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## G = D40.S3order 480 = 25·3·5

### 1st non-split extension by D40 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C120 — D40.S3
 Chief series C1 — C5 — C15 — C30 — C60 — C120 — C3×D40 — D40.S3
 Lower central C15 — C30 — C60 — C120 — D40.S3
 Upper central C1 — C2 — C4 — C8

Generators and relations for D40.S3
G = < a,b,c,d | a40=b2=c3=1, d2=a35, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Smallest permutation representation of D40.S3
On 240 points
Generators in S240
```(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 42)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(81 85)(82 84)(86 120)(87 119)(88 118)(89 117)(90 116)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(122 160)(123 159)(124 158)(125 157)(126 156)(127 155)(128 154)(129 153)(130 152)(131 151)(132 150)(133 149)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)(161 179)(162 178)(163 177)(164 176)(165 175)(166 174)(167 173)(168 172)(169 171)(180 200)(181 199)(182 198)(183 197)(184 196)(185 195)(186 194)(187 193)(188 192)(189 191)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 210)(208 209)(217 240)(218 239)(219 238)(220 237)(221 236)(222 235)(223 234)(224 233)(225 232)(226 231)(227 230)(228 229)
(1 229 62)(2 230 63)(3 231 64)(4 232 65)(5 233 66)(6 234 67)(7 235 68)(8 236 69)(9 237 70)(10 238 71)(11 239 72)(12 240 73)(13 201 74)(14 202 75)(15 203 76)(16 204 77)(17 205 78)(18 206 79)(19 207 80)(20 208 41)(21 209 42)(22 210 43)(23 211 44)(24 212 45)(25 213 46)(26 214 47)(27 215 48)(28 216 49)(29 217 50)(30 218 51)(31 219 52)(32 220 53)(33 221 54)(34 222 55)(35 223 56)(36 224 57)(37 225 58)(38 226 59)(39 227 60)(40 228 61)(81 168 159)(82 169 160)(83 170 121)(84 171 122)(85 172 123)(86 173 124)(87 174 125)(88 175 126)(89 176 127)(90 177 128)(91 178 129)(92 179 130)(93 180 131)(94 181 132)(95 182 133)(96 183 134)(97 184 135)(98 185 136)(99 186 137)(100 187 138)(101 188 139)(102 189 140)(103 190 141)(104 191 142)(105 192 143)(106 193 144)(107 194 145)(108 195 146)(109 196 147)(110 197 148)(111 198 149)(112 199 150)(113 200 151)(114 161 152)(115 162 153)(116 163 154)(117 164 155)(118 165 156)(119 166 157)(120 167 158)
(1 91 36 86 31 81 26 116 21 111 16 106 11 101 6 96)(2 92 37 87 32 82 27 117 22 112 17 107 12 102 7 97)(3 93 38 88 33 83 28 118 23 113 18 108 13 103 8 98)(4 94 39 89 34 84 29 119 24 114 19 109 14 104 9 99)(5 95 40 90 35 85 30 120 25 115 20 110 15 105 10 100)(41 197 76 192 71 187 66 182 61 177 56 172 51 167 46 162)(42 198 77 193 72 188 67 183 62 178 57 173 52 168 47 163)(43 199 78 194 73 189 68 184 63 179 58 174 53 169 48 164)(44 200 79 195 74 190 69 185 64 180 59 175 54 170 49 165)(45 161 80 196 75 191 70 186 65 181 60 176 55 171 50 166)(121 216 156 211 151 206 146 201 141 236 136 231 131 226 126 221)(122 217 157 212 152 207 147 202 142 237 137 232 132 227 127 222)(123 218 158 213 153 208 148 203 143 238 138 233 133 228 128 223)(124 219 159 214 154 209 149 204 144 239 139 234 134 229 129 224)(125 220 160 215 155 210 150 205 145 240 140 235 135 230 130 225)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 3 4A 4B 5A 5B 6A 6B 6C 8A 8B 10A 10B 12 15A 15B 16A 16B 16C 16D 20A 20B 20C 20D 24A 24B 30A 30B 40A ··· 40H 60A 60B 60C 60D 80A ··· 80P 120A ··· 120H order 1 2 2 3 4 4 5 5 6 6 6 8 8 10 10 12 15 15 16 16 16 16 20 20 20 20 24 24 30 30 40 ··· 40 60 60 60 60 80 ··· 80 120 ··· 120 size 1 1 40 2 2 120 2 2 2 40 40 2 2 2 2 4 4 4 6 6 6 6 2 2 2 2 4 4 4 4 2 ··· 2 4 4 4 4 6 ··· 6 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 S3 D4 D5 D6 D8 D10 C3⋊D4 SD32 D20 D40 C16⋊D5 D4⋊S3 S3×D5 D8.S3 C3⋊D20 C3⋊D40 D40.S3 kernel D40.S3 C5×C3⋊C16 C3×D40 Dic60 D40 C60 C3⋊C16 C40 C30 C24 C20 C15 C12 C6 C3 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 2 4 4 8 16 1 2 2 2 4 8

Matrix representation of D40.S3 in GL6(𝔽241)

 230 230 0 0 0 0 11 230 0 0 0 0 0 0 0 189 0 0 0 0 51 190 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 230 230 0 0 0 0 230 11 0 0 0 0 0 0 51 189 0 0 0 0 50 190 0 0 0 0 0 0 240 0 0 0 0 0 64 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 213 225
,
 138 41 0 0 0 0 200 138 0 0 0 0 0 0 197 85 0 0 0 0 153 44 0 0 0 0 0 0 179 224 0 0 0 0 212 62

`G:=sub<GL(6,GF(241))| [230,11,0,0,0,0,230,230,0,0,0,0,0,0,0,51,0,0,0,0,189,190,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[230,230,0,0,0,0,230,11,0,0,0,0,0,0,51,50,0,0,0,0,189,190,0,0,0,0,0,0,240,64,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,213,0,0,0,0,0,225],[138,200,0,0,0,0,41,138,0,0,0,0,0,0,197,153,0,0,0,0,85,44,0,0,0,0,0,0,179,212,0,0,0,0,224,62] >;`

D40.S3 in GAP, Magma, Sage, TeX

`D_{40}.S_3`
`% in TeX`

`G:=Group("D40.S3");`
`// GroupNames label`

`G:=SmallGroup(480,18);`
`// by ID`

`G=gap.SmallGroup(480,18);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,590,58,675,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=b^2=c^3=1,d^2=a^35,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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