direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D29, C116⋊3C22, C58.2C23, C22.9D58, D58.8C22, Dic29⋊3C22, C58⋊2(C2×C4), (C2×C116)⋊5C2, C29⋊2(C22×C4), (C2×Dic29)⋊5C2, (C2×C58).9C22, C2.1(C22×D29), (C22×D29).4C2, SmallGroup(464,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C2×C4×D29 |
Generators and relations for C2×C4×D29
G = < a,b,c,d | a2=b4=c29=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 586 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C22×C4, C29, D29, C58, C58, Dic29, C116, D58, C2×C58, C4×D29, C2×Dic29, C2×C116, C22×D29, C2×C4×D29
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D29, D58, C4×D29, C22×D29, C2×C4×D29
►On 232 points
Generators in S232
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 145)(9 117)(10 118)(11 119)(12 120)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 153)(31 154)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(43 166)(44 167)(45 168)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 182)(60 183)(61 184)(62 185)(63 186)(64 187)(65 188)(66 189)(67 190)(68 191)(69 192)(70 193)(71 194)(72 195)(73 196)(74 197)(75 198)(76 199)(77 200)(78 201)(79 202)(80 203)(81 175)(82 176)(83 177)(84 178)(85 179)(86 180)(87 181)(88 224)(89 225)(90 226)(91 227)(92 228)(93 229)(94 230)(95 231)(96 232)(97 204)(98 205)(99 206)(100 207)(101 208)(102 209)(103 210)(104 211)(105 212)(106 213)(107 214)(108 215)(109 216)(110 217)(111 218)(112 219)(113 220)(114 221)(115 222)(116 223)
(1 116 47 86)(2 88 48 87)(3 89 49 59)(4 90 50 60)(5 91 51 61)(6 92 52 62)(7 93 53 63)(8 94 54 64)(9 95 55 65)(10 96 56 66)(11 97 57 67)(12 98 58 68)(13 99 30 69)(14 100 31 70)(15 101 32 71)(16 102 33 72)(17 103 34 73)(18 104 35 74)(19 105 36 75)(20 106 37 76)(21 107 38 77)(22 108 39 78)(23 109 40 79)(24 110 41 80)(25 111 42 81)(26 112 43 82)(27 113 44 83)(28 114 45 84)(29 115 46 85)(117 231 149 188)(118 232 150 189)(119 204 151 190)(120 205 152 191)(121 206 153 192)(122 207 154 193)(123 208 155 194)(124 209 156 195)(125 210 157 196)(126 211 158 197)(127 212 159 198)(128 213 160 199)(129 214 161 200)(130 215 162 201)(131 216 163 202)(132 217 164 203)(133 218 165 175)(134 219 166 176)(135 220 167 177)(136 221 168 178)(137 222 169 179)(138 223 170 180)(139 224 171 181)(140 225 172 182)(141 226 173 183)(142 227 174 184)(143 228 146 185)(144 229 147 186)(145 230 148 187)
(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)
(1 137)(2 136)(3 135)(4 134)(5 133)(6 132)(7 131)(8 130)(9 129)(10 128)(11 127)(12 126)(13 125)(14 124)(15 123)(16 122)(17 121)(18 120)(19 119)(20 118)(21 117)(22 145)(23 144)(24 143)(25 142)(26 141)(27 140)(28 139)(29 138)(30 157)(31 156)(32 155)(33 154)(34 153)(35 152)(36 151)(37 150)(38 149)(39 148)(40 147)(41 146)(42 174)(43 173)(44 172)(45 171)(46 170)(47 169)(48 168)(49 167)(50 166)(51 165)(52 164)(53 163)(54 162)(55 161)(56 160)(57 159)(58 158)(59 177)(60 176)(61 175)(62 203)(63 202)(64 201)(65 200)(66 199)(67 198)(68 197)(69 196)(70 195)(71 194)(72 193)(73 192)(74 191)(75 190)(76 189)(77 188)(78 187)(79 186)(80 185)(81 184)(82 183)(83 182)(84 181)(85 180)(86 179)(87 178)(88 221)(89 220)(90 219)(91 218)(92 217)(93 216)(94 215)(95 214)(96 213)(97 212)(98 211)(99 210)(100 209)(101 208)(102 207)(103 206)(104 205)(105 204)(106 232)(107 231)(108 230)(109 229)(110 228)(111 227)(112 226)(113 225)(114 224)(115 223)(116 222)
G:=sub<Sym(232)| (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,153)(31,154)(32,155)(33,156)(34,157)(35,158)(36,159)(37,160)(38,161)(39,162)(40,163)(41,164)(42,165)(43,166)(44,167)(45,168)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,224)(89,225)(90,226)(91,227)(92,228)(93,229)(94,230)(95,231)(96,232)(97,204)(98,205)(99,206)(100,207)(101,208)(102,209)(103,210)(104,211)(105,212)(106,213)(107,214)(108,215)(109,216)(110,217)(111,218)(112,219)(113,220)(114,221)(115,222)(116,223), (1,116,47,86)(2,88,48,87)(3,89,49,59)(4,90,50,60)(5,91,51,61)(6,92,52,62)(7,93,53,63)(8,94,54,64)(9,95,55,65)(10,96,56,66)(11,97,57,67)(12,98,58,68)(13,99,30,69)(14,100,31,70)(15,101,32,71)(16,102,33,72)(17,103,34,73)(18,104,35,74)(19,105,36,75)(20,106,37,76)(21,107,38,77)(22,108,39,78)(23,109,40,79)(24,110,41,80)(25,111,42,81)(26,112,43,82)(27,113,44,83)(28,114,45,84)(29,115,46,85)(117,231,149,188)(118,232,150,189)(119,204,151,190)(120,205,152,191)(121,206,153,192)(122,207,154,193)(123,208,155,194)(124,209,156,195)(125,210,157,196)(126,211,158,197)(127,212,159,198)(128,213,160,199)(129,214,161,200)(130,215,162,201)(131,216,163,202)(132,217,164,203)(133,218,165,175)(134,219,166,176)(135,220,167,177)(136,221,168,178)(137,222,169,179)(138,223,170,180)(139,224,171,181)(140,225,172,182)(141,226,173,183)(142,227,174,184)(143,228,146,185)(144,229,147,186)(145,230,148,187), (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), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,145)(23,144)(24,143)(25,142)(26,141)(27,140)(28,139)(29,138)(30,157)(31,156)(32,155)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,174)(43,173)(44,172)(45,171)(46,170)(47,169)(48,168)(49,167)(50,166)(51,165)(52,164)(53,163)(54,162)(55,161)(56,160)(57,159)(58,158)(59,177)(60,176)(61,175)(62,203)(63,202)(64,201)(65,200)(66,199)(67,198)(68,197)(69,196)(70,195)(71,194)(72,193)(73,192)(74,191)(75,190)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(85,180)(86,179)(87,178)(88,221)(89,220)(90,219)(91,218)(92,217)(93,216)(94,215)(95,214)(96,213)(97,212)(98,211)(99,210)(100,209)(101,208)(102,207)(103,206)(104,205)(105,204)(106,232)(107,231)(108,230)(109,229)(110,228)(111,227)(112,226)(113,225)(114,224)(115,223)(116,222)>;
G:=Group( (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,153)(31,154)(32,155)(33,156)(34,157)(35,158)(36,159)(37,160)(38,161)(39,162)(40,163)(41,164)(42,165)(43,166)(44,167)(45,168)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,224)(89,225)(90,226)(91,227)(92,228)(93,229)(94,230)(95,231)(96,232)(97,204)(98,205)(99,206)(100,207)(101,208)(102,209)(103,210)(104,211)(105,212)(106,213)(107,214)(108,215)(109,216)(110,217)(111,218)(112,219)(113,220)(114,221)(115,222)(116,223), (1,116,47,86)(2,88,48,87)(3,89,49,59)(4,90,50,60)(5,91,51,61)(6,92,52,62)(7,93,53,63)(8,94,54,64)(9,95,55,65)(10,96,56,66)(11,97,57,67)(12,98,58,68)(13,99,30,69)(14,100,31,70)(15,101,32,71)(16,102,33,72)(17,103,34,73)(18,104,35,74)(19,105,36,75)(20,106,37,76)(21,107,38,77)(22,108,39,78)(23,109,40,79)(24,110,41,80)(25,111,42,81)(26,112,43,82)(27,113,44,83)(28,114,45,84)(29,115,46,85)(117,231,149,188)(118,232,150,189)(119,204,151,190)(120,205,152,191)(121,206,153,192)(122,207,154,193)(123,208,155,194)(124,209,156,195)(125,210,157,196)(126,211,158,197)(127,212,159,198)(128,213,160,199)(129,214,161,200)(130,215,162,201)(131,216,163,202)(132,217,164,203)(133,218,165,175)(134,219,166,176)(135,220,167,177)(136,221,168,178)(137,222,169,179)(138,223,170,180)(139,224,171,181)(140,225,172,182)(141,226,173,183)(142,227,174,184)(143,228,146,185)(144,229,147,186)(145,230,148,187), (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), (1,137)(2,136)(3,135)(4,134)(5,133)(6,132)(7,131)(8,130)(9,129)(10,128)(11,127)(12,126)(13,125)(14,124)(15,123)(16,122)(17,121)(18,120)(19,119)(20,118)(21,117)(22,145)(23,144)(24,143)(25,142)(26,141)(27,140)(28,139)(29,138)(30,157)(31,156)(32,155)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,174)(43,173)(44,172)(45,171)(46,170)(47,169)(48,168)(49,167)(50,166)(51,165)(52,164)(53,163)(54,162)(55,161)(56,160)(57,159)(58,158)(59,177)(60,176)(61,175)(62,203)(63,202)(64,201)(65,200)(66,199)(67,198)(68,197)(69,196)(70,195)(71,194)(72,193)(73,192)(74,191)(75,190)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(85,180)(86,179)(87,178)(88,221)(89,220)(90,219)(91,218)(92,217)(93,216)(94,215)(95,214)(96,213)(97,212)(98,211)(99,210)(100,209)(101,208)(102,207)(103,206)(104,205)(105,204)(106,232)(107,231)(108,230)(109,229)(110,228)(111,227)(112,226)(113,225)(114,224)(115,223)(116,222) );
G=PermutationGroup([[(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,145),(9,117),(10,118),(11,119),(12,120),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,129),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,153),(31,154),(32,155),(33,156),(34,157),(35,158),(36,159),(37,160),(38,161),(39,162),(40,163),(41,164),(42,165),(43,166),(44,167),(45,168),(46,169),(47,170),(48,171),(49,172),(50,173),(51,174),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,182),(60,183),(61,184),(62,185),(63,186),(64,187),(65,188),(66,189),(67,190),(68,191),(69,192),(70,193),(71,194),(72,195),(73,196),(74,197),(75,198),(76,199),(77,200),(78,201),(79,202),(80,203),(81,175),(82,176),(83,177),(84,178),(85,179),(86,180),(87,181),(88,224),(89,225),(90,226),(91,227),(92,228),(93,229),(94,230),(95,231),(96,232),(97,204),(98,205),(99,206),(100,207),(101,208),(102,209),(103,210),(104,211),(105,212),(106,213),(107,214),(108,215),(109,216),(110,217),(111,218),(112,219),(113,220),(114,221),(115,222),(116,223)], [(1,116,47,86),(2,88,48,87),(3,89,49,59),(4,90,50,60),(5,91,51,61),(6,92,52,62),(7,93,53,63),(8,94,54,64),(9,95,55,65),(10,96,56,66),(11,97,57,67),(12,98,58,68),(13,99,30,69),(14,100,31,70),(15,101,32,71),(16,102,33,72),(17,103,34,73),(18,104,35,74),(19,105,36,75),(20,106,37,76),(21,107,38,77),(22,108,39,78),(23,109,40,79),(24,110,41,80),(25,111,42,81),(26,112,43,82),(27,113,44,83),(28,114,45,84),(29,115,46,85),(117,231,149,188),(118,232,150,189),(119,204,151,190),(120,205,152,191),(121,206,153,192),(122,207,154,193),(123,208,155,194),(124,209,156,195),(125,210,157,196),(126,211,158,197),(127,212,159,198),(128,213,160,199),(129,214,161,200),(130,215,162,201),(131,216,163,202),(132,217,164,203),(133,218,165,175),(134,219,166,176),(135,220,167,177),(136,221,168,178),(137,222,169,179),(138,223,170,180),(139,224,171,181),(140,225,172,182),(141,226,173,183),(142,227,174,184),(143,228,146,185),(144,229,147,186),(145,230,148,187)], [(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)], [(1,137),(2,136),(3,135),(4,134),(5,133),(6,132),(7,131),(8,130),(9,129),(10,128),(11,127),(12,126),(13,125),(14,124),(15,123),(16,122),(17,121),(18,120),(19,119),(20,118),(21,117),(22,145),(23,144),(24,143),(25,142),(26,141),(27,140),(28,139),(29,138),(30,157),(31,156),(32,155),(33,154),(34,153),(35,152),(36,151),(37,150),(38,149),(39,148),(40,147),(41,146),(42,174),(43,173),(44,172),(45,171),(46,170),(47,169),(48,168),(49,167),(50,166),(51,165),(52,164),(53,163),(54,162),(55,161),(56,160),(57,159),(58,158),(59,177),(60,176),(61,175),(62,203),(63,202),(64,201),(65,200),(66,199),(67,198),(68,197),(69,196),(70,195),(71,194),(72,193),(73,192),(74,191),(75,190),(76,189),(77,188),(78,187),(79,186),(80,185),(81,184),(82,183),(83,182),(84,181),(85,180),(86,179),(87,178),(88,221),(89,220),(90,219),(91,218),(92,217),(93,216),(94,215),(95,214),(96,213),(97,212),(98,211),(99,210),(100,209),(101,208),(102,207),(103,206),(104,205),(105,204),(106,232),(107,231),(108,230),(109,229),(110,228),(111,227),(112,226),(113,225),(114,224),(115,223),(116,222)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 29A | ··· | 29N | 58A | ··· | 58AP | 116A | ··· | 116BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 29 | 29 | 29 | 29 | 1 | 1 | 1 | 1 | 29 | 29 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D29 | D58 | D58 | C4×D29 |
| kernel | C2×C4×D29 | C4×D29 | C2×Dic29 | C2×C116 | C22×D29 | D58 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 14 | 28 | 14 | 56 |
Matrix representation of C2×C4×D29 ►in GL3(𝔽233) generated by
| 232 | 0 | 0 |
| 0 | 232 | 0 |
| 0 | 0 | 232 |
| 144 | 0 | 0 |
| 0 | 232 | 0 |
| 0 | 0 | 232 |
| 1 | 0 | 0 |
| 0 | 29 | 1 |
| 0 | 39 | 146 |
| 1 | 0 | 0 |
| 0 | 152 | 58 |
| 0 | 144 | 81 |
G:=sub<GL(3,GF(233))| [232,0,0,0,232,0,0,0,232],[144,0,0,0,232,0,0,0,232],[1,0,0,0,29,39,0,1,146],[1,0,0,0,152,144,0,58,81] >;
C2×C4×D29 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{29} % in TeX
G:=Group("C2xC4xD29"); // GroupNames label
G:=SmallGroup(464,36);
// by ID
G=gap.SmallGroup(464,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,42,11204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^29=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations