metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.264D14, (C4×C8)⋊6D7, (C4×C56)⋊6C2, (C2×C4).63D28, C14.5(C4○D8), (C2×C8).288D14, (C2×C28).353D4, C28.6Q8⋊1C2, C2.D56.1C2, C28.44D4⋊1C2, C4.D28.1C2, (C2×D28).5C22, C22.93(C2×D28), C4⋊Dic7.6C22, C28.219(C4○D4), C4.103(C4○D28), C2.8(D56⋊7C2), (C2×C28).726C23, (C2×C56).348C22, (C4×C28).310C22, C14.6(C4.4D4), C2.11(C4.D28), (C2×Dic14).4C22, C7⋊1(C42.78C22), (C2×C14).109(C2×D4), (C2×C4).669(C22×D7), SmallGroup(448,231)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.264D14
G = < a,b,c,d | a4=b4=1, c14=a2b, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=b-1c13 >
Subgroups: 580 in 96 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C56, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C42.78C22, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×D28, C28.44D4, C2.D56, C4×C56, C28.6Q8, C4.D28, C42.264D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C4○D8, D28, C22×D7, C42.78C22, C2×D28, C4○D28, C4.D28, D56⋊7C2, C42.264D14
►On 224 points
Generators in S224
(1 189 108 149)(2 190 109 150)(3 191 110 151)(4 192 111 152)(5 193 112 153)(6 194 57 154)(7 195 58 155)(8 196 59 156)(9 197 60 157)(10 198 61 158)(11 199 62 159)(12 200 63 160)(13 201 64 161)(14 202 65 162)(15 203 66 163)(16 204 67 164)(17 205 68 165)(18 206 69 166)(19 207 70 167)(20 208 71 168)(21 209 72 113)(22 210 73 114)(23 211 74 115)(24 212 75 116)(25 213 76 117)(26 214 77 118)(27 215 78 119)(28 216 79 120)(29 217 80 121)(30 218 81 122)(31 219 82 123)(32 220 83 124)(33 221 84 125)(34 222 85 126)(35 223 86 127)(36 224 87 128)(37 169 88 129)(38 170 89 130)(39 171 90 131)(40 172 91 132)(41 173 92 133)(42 174 93 134)(43 175 94 135)(44 176 95 136)(45 177 96 137)(46 178 97 138)(47 179 98 139)(48 180 99 140)(49 181 100 141)(50 182 101 142)(51 183 102 143)(52 184 103 144)(53 185 104 145)(54 186 105 146)(55 187 106 147)(56 188 107 148)
(1 66 29 94)(2 67 30 95)(3 68 31 96)(4 69 32 97)(5 70 33 98)(6 71 34 99)(7 72 35 100)(8 73 36 101)(9 74 37 102)(10 75 38 103)(11 76 39 104)(12 77 40 105)(13 78 41 106)(14 79 42 107)(15 80 43 108)(16 81 44 109)(17 82 45 110)(18 83 46 111)(19 84 47 112)(20 85 48 57)(21 86 49 58)(22 87 50 59)(23 88 51 60)(24 89 52 61)(25 90 53 62)(26 91 54 63)(27 92 55 64)(28 93 56 65)(113 223 141 195)(114 224 142 196)(115 169 143 197)(116 170 144 198)(117 171 145 199)(118 172 146 200)(119 173 147 201)(120 174 148 202)(121 175 149 203)(122 176 150 204)(123 177 151 205)(124 178 152 206)(125 179 153 207)(126 180 154 208)(127 181 155 209)(128 182 156 210)(129 183 157 211)(130 184 158 212)(131 185 159 213)(132 186 160 214)(133 187 161 215)(134 188 162 216)(135 189 163 217)(136 190 164 218)(137 191 165 219)(138 192 166 220)(139 193 167 221)(140 194 168 222)
(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 42 108 93)(2 92 109 41)(3 40 110 91)(4 90 111 39)(5 38 112 89)(6 88 57 37)(7 36 58 87)(8 86 59 35)(9 34 60 85)(10 84 61 33)(11 32 62 83)(12 82 63 31)(13 30 64 81)(14 80 65 29)(15 28 66 79)(16 78 67 27)(17 26 68 77)(18 76 69 25)(19 24 70 75)(20 74 71 23)(21 22 72 73)(43 56 94 107)(44 106 95 55)(45 54 96 105)(46 104 97 53)(47 52 98 103)(48 102 99 51)(49 50 100 101)(113 182 209 142)(114 141 210 181)(115 180 211 140)(116 139 212 179)(117 178 213 138)(118 137 214 177)(119 176 215 136)(120 135 216 175)(121 174 217 134)(122 133 218 173)(123 172 219 132)(124 131 220 171)(125 170 221 130)(126 129 222 169)(127 224 223 128)(143 208 183 168)(144 167 184 207)(145 206 185 166)(146 165 186 205)(147 204 187 164)(148 163 188 203)(149 202 189 162)(150 161 190 201)(151 200 191 160)(152 159 192 199)(153 198 193 158)(154 157 194 197)(155 196 195 156)
G:=sub<Sym(224)| (1,189,108,149)(2,190,109,150)(3,191,110,151)(4,192,111,152)(5,193,112,153)(6,194,57,154)(7,195,58,155)(8,196,59,156)(9,197,60,157)(10,198,61,158)(11,199,62,159)(12,200,63,160)(13,201,64,161)(14,202,65,162)(15,203,66,163)(16,204,67,164)(17,205,68,165)(18,206,69,166)(19,207,70,167)(20,208,71,168)(21,209,72,113)(22,210,73,114)(23,211,74,115)(24,212,75,116)(25,213,76,117)(26,214,77,118)(27,215,78,119)(28,216,79,120)(29,217,80,121)(30,218,81,122)(31,219,82,123)(32,220,83,124)(33,221,84,125)(34,222,85,126)(35,223,86,127)(36,224,87,128)(37,169,88,129)(38,170,89,130)(39,171,90,131)(40,172,91,132)(41,173,92,133)(42,174,93,134)(43,175,94,135)(44,176,95,136)(45,177,96,137)(46,178,97,138)(47,179,98,139)(48,180,99,140)(49,181,100,141)(50,182,101,142)(51,183,102,143)(52,184,103,144)(53,185,104,145)(54,186,105,146)(55,187,106,147)(56,188,107,148), (1,66,29,94)(2,67,30,95)(3,68,31,96)(4,69,32,97)(5,70,33,98)(6,71,34,99)(7,72,35,100)(8,73,36,101)(9,74,37,102)(10,75,38,103)(11,76,39,104)(12,77,40,105)(13,78,41,106)(14,79,42,107)(15,80,43,108)(16,81,44,109)(17,82,45,110)(18,83,46,111)(19,84,47,112)(20,85,48,57)(21,86,49,58)(22,87,50,59)(23,88,51,60)(24,89,52,61)(25,90,53,62)(26,91,54,63)(27,92,55,64)(28,93,56,65)(113,223,141,195)(114,224,142,196)(115,169,143,197)(116,170,144,198)(117,171,145,199)(118,172,146,200)(119,173,147,201)(120,174,148,202)(121,175,149,203)(122,176,150,204)(123,177,151,205)(124,178,152,206)(125,179,153,207)(126,180,154,208)(127,181,155,209)(128,182,156,210)(129,183,157,211)(130,184,158,212)(131,185,159,213)(132,186,160,214)(133,187,161,215)(134,188,162,216)(135,189,163,217)(136,190,164,218)(137,191,165,219)(138,192,166,220)(139,193,167,221)(140,194,168,222), (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), (1,42,108,93)(2,92,109,41)(3,40,110,91)(4,90,111,39)(5,38,112,89)(6,88,57,37)(7,36,58,87)(8,86,59,35)(9,34,60,85)(10,84,61,33)(11,32,62,83)(12,82,63,31)(13,30,64,81)(14,80,65,29)(15,28,66,79)(16,78,67,27)(17,26,68,77)(18,76,69,25)(19,24,70,75)(20,74,71,23)(21,22,72,73)(43,56,94,107)(44,106,95,55)(45,54,96,105)(46,104,97,53)(47,52,98,103)(48,102,99,51)(49,50,100,101)(113,182,209,142)(114,141,210,181)(115,180,211,140)(116,139,212,179)(117,178,213,138)(118,137,214,177)(119,176,215,136)(120,135,216,175)(121,174,217,134)(122,133,218,173)(123,172,219,132)(124,131,220,171)(125,170,221,130)(126,129,222,169)(127,224,223,128)(143,208,183,168)(144,167,184,207)(145,206,185,166)(146,165,186,205)(147,204,187,164)(148,163,188,203)(149,202,189,162)(150,161,190,201)(151,200,191,160)(152,159,192,199)(153,198,193,158)(154,157,194,197)(155,196,195,156)>;
G:=Group( (1,189,108,149)(2,190,109,150)(3,191,110,151)(4,192,111,152)(5,193,112,153)(6,194,57,154)(7,195,58,155)(8,196,59,156)(9,197,60,157)(10,198,61,158)(11,199,62,159)(12,200,63,160)(13,201,64,161)(14,202,65,162)(15,203,66,163)(16,204,67,164)(17,205,68,165)(18,206,69,166)(19,207,70,167)(20,208,71,168)(21,209,72,113)(22,210,73,114)(23,211,74,115)(24,212,75,116)(25,213,76,117)(26,214,77,118)(27,215,78,119)(28,216,79,120)(29,217,80,121)(30,218,81,122)(31,219,82,123)(32,220,83,124)(33,221,84,125)(34,222,85,126)(35,223,86,127)(36,224,87,128)(37,169,88,129)(38,170,89,130)(39,171,90,131)(40,172,91,132)(41,173,92,133)(42,174,93,134)(43,175,94,135)(44,176,95,136)(45,177,96,137)(46,178,97,138)(47,179,98,139)(48,180,99,140)(49,181,100,141)(50,182,101,142)(51,183,102,143)(52,184,103,144)(53,185,104,145)(54,186,105,146)(55,187,106,147)(56,188,107,148), (1,66,29,94)(2,67,30,95)(3,68,31,96)(4,69,32,97)(5,70,33,98)(6,71,34,99)(7,72,35,100)(8,73,36,101)(9,74,37,102)(10,75,38,103)(11,76,39,104)(12,77,40,105)(13,78,41,106)(14,79,42,107)(15,80,43,108)(16,81,44,109)(17,82,45,110)(18,83,46,111)(19,84,47,112)(20,85,48,57)(21,86,49,58)(22,87,50,59)(23,88,51,60)(24,89,52,61)(25,90,53,62)(26,91,54,63)(27,92,55,64)(28,93,56,65)(113,223,141,195)(114,224,142,196)(115,169,143,197)(116,170,144,198)(117,171,145,199)(118,172,146,200)(119,173,147,201)(120,174,148,202)(121,175,149,203)(122,176,150,204)(123,177,151,205)(124,178,152,206)(125,179,153,207)(126,180,154,208)(127,181,155,209)(128,182,156,210)(129,183,157,211)(130,184,158,212)(131,185,159,213)(132,186,160,214)(133,187,161,215)(134,188,162,216)(135,189,163,217)(136,190,164,218)(137,191,165,219)(138,192,166,220)(139,193,167,221)(140,194,168,222), (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), (1,42,108,93)(2,92,109,41)(3,40,110,91)(4,90,111,39)(5,38,112,89)(6,88,57,37)(7,36,58,87)(8,86,59,35)(9,34,60,85)(10,84,61,33)(11,32,62,83)(12,82,63,31)(13,30,64,81)(14,80,65,29)(15,28,66,79)(16,78,67,27)(17,26,68,77)(18,76,69,25)(19,24,70,75)(20,74,71,23)(21,22,72,73)(43,56,94,107)(44,106,95,55)(45,54,96,105)(46,104,97,53)(47,52,98,103)(48,102,99,51)(49,50,100,101)(113,182,209,142)(114,141,210,181)(115,180,211,140)(116,139,212,179)(117,178,213,138)(118,137,214,177)(119,176,215,136)(120,135,216,175)(121,174,217,134)(122,133,218,173)(123,172,219,132)(124,131,220,171)(125,170,221,130)(126,129,222,169)(127,224,223,128)(143,208,183,168)(144,167,184,207)(145,206,185,166)(146,165,186,205)(147,204,187,164)(148,163,188,203)(149,202,189,162)(150,161,190,201)(151,200,191,160)(152,159,192,199)(153,198,193,158)(154,157,194,197)(155,196,195,156) );
G=PermutationGroup([[(1,189,108,149),(2,190,109,150),(3,191,110,151),(4,192,111,152),(5,193,112,153),(6,194,57,154),(7,195,58,155),(8,196,59,156),(9,197,60,157),(10,198,61,158),(11,199,62,159),(12,200,63,160),(13,201,64,161),(14,202,65,162),(15,203,66,163),(16,204,67,164),(17,205,68,165),(18,206,69,166),(19,207,70,167),(20,208,71,168),(21,209,72,113),(22,210,73,114),(23,211,74,115),(24,212,75,116),(25,213,76,117),(26,214,77,118),(27,215,78,119),(28,216,79,120),(29,217,80,121),(30,218,81,122),(31,219,82,123),(32,220,83,124),(33,221,84,125),(34,222,85,126),(35,223,86,127),(36,224,87,128),(37,169,88,129),(38,170,89,130),(39,171,90,131),(40,172,91,132),(41,173,92,133),(42,174,93,134),(43,175,94,135),(44,176,95,136),(45,177,96,137),(46,178,97,138),(47,179,98,139),(48,180,99,140),(49,181,100,141),(50,182,101,142),(51,183,102,143),(52,184,103,144),(53,185,104,145),(54,186,105,146),(55,187,106,147),(56,188,107,148)], [(1,66,29,94),(2,67,30,95),(3,68,31,96),(4,69,32,97),(5,70,33,98),(6,71,34,99),(7,72,35,100),(8,73,36,101),(9,74,37,102),(10,75,38,103),(11,76,39,104),(12,77,40,105),(13,78,41,106),(14,79,42,107),(15,80,43,108),(16,81,44,109),(17,82,45,110),(18,83,46,111),(19,84,47,112),(20,85,48,57),(21,86,49,58),(22,87,50,59),(23,88,51,60),(24,89,52,61),(25,90,53,62),(26,91,54,63),(27,92,55,64),(28,93,56,65),(113,223,141,195),(114,224,142,196),(115,169,143,197),(116,170,144,198),(117,171,145,199),(118,172,146,200),(119,173,147,201),(120,174,148,202),(121,175,149,203),(122,176,150,204),(123,177,151,205),(124,178,152,206),(125,179,153,207),(126,180,154,208),(127,181,155,209),(128,182,156,210),(129,183,157,211),(130,184,158,212),(131,185,159,213),(132,186,160,214),(133,187,161,215),(134,188,162,216),(135,189,163,217),(136,190,164,218),(137,191,165,219),(138,192,166,220),(139,193,167,221),(140,194,168,222)], [(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)], [(1,42,108,93),(2,92,109,41),(3,40,110,91),(4,90,111,39),(5,38,112,89),(6,88,57,37),(7,36,58,87),(8,86,59,35),(9,34,60,85),(10,84,61,33),(11,32,62,83),(12,82,63,31),(13,30,64,81),(14,80,65,29),(15,28,66,79),(16,78,67,27),(17,26,68,77),(18,76,69,25),(19,24,70,75),(20,74,71,23),(21,22,72,73),(43,56,94,107),(44,106,95,55),(45,54,96,105),(46,104,97,53),(47,52,98,103),(48,102,99,51),(49,50,100,101),(113,182,209,142),(114,141,210,181),(115,180,211,140),(116,139,212,179),(117,178,213,138),(118,137,214,177),(119,176,215,136),(120,135,216,175),(121,174,217,134),(122,133,218,173),(123,172,219,132),(124,131,220,171),(125,170,221,130),(126,129,222,169),(127,224,223,128),(143,208,183,168),(144,167,184,207),(145,206,185,166),(146,165,186,205),(147,204,187,164),(148,163,188,203),(149,202,189,162),(150,161,190,201),(151,200,191,160),(152,159,192,199),(153,198,193,158),(154,157,194,197),(155,196,195,156)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 2 | ··· | 2 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C4○D8 | D28 | C4○D28 | D56⋊7C2 |
| kernel | C42.264D14 | C28.44D4 | C2.D56 | C4×C56 | C28.6Q8 | C4.D28 | C2×C28 | C4×C8 | C28 | C42 | C2×C8 | C14 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of C42.264D14 ►in GL4(𝔽113) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 30 |
| 0 | 0 | 98 | 15 |
| 17 | 8 | 0 | 0 |
| 105 | 96 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 1 | 112 |
| 23 | 56 | 0 | 0 |
| 57 | 11 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 31 | 51 |
| 56 | 23 | 0 | 0 |
| 11 | 57 | 0 | 0 |
| 0 | 0 | 51 | 62 |
| 0 | 0 | 82 | 62 |
G:=sub<GL(4,GF(113))| [98,0,0,0,0,98,0,0,0,0,98,98,0,0,30,15],[17,105,0,0,8,96,0,0,0,0,1,1,0,0,111,112],[23,57,0,0,56,11,0,0,0,0,0,31,0,0,51,51],[56,11,0,0,23,57,0,0,0,0,51,82,0,0,62,62] >;
C42.264D14 in GAP, Magma, Sage, TeX
C_4^2._{264}D_{14} % in TeX
G:=Group("C4^2.264D14"); // GroupNames label
G:=SmallGroup(448,231);
// by ID
G=gap.SmallGroup(448,231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2*b,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^13>;
// generators/relations