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## G = C3×C5⋊SD32order 480 = 25·3·5

### Direct product of C3 and C5⋊SD32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C3×C5⋊SD32
 Chief series C1 — C5 — C10 — C20 — C40 — C120 — C3×D40 — C3×C5⋊SD32
 Lower central C5 — C10 — C20 — C40 — C3×C5⋊SD32
 Upper central C1 — C6 — C12 — C24 — C3×Q16

Generators and relations for C3×C5⋊SD32
G = < a,b,c,d | a3=b5=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c7 >

Smallest permutation representation of C3×C5⋊SD32
On 240 points
Generators in S240
(1 150 89)(2 151 90)(3 152 91)(4 153 92)(5 154 93)(6 155 94)(7 156 95)(8 157 96)(9 158 81)(10 159 82)(11 160 83)(12 145 84)(13 146 85)(14 147 86)(15 148 87)(16 149 88)(17 204 188)(18 205 189)(19 206 190)(20 207 191)(21 208 192)(22 193 177)(23 194 178)(24 195 179)(25 196 180)(26 197 181)(27 198 182)(28 199 183)(29 200 184)(30 201 185)(31 202 186)(32 203 187)(33 212 72)(34 213 73)(35 214 74)(36 215 75)(37 216 76)(38 217 77)(39 218 78)(40 219 79)(41 220 80)(42 221 65)(43 222 66)(44 223 67)(45 224 68)(46 209 69)(47 210 70)(48 211 71)(49 109 174)(50 110 175)(51 111 176)(52 112 161)(53 97 162)(54 98 163)(55 99 164)(56 100 165)(57 101 166)(58 102 167)(59 103 168)(60 104 169)(61 105 170)(62 106 171)(63 107 172)(64 108 173)(113 235 131)(114 236 132)(115 237 133)(116 238 134)(117 239 135)(118 240 136)(119 225 137)(120 226 138)(121 227 139)(122 228 140)(123 229 141)(124 230 142)(125 231 143)(126 232 144)(127 233 129)(128 234 130)
(1 205 61 133 220)(2 221 134 62 206)(3 207 63 135 222)(4 223 136 64 208)(5 193 49 137 224)(6 209 138 50 194)(7 195 51 139 210)(8 211 140 52 196)(9 197 53 141 212)(10 213 142 54 198)(11 199 55 143 214)(12 215 144 56 200)(13 201 57 129 216)(14 217 130 58 202)(15 203 59 131 218)(16 219 132 60 204)(17 88 40 236 169)(18 170 237 41 89)(19 90 42 238 171)(20 172 239 43 91)(21 92 44 240 173)(22 174 225 45 93)(23 94 46 226 175)(24 176 227 47 95)(25 96 48 228 161)(26 162 229 33 81)(27 82 34 230 163)(28 164 231 35 83)(29 84 36 232 165)(30 166 233 37 85)(31 86 38 234 167)(32 168 235 39 87)(65 116 106 190 151)(66 152 191 107 117)(67 118 108 192 153)(68 154 177 109 119)(69 120 110 178 155)(70 156 179 111 121)(71 122 112 180 157)(72 158 181 97 123)(73 124 98 182 159)(74 160 183 99 125)(75 126 100 184 145)(76 146 185 101 127)(77 128 102 186 147)(78 148 187 103 113)(79 114 104 188 149)(80 150 189 105 115)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 34)(18 41)(19 48)(20 39)(21 46)(22 37)(23 44)(24 35)(25 42)(26 33)(27 40)(28 47)(29 38)(30 45)(31 36)(32 43)(49 129)(50 136)(51 143)(52 134)(53 141)(54 132)(55 139)(56 130)(57 137)(58 144)(59 135)(60 142)(61 133)(62 140)(63 131)(64 138)(65 180)(66 187)(67 178)(68 185)(69 192)(70 183)(71 190)(72 181)(73 188)(74 179)(75 186)(76 177)(77 184)(78 191)(79 182)(80 189)(82 88)(83 95)(84 86)(85 93)(87 91)(90 96)(92 94)(97 123)(98 114)(99 121)(100 128)(101 119)(102 126)(103 117)(104 124)(105 115)(106 122)(107 113)(108 120)(109 127)(110 118)(111 125)(112 116)(145 147)(146 154)(148 152)(149 159)(151 157)(153 155)(156 160)(161 238)(162 229)(163 236)(164 227)(165 234)(166 225)(167 232)(168 239)(169 230)(170 237)(171 228)(172 235)(173 226)(174 233)(175 240)(176 231)(193 216)(194 223)(195 214)(196 221)(197 212)(198 219)(199 210)(200 217)(201 224)(202 215)(203 222)(204 213)(205 220)(206 211)(207 218)(208 209)

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

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

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

75 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 30C 30D 40A 40B 40C 40D 48A ··· 48H 60A 60B 60C 60D 60E ··· 60L 120A ··· 120H order 1 2 2 3 3 4 4 5 5 6 6 6 6 8 8 10 10 12 12 12 12 15 15 15 15 16 16 16 16 20 20 20 20 20 20 24 24 24 24 30 30 30 30 40 40 40 40 48 ··· 48 60 60 60 60 60 ··· 60 120 ··· 120 size 1 1 40 1 1 2 8 2 2 1 1 40 40 2 2 2 2 2 2 8 8 2 2 2 2 10 10 10 10 4 4 8 8 8 8 2 2 2 2 2 2 2 2 4 4 4 4 10 ··· 10 4 4 4 4 8 ··· 8 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 SD32 C5⋊D4 C3×D8 C6×D5 C3×SD32 C3×C5⋊D4 D4⋊D5 C5⋊SD32 C3×D4⋊D5 C3×C5⋊SD32 kernel C3×C5⋊SD32 C3×C5⋊2C16 C3×D40 C15×Q16 C5⋊SD32 C5⋊2C16 D40 C5×Q16 C60 C3×Q16 C30 C24 C20 Q16 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⋊SD32 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 1 0 0 0 0 1
,
 240 188 0 0 1 52 0 0 0 0 1 0 0 0 0 1
,
 145 42 0 0 165 96 0 0 0 0 179 160 0 0 223 97
,
 189 189 0 0 1 52 0 0 0 0 1 0 0 0 123 240
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,1,0,0,0,0,1],[240,1,0,0,188,52,0,0,0,0,1,0,0,0,0,1],[145,165,0,0,42,96,0,0,0,0,179,223,0,0,160,97],[189,1,0,0,189,52,0,0,0,0,1,123,0,0,0,240] >;

C3×C5⋊SD32 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes {\rm SD}_{32}
% in TeX

G:=Group("C3xC5:SD32");
// GroupNames label

G:=SmallGroup(480,106);
// by ID

G=gap.SmallGroup(480,106);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,197,344,1011,514,192,2524,1271,102,18822]);
// 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^7>;
// generators/relations

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