137,209,170,163,224,217,178,129)(12,84,233,38,31,92,64,46,118,100,72)(13,85,234,39,32,93,65,47,119,101,73)(14,86,235,40,33,94,66,48,120,102,74)(15,87,236,41,23,95,56,49,121,103,75)(16,88,237,42,24,96,57,50,111,104,76)(17,78,238,43,25,97,58,51,112,105,77)(18,79,239,44,26,98,59,52,113,106,67)(19,80,240,34,27,99,60,53,114,107,68)(20,81,241,35,28,89,61,54,115,108,69)(21,82,242,36,29,90,62,55,116,109,70)(22,83,232,37,30,91,63,45,117,110,71), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(11,22)(23,231)(24,230)(25,229)(26,228)(27,227)(28,226)(29,225)(30,224)(31,223)(32,222)(33,221)(34,220)(35,219)(36,218)(37,217)(38,216)(39,215)(40,214)(41,213)(42,212)(43,211)(44,210)(45,209)(46,208)(47,207)(48,206)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,166)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,168)(66,167)(67,190)(68,189)(69,188)(70,198)(71,197)(72,196)(73,195)(74,194)(75,193)(76,192)(77,191)(78,123)(79,122)(80,132)(81,131)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,165)(90,164)(91,163)(92,162)(93,161)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(177,233)(178,232)(179,242)(180,241)(181,240)(182,239)(183,238)(184,237)(185,236)(186,235)(187,234)>;
G:=Group( (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,78)(9,79)(10,80)(11,81)(12,132)(13,122)(14,123)(15,124)(16,125)(17,126)(18,127)(19,128)(20,129)(21,130)(22,131)(23,143)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,140)(32,141)(33,142)(34,154)(35,144)(36,145)(37,146)(38,147)(39,148)(40,149)(41,150)(42,151)(43,152)(44,153)(45,165)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,176)(57,166)(58,167)(59,168)(60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,187)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,183)(75,184)(76,185)(77,186)(89,209)(90,199)(91,200)(92,201)(93,202)(94,203)(95,204)(96,205)(97,206)(98,207)(99,208)(100,220)(101,210)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,217)(109,218)(110,219)(111,231)(112,221)(113,222)(114,223)(115,224)(116,225)(117,226)(118,227)(119,228)(120,229)(121,230)(188,232)(189,233)(190,234)(191,235)(192,236)(193,237)(194,238)(195,239)(196,240)(197,241)(198,242), (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)(221,222,223,224,225,226,227,228,229,230,231)(232,233,234,235,236,237,238,239,240,241,242), (1,198,145,138,199,171,164,225,218,179,130)(2,188,146,139,200,172,165,226,219,180,131)(3,189,147,140,201,173,155,227,220,181,132)(4,190,148,141,202,174,156,228,210,182,122)(5,191,149,142,203,175,157,229,211,183,123)(6,192,150,143,204,176,158,230,212,184,124)(7,193,151,133,205,166,159,231,213,185,125)(8,194,152,134,206,167,160,221,214,186,126)(9,195,153,135,207,168,161,222,215,187,127)(10,196,154,136,208,169,162,223,216,177,128)(11,197,144,137,209,170,163,224,217,178,129)(12,84,233,38,31,92,64,46,118,100,72)(13,85,234,39,32,93,65,47,119,101,73)(14,86,235,40,33,94,66,48,120,102,74)(15,87,236,41,23,95,56,49,121,103,75)(16,88,237,42,24,96,57,50,111,104,76)(17,78,238,43,25,97,58,51,112,105,77)(18,79,239,44,26,98,59,52,113,106,67)(19,80,240,34,27,99,60,53,114,107,68)(20,81,241,35,28,89,61,54,115,108,69)(21,82,242,36,29,90,62,55,116,109,70)(22,83,232,37,30,91,63,45,117,110,71), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(11,22)(23,231)(24,230)(25,229)(26,228)(27,227)(28,226)(29,225)(30,224)(31,223)(32,222)(33,221)(34,220)(35,219)(36,218)(37,217)(38,216)(39,215)(40,214)(41,213)(42,212)(43,211)(44,210)(45,209)(46,208)(47,207)(48,206)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,166)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,168)(66,167)(67,190)(68,189)(69,188)(70,198)(71,197)(72,196)(73,195)(74,194)(75,193)(76,192)(77,191)(78,123)(79,122)(80,132)(81,131)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,165)(90,164)(91,163)(92,162)(93,161)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(177,233)(178,232)(179,242)(180,241)(181,240)(182,239)(183,238)(184,237)(185,236)(186,235)(187,234) );
G=PermutationGroup([[(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,78),(9,79),(10,80),(11,81),(12,132),(13,122),(14,123),(15,124),(16,125),(17,126),(18,127),(19,128),(20,129),(21,130),(22,131),(23,143),(24,133),(25,134),(26,135),(27,136),(28,137),(29,138),(30,139),(31,140),(32,141),(33,142),(34,154),(35,144),(36,145),(37,146),(38,147),(39,148),(40,149),(41,150),(42,151),(43,152),(44,153),(45,165),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,161),(53,162),(54,163),(55,164),(56,176),(57,166),(58,167),(59,168),(60,169),(61,170),(62,171),(63,172),(64,173),(65,174),(66,175),(67,187),(68,177),(69,178),(70,179),(71,180),(72,181),(73,182),(74,183),(75,184),(76,185),(77,186),(89,209),(90,199),(91,200),(92,201),(93,202),(94,203),(95,204),(96,205),(97,206),(98,207),(99,208),(100,220),(101,210),(102,211),(103,212),(104,213),(105,214),(106,215),(107,216),(108,217),(109,218),(110,219),(111,231),(112,221),(113,222),(114,223),(115,224),(116,225),(117,226),(118,227),(119,228),(120,229),(121,230),(188,232),(189,233),(190,234),(191,235),(192,236),(193,237),(194,238),(195,239),(196,240),(197,241),(198,242)], [(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),(221,222,223,224,225,226,227,228,229,230,231),(232,233,234,235,236,237,238,239,240,241,242)], [(1,198,145,138,199,171,164,225,218,179,130),(2,188,146,139,200,172,165,226,219,180,131),(3,189,147,140,201,173,155,227,220,181,132),(4,190,148,141,202,174,156,228,210,182,122),(5,191,149,142,203,175,157,229,211,183,123),(6,192,150,143,204,176,158,230,212,184,124),(7,193,151,133,205,166,159,231,213,185,125),(8,194,152,134,206,167,160,221,214,186,126),(9,195,153,135,207,168,161,222,215,187,127),(10,196,154,136,208,169,162,223,216,177,128),(11,197,144,137,209,170,163,224,217,178,129),(12,84,233,38,31,92,64,46,118,100,72),(13,85,234,39,32,93,65,47,119,101,73),(14,86,235,40,33,94,66,48,120,102,74),(15,87,236,41,23,95,56,49,121,103,75),(16,88,237,42,24,96,57,50,111,104,76),(17,78,238,43,25,97,58,51,112,105,77),(18,79,239,44,26,98,59,52,113,106,67),(19,80,240,34,27,99,60,53,114,107,68),(20,81,241,35,28,89,61,54,115,108,69),(21,82,242,36,29,90,62,55,116,109,70),(22,83,232,37,30,91,63,45,117,110,71)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(11,22),(23,231),(24,230),(25,229),(26,228),(27,227),(28,226),(29,225),(30,224),(31,223),(32,222),(33,221),(34,220),(35,219),(36,218),(37,217),(38,216),(39,215),(40,214),(41,213),(42,212),(43,211),(44,210),(45,209),(46,208),(47,207),(48,206),(49,205),(50,204),(51,203),(52,202),(53,201),(54,200),(55,199),(56,166),(57,176),(58,175),(59,174),(60,173),(61,172),(62,171),(63,170),(64,169),(65,168),(66,167),(67,190),(68,189),(69,188),(70,198),(71,197),(72,196),(73,195),(74,194),(75,193),(76,192),(77,191),(78,123),(79,122),(80,132),(81,131),(82,130),(83,129),(84,128),(85,127),(86,126),(87,125),(88,124),(89,165),(90,164),(91,163),(92,162),(93,161),(94,160),(95,159),(96,158),(97,157),(98,156),(99,155),(100,154),(101,153),(102,152),(103,151),(104,150),(105,149),(106,148),(107,147),(108,146),(109,145),(110,144),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,133),(177,233),(178,232),(179,242),(180,241),(181,240),(182,239),(183,238),(184,237),(185,236),(186,235),(187,234)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 11A | ··· | 11BH | 22A | ··· | 22BH |
| order | 1 | 2 | 2 | 2 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 1 | 121 | 121 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D11 | D22 |
| kernel | C2×C11⋊D11 | C11⋊D11 | C11×C22 | C22 | C11 |
| # reps | 1 | 2 | 1 | 60 | 60 |
Matrix representation of C2×C11⋊D11 ►in GL4(𝔽23) generated by
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 8 | 0 | 0 |
| 11 | 17 | 0 | 0 |
| 0 | 0 | 11 | 20 |
| 0 | 0 | 3 | 16 |
| 6 | 1 | 0 | 0 |
| 10 | 21 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 22 | 6 |
| 11 | 15 | 0 | 0 |
| 15 | 12 | 0 | 0 |
| 0 | 0 | 11 | 6 |
| 0 | 0 | 3 | 12 |
G:=sub<GL(4,GF(23))| [22,0,0,0,0,22,0,0,0,0,1,0,0,0,0,1],[12,11,0,0,8,17,0,0,0,0,11,3,0,0,20,16],[6,10,0,0,1,21,0,0,0,0,0,22,0,0,1,6],[11,15,0,0,15,12,0,0,0,0,11,3,0,0,6,12] >;
C2×C11⋊D11 in GAP, Magma, Sage, TeX
C_2\times C_{11}\rtimes D_{11} % in TeX
G:=Group("C2xC11:D11"); // GroupNames label
G:=SmallGroup(484,11);
// by ID
G=gap.SmallGroup(484,11);
# by ID
G:=PCGroup([4,-2,-2,-11,-11,482,7043]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^11=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations