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## G = C5×Dic3⋊C4order 240 = 24·3·5

### Direct product of C5 and Dic3⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×Dic3⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — C5×Dic3⋊C4
 Lower central C3 — C6 — C5×Dic3⋊C4
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C5×Dic3⋊C4
G = < a,b,c,d | a5=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Smallest permutation representation of C5×Dic3⋊C4
Regular action on 240 points
Generators in S240
(1 49 37 25 13)(2 50 38 26 14)(3 51 39 27 15)(4 52 40 28 16)(5 53 41 29 17)(6 54 42 30 18)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(121 169 157 145 133)(122 170 158 146 134)(123 171 159 147 135)(124 172 160 148 136)(125 173 161 149 137)(126 174 162 150 138)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(181 229 217 205 193)(182 230 218 206 194)(183 231 219 207 195)(184 232 220 208 196)(185 233 221 209 197)(186 234 222 210 198)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(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)
(1 128 4 131)(2 127 5 130)(3 132 6 129)(7 122 10 125)(8 121 11 124)(9 126 12 123)(13 140 16 143)(14 139 17 142)(15 144 18 141)(19 134 22 137)(20 133 23 136)(21 138 24 135)(25 152 28 155)(26 151 29 154)(27 156 30 153)(31 146 34 149)(32 145 35 148)(33 150 36 147)(37 164 40 167)(38 163 41 166)(39 168 42 165)(43 158 46 161)(44 157 47 160)(45 162 48 159)(49 176 52 179)(50 175 53 178)(51 180 54 177)(55 170 58 173)(56 169 59 172)(57 174 60 171)(61 188 64 191)(62 187 65 190)(63 192 66 189)(67 182 70 185)(68 181 71 184)(69 186 72 183)(73 200 76 203)(74 199 77 202)(75 204 78 201)(79 194 82 197)(80 193 83 196)(81 198 84 195)(85 212 88 215)(86 211 89 214)(87 216 90 213)(91 206 94 209)(92 205 95 208)(93 210 96 207)(97 224 100 227)(98 223 101 226)(99 228 102 225)(103 218 106 221)(104 217 107 220)(105 222 108 219)(109 236 112 239)(110 235 113 238)(111 240 114 237)(115 230 118 233)(116 229 119 232)(117 234 120 231)
(1 67 7 61)(2 68 8 62)(3 69 9 63)(4 70 10 64)(5 71 11 65)(6 72 12 66)(13 79 19 73)(14 80 20 74)(15 81 21 75)(16 82 22 76)(17 83 23 77)(18 84 24 78)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 103 43 97)(38 104 44 98)(39 105 45 99)(40 106 46 100)(41 107 47 101)(42 108 48 102)(49 115 55 109)(50 116 56 110)(51 117 57 111)(52 118 58 112)(53 119 59 113)(54 120 60 114)(121 190 127 184)(122 191 128 185)(123 192 129 186)(124 187 130 181)(125 188 131 182)(126 189 132 183)(133 202 139 196)(134 203 140 197)(135 204 141 198)(136 199 142 193)(137 200 143 194)(138 201 144 195)(145 214 151 208)(146 215 152 209)(147 216 153 210)(148 211 154 205)(149 212 155 206)(150 213 156 207)(157 226 163 220)(158 227 164 221)(159 228 165 222)(160 223 166 217)(161 224 167 218)(162 225 168 219)(169 238 175 232)(170 239 176 233)(171 240 177 234)(172 235 178 229)(173 236 179 230)(174 237 180 231)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I ··· 20X 30A ··· 30L 60A ··· 60P order 1 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 6 6 6 6 1 1 1 1 2 2 2 1 ··· 1 2 2 2 2 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C4 C5 C10 C10 C20 S3 D4 Q8 D6 Dic6 C4×S3 C3⋊D4 C5×S3 C5×D4 C5×Q8 S3×C10 C5×Dic6 S3×C20 C5×C3⋊D4 kernel C5×Dic3⋊C4 C10×Dic3 C2×C60 C5×Dic3 Dic3⋊C4 C2×Dic3 C2×C12 Dic3 C2×C20 C30 C30 C2×C10 C10 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 4 4 8 4 16 1 1 1 1 2 2 2 4 4 4 4 8 8 8

Matrix representation of C5×Dic3⋊C4 in GL5(𝔽61)

 1 0 0 0 0 0 58 0 0 0 0 0 58 0 0 0 0 0 20 0 0 0 0 0 20
,
 1 0 0 0 0 0 0 60 0 0 0 1 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 60 0 0 0 0 0 1 60 0 0 0 0 60 0 0 0 0 0 58 4 0 0 0 28 3
,
 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 59 0 0 0 0 50

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,58,0,0,0,0,0,58,0,0,0,0,0,20,0,0,0,0,0,20],[1,0,0,0,0,0,0,1,0,0,0,60,60,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,58,28,0,0,0,4,3],[11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,59,50] >;

C5×Dic3⋊C4 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("C5xDic3:C4");
// GroupNames label

G:=SmallGroup(240,57);
// by ID

G=gap.SmallGroup(240,57);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,240,505,127,5765]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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