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## G = C3×C16⋊D5order 480 = 25·3·5

### Direct product of C3 and C16⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C3×C16⋊D5
 Chief series C1 — C5 — C10 — C20 — C40 — C120 — C3×D40 — C3×C16⋊D5
 Lower central C5 — C10 — C20 — C40 — C3×C16⋊D5
 Upper central C1 — C6 — C12 — C24 — C48

Generators and relations for C3×C16⋊D5
G = < a,b,c,d | a3=b16=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b7, dcd=c-1 >

Smallest permutation representation of C3×C16⋊D5
On 240 points
Generators in S240
(1 187 116)(2 188 117)(3 189 118)(4 190 119)(5 191 120)(6 192 121)(7 177 122)(8 178 123)(9 179 124)(10 180 125)(11 181 126)(12 182 127)(13 183 128)(14 184 113)(15 185 114)(16 186 115)(17 215 86)(18 216 87)(19 217 88)(20 218 89)(21 219 90)(22 220 91)(23 221 92)(24 222 93)(25 223 94)(26 224 95)(27 209 96)(28 210 81)(29 211 82)(30 212 83)(31 213 84)(32 214 85)(33 233 138)(34 234 139)(35 235 140)(36 236 141)(37 237 142)(38 238 143)(39 239 144)(40 240 129)(41 225 130)(42 226 131)(43 227 132)(44 228 133)(45 229 134)(46 230 135)(47 231 136)(48 232 137)(49 77 154)(50 78 155)(51 79 156)(52 80 157)(53 65 158)(54 66 159)(55 67 160)(56 68 145)(57 69 146)(58 70 147)(59 71 148)(60 72 149)(61 73 150)(62 74 151)(63 75 152)(64 76 153)(97 193 172)(98 194 173)(99 195 174)(100 196 175)(101 197 176)(102 198 161)(103 199 162)(104 200 163)(105 201 164)(106 202 165)(107 203 166)(108 204 167)(109 205 168)(110 206 169)(111 207 170)(112 208 171)
(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 66 41 83 195)(2 67 42 84 196)(3 68 43 85 197)(4 69 44 86 198)(5 70 45 87 199)(6 71 46 88 200)(7 72 47 89 201)(8 73 48 90 202)(9 74 33 91 203)(10 75 34 92 204)(11 76 35 93 205)(12 77 36 94 206)(13 78 37 95 207)(14 79 38 96 208)(15 80 39 81 193)(16 65 40 82 194)(17 161 190 146 228)(18 162 191 147 229)(19 163 192 148 230)(20 164 177 149 231)(21 165 178 150 232)(22 166 179 151 233)(23 167 180 152 234)(24 168 181 153 235)(25 169 182 154 236)(26 170 183 155 237)(27 171 184 156 238)(28 172 185 157 239)(29 173 186 158 240)(30 174 187 159 225)(31 175 188 160 226)(32 176 189 145 227)(49 141 223 110 127)(50 142 224 111 128)(51 143 209 112 113)(52 144 210 97 114)(53 129 211 98 115)(54 130 212 99 116)(55 131 213 100 117)(56 132 214 101 118)(57 133 215 102 119)(58 134 216 103 120)(59 135 217 104 121)(60 136 218 105 122)(61 137 219 106 123)(62 138 220 107 124)(63 139 221 108 125)(64 140 222 109 126)
(1 195)(2 202)(3 193)(4 200)(5 207)(6 198)(7 205)(8 196)(9 203)(10 194)(11 201)(12 208)(13 199)(14 206)(15 197)(16 204)(17 148)(18 155)(19 146)(20 153)(21 160)(22 151)(23 158)(24 149)(25 156)(26 147)(27 154)(28 145)(29 152)(30 159)(31 150)(32 157)(34 40)(35 47)(36 38)(37 45)(39 43)(42 48)(44 46)(49 209)(50 216)(51 223)(52 214)(53 221)(54 212)(55 219)(56 210)(57 217)(58 224)(59 215)(60 222)(61 213)(62 220)(63 211)(64 218)(65 92)(66 83)(67 90)(68 81)(69 88)(70 95)(71 86)(72 93)(73 84)(74 91)(75 82)(76 89)(77 96)(78 87)(79 94)(80 85)(97 118)(98 125)(99 116)(100 123)(101 114)(102 121)(103 128)(104 119)(105 126)(106 117)(107 124)(108 115)(109 122)(110 113)(111 120)(112 127)(129 139)(131 137)(132 144)(133 135)(134 142)(136 140)(141 143)(161 192)(162 183)(163 190)(164 181)(165 188)(166 179)(167 186)(168 177)(169 184)(170 191)(171 182)(172 189)(173 180)(174 187)(175 178)(176 185)(226 232)(227 239)(228 230)(229 237)(231 235)(234 240)(236 238)

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

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

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

129 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 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 48A ··· 48H 60A ··· 60H 80A ··· 80P 120A ··· 120P 240A ··· 240AF 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 24 24 24 24 30 30 30 30 40 ··· 40 48 ··· 48 60 ··· 60 80 ··· 80 120 ··· 120 240 ··· 240 size 1 1 40 1 1 2 40 2 2 1 1 40 40 2 2 2 2 2 2 40 40 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

129 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 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 D8 D10 C3×D4 C3×D5 SD32 D20 C3×D8 C6×D5 D40 C3×SD32 C3×D20 C16⋊D5 C3×D40 C3×C16⋊D5 kernel C3×C16⋊D5 C240 C3×D40 C3×Dic20 C16⋊D5 C80 D40 Dic20 C60 C48 C30 C24 C20 C16 C15 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 2 4 4 4 4 4 8 8 8 16 16 32

Matrix representation of C3×C16⋊D5 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 1 0 0 0 0 1
,
 221 33 0 0 179 42 0 0 0 0 179 138 0 0 206 144
,
 52 240 0 0 53 240 0 0 0 0 1 0 0 0 0 1
,
 0 51 0 0 52 0 0 0 0 0 1 1 0 0 0 240
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[221,179,0,0,33,42,0,0,0,0,179,206,0,0,138,144],[52,53,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[0,52,0,0,51,0,0,0,0,0,1,0,0,0,1,240] >;

C3×C16⋊D5 in GAP, Magma, Sage, TeX

C_3\times C_{16}\rtimes D_5
% in TeX

G:=Group("C3xC16:D5");
// GroupNames label

G:=SmallGroup(480,78);
// by ID

G=gap.SmallGroup(480,78);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,197,260,2355,80,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^16=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^7,d*c*d=c^-1>;
// generators/relations

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