metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.6Dic7, C42.285D14, C28.30M4(2), (C2×C28)⋊6C8, (C4×C28).21C4, C28.38(C2×C8), C28⋊C8⋊37C2, (C2×C42).11D7, (C22×C28).26C4, C14.23(C22×C8), C28.248(C4○D4), C4.132(C4○D28), (C2×C28).843C23, (C4×C28).345C22, C7⋊4(C42.12C4), (C22×C4).392D14, C14.38(C2×M4(2)), C4.12(C4.Dic7), C23.26(C2×Dic7), (C22×C4).16Dic7, C28.55D4.18C2, C14.39(C42⋊C2), (C22×C28).553C22, C22.15(C22×Dic7), C2.1(C23.21D14), (C4×C7⋊C8)⋊24C2, (C2×C4)⋊4(C7⋊C8), C4.16(C2×C7⋊C8), (C2×C4×C28).19C2, C22.5(C2×C7⋊C8), C2.4(C22×C7⋊C8), (C2×C14).35(C2×C8), (C2×C28).315(C2×C4), C2.4(C2×C4.Dic7), (C2×C7⋊C8).312C22, (C2×C4).98(C2×Dic7), (C2×C4).785(C22×D7), (C22×C14).131(C2×C4), (C2×C14).172(C22×C4), SmallGroup(448,459)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.6Dic7
G = < a,b,c,d | a4=b4=1, c14=b2, d2=b2c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c13 >
Subgroups: 260 in 118 conjugacy classes, 79 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C42.12C4, C2×C7⋊C8, C4×C28, C22×C28, C4×C7⋊C8, C28⋊C8, C28.55D4, C2×C4×C28, C42.6Dic7
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, M4(2), C22×C4, C4○D4, Dic7, D14, C42⋊C2, C22×C8, C2×M4(2), C7⋊C8, C2×Dic7, C22×D7, C42.12C4, C2×C7⋊C8, C4.Dic7, C4○D28, C22×Dic7, C22×C7⋊C8, C2×C4.Dic7, C23.21D14, C42.6Dic7
►On 224 points
Generators in S224
(1 106 182 203)(2 107 183 204)(3 108 184 205)(4 109 185 206)(5 110 186 207)(6 111 187 208)(7 112 188 209)(8 85 189 210)(9 86 190 211)(10 87 191 212)(11 88 192 213)(12 89 193 214)(13 90 194 215)(14 91 195 216)(15 92 196 217)(16 93 169 218)(17 94 170 219)(18 95 171 220)(19 96 172 221)(20 97 173 222)(21 98 174 223)(22 99 175 224)(23 100 176 197)(24 101 177 198)(25 102 178 199)(26 103 179 200)(27 104 180 201)(28 105 181 202)(29 113 151 80)(30 114 152 81)(31 115 153 82)(32 116 154 83)(33 117 155 84)(34 118 156 57)(35 119 157 58)(36 120 158 59)(37 121 159 60)(38 122 160 61)(39 123 161 62)(40 124 162 63)(41 125 163 64)(42 126 164 65)(43 127 165 66)(44 128 166 67)(45 129 167 68)(46 130 168 69)(47 131 141 70)(48 132 142 71)(49 133 143 72)(50 134 144 73)(51 135 145 74)(52 136 146 75)(53 137 147 76)(54 138 148 77)(55 139 149 78)(56 140 150 79)
(1 8 15 22)(2 9 16 23)(3 10 17 24)(4 11 18 25)(5 12 19 26)(6 13 20 27)(7 14 21 28)(29 158 43 144)(30 159 44 145)(31 160 45 146)(32 161 46 147)(33 162 47 148)(34 163 48 149)(35 164 49 150)(36 165 50 151)(37 166 51 152)(38 167 52 153)(39 168 53 154)(40 141 54 155)(41 142 55 156)(42 143 56 157)(57 125 71 139)(58 126 72 140)(59 127 73 113)(60 128 74 114)(61 129 75 115)(62 130 76 116)(63 131 77 117)(64 132 78 118)(65 133 79 119)(66 134 80 120)(67 135 81 121)(68 136 82 122)(69 137 83 123)(70 138 84 124)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)(169 176 183 190)(170 177 184 191)(171 178 185 192)(172 179 186 193)(173 180 187 194)(174 181 188 195)(175 182 189 196)(197 204 211 218)(198 205 212 219)(199 206 213 220)(200 207 214 221)(201 208 215 222)(202 209 216 223)(203 210 217 224)
(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 32 22 53 15 46 8 39)(2 45 23 38 16 31 9 52)(3 30 24 51 17 44 10 37)(4 43 25 36 18 29 11 50)(5 56 26 49 19 42 12 35)(6 41 27 34 20 55 13 48)(7 54 28 47 21 40 14 33)(57 222 78 215 71 208 64 201)(58 207 79 200 72 221 65 214)(59 220 80 213 73 206 66 199)(60 205 81 198 74 219 67 212)(61 218 82 211 75 204 68 197)(62 203 83 224 76 217 69 210)(63 216 84 209 77 202 70 223)(85 123 106 116 99 137 92 130)(86 136 107 129 100 122 93 115)(87 121 108 114 101 135 94 128)(88 134 109 127 102 120 95 113)(89 119 110 140 103 133 96 126)(90 132 111 125 104 118 97 139)(91 117 112 138 105 131 98 124)(141 174 162 195 155 188 148 181)(142 187 163 180 156 173 149 194)(143 172 164 193 157 186 150 179)(144 185 165 178 158 171 151 192)(145 170 166 191 159 184 152 177)(146 183 167 176 160 169 153 190)(147 196 168 189 161 182 154 175)
G:=sub<Sym(224)| (1,106,182,203)(2,107,183,204)(3,108,184,205)(4,109,185,206)(5,110,186,207)(6,111,187,208)(7,112,188,209)(8,85,189,210)(9,86,190,211)(10,87,191,212)(11,88,192,213)(12,89,193,214)(13,90,194,215)(14,91,195,216)(15,92,196,217)(16,93,169,218)(17,94,170,219)(18,95,171,220)(19,96,172,221)(20,97,173,222)(21,98,174,223)(22,99,175,224)(23,100,176,197)(24,101,177,198)(25,102,178,199)(26,103,179,200)(27,104,180,201)(28,105,181,202)(29,113,151,80)(30,114,152,81)(31,115,153,82)(32,116,154,83)(33,117,155,84)(34,118,156,57)(35,119,157,58)(36,120,158,59)(37,121,159,60)(38,122,160,61)(39,123,161,62)(40,124,162,63)(41,125,163,64)(42,126,164,65)(43,127,165,66)(44,128,166,67)(45,129,167,68)(46,130,168,69)(47,131,141,70)(48,132,142,71)(49,133,143,72)(50,134,144,73)(51,135,145,74)(52,136,146,75)(53,137,147,76)(54,138,148,77)(55,139,149,78)(56,140,150,79), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,158,43,144)(30,159,44,145)(31,160,45,146)(32,161,46,147)(33,162,47,148)(34,163,48,149)(35,164,49,150)(36,165,50,151)(37,166,51,152)(38,167,52,153)(39,168,53,154)(40,141,54,155)(41,142,55,156)(42,143,56,157)(57,125,71,139)(58,126,72,140)(59,127,73,113)(60,128,74,114)(61,129,75,115)(62,130,76,116)(63,131,77,117)(64,132,78,118)(65,133,79,119)(66,134,80,120)(67,135,81,121)(68,136,82,122)(69,137,83,123)(70,138,84,124)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112)(169,176,183,190)(170,177,184,191)(171,178,185,192)(172,179,186,193)(173,180,187,194)(174,181,188,195)(175,182,189,196)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (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), (1,32,22,53,15,46,8,39)(2,45,23,38,16,31,9,52)(3,30,24,51,17,44,10,37)(4,43,25,36,18,29,11,50)(5,56,26,49,19,42,12,35)(6,41,27,34,20,55,13,48)(7,54,28,47,21,40,14,33)(57,222,78,215,71,208,64,201)(58,207,79,200,72,221,65,214)(59,220,80,213,73,206,66,199)(60,205,81,198,74,219,67,212)(61,218,82,211,75,204,68,197)(62,203,83,224,76,217,69,210)(63,216,84,209,77,202,70,223)(85,123,106,116,99,137,92,130)(86,136,107,129,100,122,93,115)(87,121,108,114,101,135,94,128)(88,134,109,127,102,120,95,113)(89,119,110,140,103,133,96,126)(90,132,111,125,104,118,97,139)(91,117,112,138,105,131,98,124)(141,174,162,195,155,188,148,181)(142,187,163,180,156,173,149,194)(143,172,164,193,157,186,150,179)(144,185,165,178,158,171,151,192)(145,170,166,191,159,184,152,177)(146,183,167,176,160,169,153,190)(147,196,168,189,161,182,154,175)>;
G:=Group( (1,106,182,203)(2,107,183,204)(3,108,184,205)(4,109,185,206)(5,110,186,207)(6,111,187,208)(7,112,188,209)(8,85,189,210)(9,86,190,211)(10,87,191,212)(11,88,192,213)(12,89,193,214)(13,90,194,215)(14,91,195,216)(15,92,196,217)(16,93,169,218)(17,94,170,219)(18,95,171,220)(19,96,172,221)(20,97,173,222)(21,98,174,223)(22,99,175,224)(23,100,176,197)(24,101,177,198)(25,102,178,199)(26,103,179,200)(27,104,180,201)(28,105,181,202)(29,113,151,80)(30,114,152,81)(31,115,153,82)(32,116,154,83)(33,117,155,84)(34,118,156,57)(35,119,157,58)(36,120,158,59)(37,121,159,60)(38,122,160,61)(39,123,161,62)(40,124,162,63)(41,125,163,64)(42,126,164,65)(43,127,165,66)(44,128,166,67)(45,129,167,68)(46,130,168,69)(47,131,141,70)(48,132,142,71)(49,133,143,72)(50,134,144,73)(51,135,145,74)(52,136,146,75)(53,137,147,76)(54,138,148,77)(55,139,149,78)(56,140,150,79), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,158,43,144)(30,159,44,145)(31,160,45,146)(32,161,46,147)(33,162,47,148)(34,163,48,149)(35,164,49,150)(36,165,50,151)(37,166,51,152)(38,167,52,153)(39,168,53,154)(40,141,54,155)(41,142,55,156)(42,143,56,157)(57,125,71,139)(58,126,72,140)(59,127,73,113)(60,128,74,114)(61,129,75,115)(62,130,76,116)(63,131,77,117)(64,132,78,118)(65,133,79,119)(66,134,80,120)(67,135,81,121)(68,136,82,122)(69,137,83,123)(70,138,84,124)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112)(169,176,183,190)(170,177,184,191)(171,178,185,192)(172,179,186,193)(173,180,187,194)(174,181,188,195)(175,182,189,196)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (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), (1,32,22,53,15,46,8,39)(2,45,23,38,16,31,9,52)(3,30,24,51,17,44,10,37)(4,43,25,36,18,29,11,50)(5,56,26,49,19,42,12,35)(6,41,27,34,20,55,13,48)(7,54,28,47,21,40,14,33)(57,222,78,215,71,208,64,201)(58,207,79,200,72,221,65,214)(59,220,80,213,73,206,66,199)(60,205,81,198,74,219,67,212)(61,218,82,211,75,204,68,197)(62,203,83,224,76,217,69,210)(63,216,84,209,77,202,70,223)(85,123,106,116,99,137,92,130)(86,136,107,129,100,122,93,115)(87,121,108,114,101,135,94,128)(88,134,109,127,102,120,95,113)(89,119,110,140,103,133,96,126)(90,132,111,125,104,118,97,139)(91,117,112,138,105,131,98,124)(141,174,162,195,155,188,148,181)(142,187,163,180,156,173,149,194)(143,172,164,193,157,186,150,179)(144,185,165,178,158,171,151,192)(145,170,166,191,159,184,152,177)(146,183,167,176,160,169,153,190)(147,196,168,189,161,182,154,175) );
G=PermutationGroup([[(1,106,182,203),(2,107,183,204),(3,108,184,205),(4,109,185,206),(5,110,186,207),(6,111,187,208),(7,112,188,209),(8,85,189,210),(9,86,190,211),(10,87,191,212),(11,88,192,213),(12,89,193,214),(13,90,194,215),(14,91,195,216),(15,92,196,217),(16,93,169,218),(17,94,170,219),(18,95,171,220),(19,96,172,221),(20,97,173,222),(21,98,174,223),(22,99,175,224),(23,100,176,197),(24,101,177,198),(25,102,178,199),(26,103,179,200),(27,104,180,201),(28,105,181,202),(29,113,151,80),(30,114,152,81),(31,115,153,82),(32,116,154,83),(33,117,155,84),(34,118,156,57),(35,119,157,58),(36,120,158,59),(37,121,159,60),(38,122,160,61),(39,123,161,62),(40,124,162,63),(41,125,163,64),(42,126,164,65),(43,127,165,66),(44,128,166,67),(45,129,167,68),(46,130,168,69),(47,131,141,70),(48,132,142,71),(49,133,143,72),(50,134,144,73),(51,135,145,74),(52,136,146,75),(53,137,147,76),(54,138,148,77),(55,139,149,78),(56,140,150,79)], [(1,8,15,22),(2,9,16,23),(3,10,17,24),(4,11,18,25),(5,12,19,26),(6,13,20,27),(7,14,21,28),(29,158,43,144),(30,159,44,145),(31,160,45,146),(32,161,46,147),(33,162,47,148),(34,163,48,149),(35,164,49,150),(36,165,50,151),(37,166,51,152),(38,167,52,153),(39,168,53,154),(40,141,54,155),(41,142,55,156),(42,143,56,157),(57,125,71,139),(58,126,72,140),(59,127,73,113),(60,128,74,114),(61,129,75,115),(62,130,76,116),(63,131,77,117),(64,132,78,118),(65,133,79,119),(66,134,80,120),(67,135,81,121),(68,136,82,122),(69,137,83,123),(70,138,84,124),(85,92,99,106),(86,93,100,107),(87,94,101,108),(88,95,102,109),(89,96,103,110),(90,97,104,111),(91,98,105,112),(169,176,183,190),(170,177,184,191),(171,178,185,192),(172,179,186,193),(173,180,187,194),(174,181,188,195),(175,182,189,196),(197,204,211,218),(198,205,212,219),(199,206,213,220),(200,207,214,221),(201,208,215,222),(202,209,216,223),(203,210,217,224)], [(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)], [(1,32,22,53,15,46,8,39),(2,45,23,38,16,31,9,52),(3,30,24,51,17,44,10,37),(4,43,25,36,18,29,11,50),(5,56,26,49,19,42,12,35),(6,41,27,34,20,55,13,48),(7,54,28,47,21,40,14,33),(57,222,78,215,71,208,64,201),(58,207,79,200,72,221,65,214),(59,220,80,213,73,206,66,199),(60,205,81,198,74,219,67,212),(61,218,82,211,75,204,68,197),(62,203,83,224,76,217,69,210),(63,216,84,209,77,202,70,223),(85,123,106,116,99,137,92,130),(86,136,107,129,100,122,93,115),(87,121,108,114,101,135,94,128),(88,134,109,127,102,120,95,113),(89,119,110,140,103,133,96,126),(90,132,111,125,104,118,97,139),(91,117,112,138,105,131,98,124),(141,174,162,195,155,188,148,181),(142,187,163,180,156,173,149,194),(143,172,164,193,157,186,150,179),(144,185,165,178,158,171,151,192),(145,170,166,191,159,184,152,177),(146,183,167,176,160,169,153,190),(147,196,168,189,161,182,154,175)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 7A | 7B | 7C | 8A | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D7 | M4(2) | C4○D4 | Dic7 | D14 | Dic7 | D14 | C7⋊C8 | C4.Dic7 | C4○D28 |
| kernel | C42.6Dic7 | C4×C7⋊C8 | C28⋊C8 | C28.55D4 | C2×C4×C28 | C4×C28 | C22×C28 | C2×C28 | C2×C42 | C28 | C28 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 16 | 3 | 4 | 4 | 6 | 6 | 6 | 3 | 24 | 24 | 24 |
Matrix representation of C42.6Dic7 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 98 |
| 98 | 0 | 0 |
| 0 | 98 | 74 |
| 0 | 0 | 15 |
| 98 | 0 | 0 |
| 0 | 53 | 54 |
| 0 | 0 | 81 |
| 18 | 0 | 0 |
| 0 | 95 | 21 |
| 0 | 66 | 18 |
G:=sub<GL(3,GF(113))| [112,0,0,0,98,0,0,0,98],[98,0,0,0,98,0,0,74,15],[98,0,0,0,53,0,0,54,81],[18,0,0,0,95,66,0,21,18] >;
C42.6Dic7 in GAP, Magma, Sage, TeX
C_4^2._6{\rm Dic}_7 % in TeX
G:=Group("C4^2.6Dic7"); // GroupNames label
G:=SmallGroup(448,459);
// by ID
G=gap.SmallGroup(448,459);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,120,422,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^13>;
// generators/relations