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## G = C3×D40⋊7C2order 480 = 25·3·5

### Direct product of C3 and D40⋊7C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D40⋊7C2
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C3×C4○D20 — C3×D40⋊7C2
 Lower central C5 — C10 — C20 — C3×D40⋊7C2
 Upper central C1 — C12 — C2×C12 — C2×C24

Generators and relations for C3×D407C2
G = < a,b,c,d | a3=b40=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b20c >

Subgroups: 464 in 124 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×4], C3×Q8 [×2], C3×D5 [×2], C30, C30, C4○D8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C3×C4○D4 [×2], C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C2×C30, C40⋊C2 [×2], D40, Dic20, C2×C40, C4○D20 [×2], C3×C4○D8, C120 [×2], C3×Dic10 [×2], D5×C12 [×2], C3×D20 [×2], C3×C5⋊D4 [×2], C2×C60, D407C2, C3×C40⋊C2 [×2], C3×D40, C3×Dic20, C2×C120, C3×C4○D20 [×2], C3×D407C2
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C4○D8, D20 [×2], C22×D5, C6×D4, C6×D5 [×3], C2×D20, C3×C4○D8, C3×D20 [×2], D5×C2×C6, D407C2, C6×D20, C3×D407C2

Smallest permutation representation of C3×D407C2
On 240 points
Generators in S240
(1 219 42)(2 220 43)(3 221 44)(4 222 45)(5 223 46)(6 224 47)(7 225 48)(8 226 49)(9 227 50)(10 228 51)(11 229 52)(12 230 53)(13 231 54)(14 232 55)(15 233 56)(16 234 57)(17 235 58)(18 236 59)(19 237 60)(20 238 61)(21 239 62)(22 240 63)(23 201 64)(24 202 65)(25 203 66)(26 204 67)(27 205 68)(28 206 69)(29 207 70)(30 208 71)(31 209 72)(32 210 73)(33 211 74)(34 212 75)(35 213 76)(36 214 77)(37 215 78)(38 216 79)(39 217 80)(40 218 41)(81 143 171)(82 144 172)(83 145 173)(84 146 174)(85 147 175)(86 148 176)(87 149 177)(88 150 178)(89 151 179)(90 152 180)(91 153 181)(92 154 182)(93 155 183)(94 156 184)(95 157 185)(96 158 186)(97 159 187)(98 160 188)(99 121 189)(100 122 190)(101 123 191)(102 124 192)(103 125 193)(104 126 194)(105 127 195)(106 128 196)(107 129 197)(108 130 198)(109 131 199)(110 132 200)(111 133 161)(112 134 162)(113 135 163)(114 136 164)(115 137 165)(116 138 166)(117 139 167)(118 140 168)(119 141 169)(120 142 170)
(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)
(1 150)(2 149)(3 148)(4 147)(5 146)(6 145)(7 144)(8 143)(9 142)(10 141)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(161 236)(162 235)(163 234)(164 233)(165 232)(166 231)(167 230)(168 229)(169 228)(170 227)(171 226)(172 225)(173 224)(174 223)(175 222)(176 221)(177 220)(178 219)(179 218)(180 217)(181 216)(182 215)(183 214)(184 213)(185 212)(186 211)(187 210)(188 209)(189 208)(190 207)(191 206)(192 205)(193 204)(194 203)(195 202)(196 201)(197 240)(198 239)(199 238)(200 237)
(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)(161 181)(162 182)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 189)(170 190)(171 191)(172 192)(173 193)(174 194)(175 195)(176 196)(177 197)(178 198)(179 199)(180 200)

G:=sub<Sym(240)| (1,219,42)(2,220,43)(3,221,44)(4,222,45)(5,223,46)(6,224,47)(7,225,48)(8,226,49)(9,227,50)(10,228,51)(11,229,52)(12,230,53)(13,231,54)(14,232,55)(15,233,56)(16,234,57)(17,235,58)(18,236,59)(19,237,60)(20,238,61)(21,239,62)(22,240,63)(23,201,64)(24,202,65)(25,203,66)(26,204,67)(27,205,68)(28,206,69)(29,207,70)(30,208,71)(31,209,72)(32,210,73)(33,211,74)(34,212,75)(35,213,76)(36,214,77)(37,215,78)(38,216,79)(39,217,80)(40,218,41)(81,143,171)(82,144,172)(83,145,173)(84,146,174)(85,147,175)(86,148,176)(87,149,177)(88,150,178)(89,151,179)(90,152,180)(91,153,181)(92,154,182)(93,155,183)(94,156,184)(95,157,185)(96,158,186)(97,159,187)(98,160,188)(99,121,189)(100,122,190)(101,123,191)(102,124,192)(103,125,193)(104,126,194)(105,127,195)(106,128,196)(107,129,197)(108,130,198)(109,131,199)(110,132,200)(111,133,161)(112,134,162)(113,135,163)(114,136,164)(115,137,165)(116,138,166)(117,139,167)(118,140,168)(119,141,169)(120,142,170), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(161,236)(162,235)(163,234)(164,233)(165,232)(166,231)(167,230)(168,229)(169,228)(170,227)(171,226)(172,225)(173,224)(174,223)(175,222)(176,221)(177,220)(178,219)(179,218)(180,217)(181,216)(182,215)(183,214)(184,213)(185,212)(186,211)(187,210)(188,209)(189,208)(190,207)(191,206)(192,205)(193,204)(194,203)(195,202)(196,201)(197,240)(198,239)(199,238)(200,237), (81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(161,181)(162,182)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200)>;

G:=Group( (1,219,42)(2,220,43)(3,221,44)(4,222,45)(5,223,46)(6,224,47)(7,225,48)(8,226,49)(9,227,50)(10,228,51)(11,229,52)(12,230,53)(13,231,54)(14,232,55)(15,233,56)(16,234,57)(17,235,58)(18,236,59)(19,237,60)(20,238,61)(21,239,62)(22,240,63)(23,201,64)(24,202,65)(25,203,66)(26,204,67)(27,205,68)(28,206,69)(29,207,70)(30,208,71)(31,209,72)(32,210,73)(33,211,74)(34,212,75)(35,213,76)(36,214,77)(37,215,78)(38,216,79)(39,217,80)(40,218,41)(81,143,171)(82,144,172)(83,145,173)(84,146,174)(85,147,175)(86,148,176)(87,149,177)(88,150,178)(89,151,179)(90,152,180)(91,153,181)(92,154,182)(93,155,183)(94,156,184)(95,157,185)(96,158,186)(97,159,187)(98,160,188)(99,121,189)(100,122,190)(101,123,191)(102,124,192)(103,125,193)(104,126,194)(105,127,195)(106,128,196)(107,129,197)(108,130,198)(109,131,199)(110,132,200)(111,133,161)(112,134,162)(113,135,163)(114,136,164)(115,137,165)(116,138,166)(117,139,167)(118,140,168)(119,141,169)(120,142,170), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(161,236)(162,235)(163,234)(164,233)(165,232)(166,231)(167,230)(168,229)(169,228)(170,227)(171,226)(172,225)(173,224)(174,223)(175,222)(176,221)(177,220)(178,219)(179,218)(180,217)(181,216)(182,215)(183,214)(184,213)(185,212)(186,211)(187,210)(188,209)(189,208)(190,207)(191,206)(192,205)(193,204)(194,203)(195,202)(196,201)(197,240)(198,239)(199,238)(200,237), (81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(161,181)(162,182)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,189)(170,190)(171,191)(172,192)(173,193)(174,194)(175,195)(176,196)(177,197)(178,198)(179,199)(180,200) );

G=PermutationGroup([(1,219,42),(2,220,43),(3,221,44),(4,222,45),(5,223,46),(6,224,47),(7,225,48),(8,226,49),(9,227,50),(10,228,51),(11,229,52),(12,230,53),(13,231,54),(14,232,55),(15,233,56),(16,234,57),(17,235,58),(18,236,59),(19,237,60),(20,238,61),(21,239,62),(22,240,63),(23,201,64),(24,202,65),(25,203,66),(26,204,67),(27,205,68),(28,206,69),(29,207,70),(30,208,71),(31,209,72),(32,210,73),(33,211,74),(34,212,75),(35,213,76),(36,214,77),(37,215,78),(38,216,79),(39,217,80),(40,218,41),(81,143,171),(82,144,172),(83,145,173),(84,146,174),(85,147,175),(86,148,176),(87,149,177),(88,150,178),(89,151,179),(90,152,180),(91,153,181),(92,154,182),(93,155,183),(94,156,184),(95,157,185),(96,158,186),(97,159,187),(98,160,188),(99,121,189),(100,122,190),(101,123,191),(102,124,192),(103,125,193),(104,126,194),(105,127,195),(106,128,196),(107,129,197),(108,130,198),(109,131,199),(110,132,200),(111,133,161),(112,134,162),(113,135,163),(114,136,164),(115,137,165),(116,138,166),(117,139,167),(118,140,168),(119,141,169),(120,142,170)], [(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)], [(1,150),(2,149),(3,148),(4,147),(5,146),(6,145),(7,144),(8,143),(9,142),(10,141),(11,140),(12,139),(13,138),(14,137),(15,136),(16,135),(17,134),(18,133),(19,132),(20,131),(21,130),(22,129),(23,128),(24,127),(25,126),(26,125),(27,124),(28,123),(29,122),(30,121),(31,160),(32,159),(33,158),(34,157),(35,156),(36,155),(37,154),(38,153),(39,152),(40,151),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,81),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(161,236),(162,235),(163,234),(164,233),(165,232),(166,231),(167,230),(168,229),(169,228),(170,227),(171,226),(172,225),(173,224),(174,223),(175,222),(176,221),(177,220),(178,219),(179,218),(180,217),(181,216),(182,215),(183,214),(184,213),(185,212),(186,211),(187,210),(188,209),(189,208),(190,207),(191,206),(192,205),(193,204),(194,203),(195,202),(196,201),(197,240),(198,239),(199,238),(200,237)], [(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160),(161,181),(162,182),(163,183),(164,184),(165,185),(166,186),(167,187),(168,188),(169,189),(170,190),(171,191),(172,192),(173,193),(174,194),(175,195),(176,196),(177,197),(178,198),(179,199),(180,200)])

138 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 3 3 4 4 4 4 4 5 5 6 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 2 20 20 1 1 1 1 2 20 20 2 2 1 1 2 2 20 20 20 20 2 2 2 2 2 ··· 2 1 1 1 1 2 2 20 20 20 20 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

138 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C4○D8 D20 D20 C6×D5 C6×D5 C3×C4○D8 C3×D20 C3×D20 D40⋊7C2 C3×D40⋊7C2 kernel C3×D40⋊7C2 C3×C40⋊C2 C3×D40 C3×Dic20 C2×C120 C3×C4○D20 D40⋊7C2 C40⋊C2 D40 Dic20 C2×C40 C4○D20 C60 C2×C30 C2×C24 C24 C2×C12 C20 C2×C10 C2×C8 C15 C12 C2×C6 C8 C2×C4 C5 C4 C22 C3 C1 # reps 1 2 1 1 1 2 2 4 2 2 2 4 1 1 2 4 2 2 2 4 4 4 4 8 4 8 8 8 16 32

Matrix representation of C3×D407C2 in GL2(𝔽241) generated by

 225 0 0 225
,
 27 0 75 125
,
 23 210 87 218
,
 1 0 87 240
G:=sub<GL(2,GF(241))| [225,0,0,225],[27,75,0,125],[23,87,210,218],[1,87,0,240] >;

C3×D407C2 in GAP, Magma, Sage, TeX

C_3\times D_{40}\rtimes_7C_2
% in TeX

G:=Group("C3xD40:7C2");
// GroupNames label

G:=SmallGroup(480,697);
// by ID

G=gap.SmallGroup(480,697);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,142,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^40=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^20*c>;
// generators/relations

׿
×
𝔽