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## G = C3×C8.F5order 480 = 25·3·5

### Direct product of C3 and C8.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C8.F5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C3×C5⋊2C8 — C3×C5⋊C16 — C3×C8.F5
 Lower central C5 — C10 — C3×C8.F5
 Upper central C1 — C12 — C24

Generators and relations for C3×C8.F5
G = < a,b,c,d | a3=b8=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c3 >

Smallest permutation representation of C3×C8.F5
On 240 points
Generators in S240
(1 120 48)(2 121 33)(3 122 34)(4 123 35)(5 124 36)(6 125 37)(7 126 38)(8 127 39)(9 128 40)(10 113 41)(11 114 42)(12 115 43)(13 116 44)(14 117 45)(15 118 46)(16 119 47)(17 78 228)(18 79 229)(19 80 230)(20 65 231)(21 66 232)(22 67 233)(23 68 234)(24 69 235)(25 70 236)(26 71 237)(27 72 238)(28 73 239)(29 74 240)(30 75 225)(31 76 226)(32 77 227)(49 157 140)(50 158 141)(51 159 142)(52 160 143)(53 145 144)(54 146 129)(55 147 130)(56 148 131)(57 149 132)(58 150 133)(59 151 134)(60 152 135)(61 153 136)(62 154 137)(63 155 138)(64 156 139)(81 206 169)(82 207 170)(83 208 171)(84 193 172)(85 194 173)(86 195 174)(87 196 175)(88 197 176)(89 198 161)(90 199 162)(91 200 163)(92 201 164)(93 202 165)(94 203 166)(95 204 167)(96 205 168)(97 211 186)(98 212 187)(99 213 188)(100 214 189)(101 215 190)(102 216 191)(103 217 192)(104 218 177)(105 219 178)(106 220 179)(107 221 180)(108 222 181)(109 223 182)(110 224 183)(111 209 184)(112 210 185)
(1 11 5 15 9 3 13 7)(2 4 6 8 10 12 14 16)(17 209 21 213 25 217 29 221)(18 218 22 222 26 210 30 214)(19 211 23 215 27 219 31 223)(20 220 24 224 28 212 32 216)(33 35 37 39 41 43 45 47)(34 44 38 48 42 36 46 40)(49 171 53 175 57 163 61 167)(50 164 54 168 58 172 62 176)(51 173 55 161 59 165 63 169)(52 166 56 170 60 174 64 162)(65 179 69 183 73 187 77 191)(66 188 70 192 74 180 78 184)(67 181 71 185 75 189 79 177)(68 190 72 178 76 182 80 186)(81 159 85 147 89 151 93 155)(82 152 86 156 90 160 94 148)(83 145 87 149 91 153 95 157)(84 154 88 158 92 146 96 150)(97 234 101 238 105 226 109 230)(98 227 102 231 106 235 110 239)(99 236 103 240 107 228 111 232)(100 229 104 233 108 237 112 225)(113 115 117 119 121 123 125 127)(114 124 118 128 122 116 126 120)(129 205 133 193 137 197 141 201)(130 198 134 202 138 206 142 194)(131 207 135 195 139 199 143 203)(132 200 136 204 140 208 144 196)
(1 139 224 18 205)(2 19 140 206 209)(3 207 20 210 141)(4 211 208 142 21)(5 143 212 22 193)(6 23 144 194 213)(7 195 24 214 129)(8 215 196 130 25)(9 131 216 26 197)(10 27 132 198 217)(11 199 28 218 133)(12 219 200 134 29)(13 135 220 30 201)(14 31 136 202 221)(15 203 32 222 137)(16 223 204 138 17)(33 230 157 81 111)(34 82 231 112 158)(35 97 83 159 232)(36 160 98 233 84)(37 234 145 85 99)(38 86 235 100 146)(39 101 87 147 236)(40 148 102 237 88)(41 238 149 89 103)(42 90 239 104 150)(43 105 91 151 240)(44 152 106 225 92)(45 226 153 93 107)(46 94 227 108 154)(47 109 95 155 228)(48 156 110 229 96)(49 169 184 121 80)(50 122 170 65 185)(51 66 123 186 171)(52 187 67 172 124)(53 173 188 125 68)(54 126 174 69 189)(55 70 127 190 175)(56 191 71 176 128)(57 161 192 113 72)(58 114 162 73 177)(59 74 115 178 163)(60 179 75 164 116)(61 165 180 117 76)(62 118 166 77 181)(63 78 119 182 167)(64 183 79 168 120)
(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)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C24 C24 M5(2) C3×M5(2) F5 C2×F5 C3×F5 D5⋊C8 C6×F5 C8.F5 C3×D5⋊C8 C3×C8.F5 kernel C3×C8.F5 C3×C5⋊C16 D5×C24 C8.F5 C120 D5×C12 C5⋊C16 C8×D5 C3×Dic5 C6×D5 C40 C4×D5 Dic5 D10 C15 C5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 4 4 8 8 4 8 1 1 2 2 2 4 4 8

Matrix representation of C3×C8.F5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 30 0 0 0 0 0 0 211 0 0 0 0 0 0 177 0 0 0 0 0 0 177 0 0 0 0 0 0 177 0 0 0 0 0 0 177
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 0 1 0 0 0 0 211 0 0 0 0 0 0 0 0 134 219 134 0 0 85 0 107 107 0 0 107 107 0 85 0 0 134 219 134 0

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[30,0,0,0,0,0,0,211,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[0,211,0,0,0,0,1,0,0,0,0,0,0,0,0,85,107,134,0,0,134,0,107,219,0,0,219,107,0,134,0,0,134,107,85,0] >;

C3×C8.F5 in GAP, Magma, Sage, TeX

C_3\times C_8.F_5
% in TeX

G:=Group("C3xC8.F5");
// GroupNames label

G:=SmallGroup(480,270);
// by ID

G=gap.SmallGroup(480,270);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,80,102,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations

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