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## G = C3×C5⋊D16order 480 = 25·3·5

### Direct product of C3 and C5⋊D16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C3×C5⋊D16
 Chief series C1 — C5 — C10 — C20 — C40 — C120 — C3×D40 — C3×C5⋊D16
 Lower central C5 — C10 — C20 — C40 — C3×C5⋊D16
 Upper central C1 — C6 — C12 — C24 — C3×D8

Generators and relations for C3×C5⋊D16
G = < a,b,c,d | a3=b5=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C3×C5⋊D16
On 240 points
Generators in S240
(1 133 44)(2 134 45)(3 135 46)(4 136 47)(5 137 48)(6 138 33)(7 139 34)(8 140 35)(9 141 36)(10 142 37)(11 143 38)(12 144 39)(13 129 40)(14 130 41)(15 131 42)(16 132 43)(17 219 128)(18 220 113)(19 221 114)(20 222 115)(21 223 116)(22 224 117)(23 209 118)(24 210 119)(25 211 120)(26 212 121)(27 213 122)(28 214 123)(29 215 124)(30 216 125)(31 217 126)(32 218 127)(49 175 153)(50 176 154)(51 161 155)(52 162 156)(53 163 157)(54 164 158)(55 165 159)(56 166 160)(57 167 145)(58 168 146)(59 169 147)(60 170 148)(61 171 149)(62 172 150)(63 173 151)(64 174 152)(65 237 201)(66 238 202)(67 239 203)(68 240 204)(69 225 205)(70 226 206)(71 227 207)(72 228 208)(73 229 193)(74 230 194)(75 231 195)(76 232 196)(77 233 197)(78 234 198)(79 235 199)(80 236 200)(81 106 184)(82 107 185)(83 108 186)(84 109 187)(85 110 188)(86 111 189)(87 112 190)(88 97 191)(89 98 192)(90 99 177)(91 100 178)(92 101 179)(93 102 180)(94 103 181)(95 104 182)(96 105 183)
(1 165 186 80 224)(2 209 65 187 166)(3 167 188 66 210)(4 211 67 189 168)(5 169 190 68 212)(6 213 69 191 170)(7 171 192 70 214)(8 215 71 177 172)(9 173 178 72 216)(10 217 73 179 174)(11 175 180 74 218)(12 219 75 181 176)(13 161 182 76 220)(14 221 77 183 162)(15 163 184 78 222)(16 223 79 185 164)(17 195 103 50 39)(18 40 51 104 196)(19 197 105 52 41)(20 42 53 106 198)(21 199 107 54 43)(22 44 55 108 200)(23 201 109 56 45)(24 46 57 110 202)(25 203 111 58 47)(26 48 59 112 204)(27 205 97 60 33)(28 34 61 98 206)(29 207 99 62 35)(30 36 63 100 208)(31 193 101 64 37)(32 38 49 102 194)(81 234 115 131 157)(82 158 132 116 235)(83 236 117 133 159)(84 160 134 118 237)(85 238 119 135 145)(86 146 136 120 239)(87 240 121 137 147)(88 148 138 122 225)(89 226 123 139 149)(90 150 140 124 227)(91 228 125 141 151)(92 152 142 126 229)(93 230 127 143 153)(94 154 144 128 231)(95 232 113 129 155)(96 156 130 114 233)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 64)(30 63)(31 62)(32 61)(33 39)(34 38)(35 37)(40 48)(41 47)(42 46)(43 45)(65 185)(66 184)(67 183)(68 182)(69 181)(70 180)(71 179)(72 178)(73 177)(74 192)(75 191)(76 190)(77 189)(78 188)(79 187)(80 186)(81 238)(82 237)(83 236)(84 235)(85 234)(86 233)(87 232)(88 231)(89 230)(90 229)(91 228)(92 227)(93 226)(94 225)(95 240)(96 239)(97 195)(98 194)(99 193)(100 208)(101 207)(102 206)(103 205)(104 204)(105 203)(106 202)(107 201)(108 200)(109 199)(110 198)(111 197)(112 196)(113 147)(114 146)(115 145)(116 160)(117 159)(118 158)(119 157)(120 156)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 137)(130 136)(131 135)(132 134)(138 144)(139 143)(140 142)(161 212)(162 211)(163 210)(164 209)(165 224)(166 223)(167 222)(168 221)(169 220)(170 219)(171 218)(172 217)(173 216)(174 215)(175 214)(176 213)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4 5A 5B 6A 6B 6C 6D 6E 6F 8A 8B 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 15C 15D 16A 16B 16C 16D 20A 20B 24A 24B 24C 24D 30A 30B 30C 30D 30E ··· 30L 40A 40B 40C 40D 48A ··· 48H 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 3 4 5 5 6 6 6 6 6 6 8 8 10 10 10 10 10 10 12 12 15 15 15 15 16 16 16 16 20 20 24 24 24 24 30 30 30 30 30 ··· 30 40 40 40 40 48 ··· 48 60 60 60 60 120 ··· 120 size 1 1 8 40 1 1 2 2 2 1 1 8 8 40 40 2 2 2 2 8 8 8 8 2 2 2 2 2 2 10 10 10 10 4 4 2 2 2 2 2 2 2 2 8 ··· 8 4 4 4 4 10 ··· 10 4 4 4 4 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 D8 D10 C3×D4 C3×D5 D16 C5⋊D4 C3×D8 C6×D5 C3×D16 C3×C5⋊D4 D4⋊D5 C5⋊D16 C3×D4⋊D5 C3×C5⋊D16 kernel C3×C5⋊D16 C3×C5⋊2C16 C3×D40 C15×D8 C5⋊D16 C5⋊2C16 D40 C5×D8 C60 C3×D8 C30 C24 C20 D8 C15 C12 C10 C8 C5 C4 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 2 4 4 4 4 4 8 8 2 4 4 8

Matrix representation of C3×C5⋊D16 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 1 0 0 0 0 1
,
 189 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 96 27 0 0 96 145 0 0 0 0 156 27 0 0 214 156
,
 189 240 0 0 52 52 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,1,0,0,0,0,1],[189,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[96,96,0,0,27,145,0,0,0,0,156,214,0,0,27,156],[189,52,0,0,240,52,0,0,0,0,0,1,0,0,1,0] >;

C3×C5⋊D16 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes D_{16}
% in TeX

G:=Group("C3xC5:D16");
// GroupNames label

G:=SmallGroup(480,104);
// by ID

G=gap.SmallGroup(480,104);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,197,1011,514,192,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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