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## G = C5×C12.55D4order 480 = 25·3·5

### Direct product of C5 and C12.55D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C12.55D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C10×C3⋊C8 — C5×C12.55D4
 Lower central C3 — C6 — C5×C12.55D4
 Upper central C1 — C2×C20 — C22×C20

Generators and relations for C5×C12.55D4
G = < a,b,c,d | a5=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b3c3 >

Subgroups: 164 in 100 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C30, C30, C22⋊C8, C40, C2×C20, C2×C20, C22×C10, C2×C3⋊C8, C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C2×C40, C22×C20, C12.55D4, C5×C3⋊C8, C2×C60, C2×C60, C22×C30, C5×C22⋊C8, C10×C3⋊C8, C22×C60, C5×C12.55D4
Quotients:

Smallest permutation representation of C5×C12.55D4
On 240 points
Generators in S240
(1 59 160 132 113)(2 60 161 121 114)(3 49 162 122 115)(4 50 163 123 116)(5 51 164 124 117)(6 52 165 125 118)(7 53 166 126 119)(8 54 167 127 120)(9 55 168 128 109)(10 56 157 129 110)(11 57 158 130 111)(12 58 159 131 112)(13 191 61 102 78)(14 192 62 103 79)(15 181 63 104 80)(16 182 64 105 81)(17 183 65 106 82)(18 184 66 107 83)(19 185 67 108 84)(20 186 68 97 73)(21 187 69 98 74)(22 188 70 99 75)(23 189 71 100 76)(24 190 72 101 77)(25 42 231 220 139)(26 43 232 221 140)(27 44 233 222 141)(28 45 234 223 142)(29 46 235 224 143)(30 47 236 225 144)(31 48 237 226 133)(32 37 238 227 134)(33 38 239 228 135)(34 39 240 217 136)(35 40 229 218 137)(36 41 230 219 138)(85 213 169 200 151)(86 214 170 201 152)(87 215 171 202 153)(88 216 172 203 154)(89 205 173 204 155)(90 206 174 193 156)(91 207 175 194 145)(92 208 176 195 146)(93 209 177 196 147)(94 210 178 197 148)(95 211 179 198 149)(96 212 180 199 150)
(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 70 138 177 7 64 144 171)(2 63 139 170 8 69 133 176)(3 68 140 175 9 62 134 169)(4 61 141 180 10 67 135 174)(5 66 142 173 11 72 136 179)(6 71 143 178 12 65 137 172)(13 233 96 129 19 239 90 123)(14 238 85 122 20 232 91 128)(15 231 86 127 21 237 92 121)(16 236 87 132 22 230 93 126)(17 229 88 125 23 235 94 131)(18 234 89 130 24 240 95 124)(25 201 54 98 31 195 60 104)(26 194 55 103 32 200 49 97)(27 199 56 108 33 193 50 102)(28 204 57 101 34 198 51 107)(29 197 58 106 35 203 52 100)(30 202 59 99 36 196 53 105)(37 151 162 73 43 145 168 79)(38 156 163 78 44 150 157 84)(39 149 164 83 45 155 158 77)(40 154 165 76 46 148 159 82)(41 147 166 81 47 153 160 75)(42 152 167 74 48 146 161 80)(109 192 227 213 115 186 221 207)(110 185 228 206 116 191 222 212)(111 190 217 211 117 184 223 205)(112 183 218 216 118 189 224 210)(113 188 219 209 119 182 225 215)(114 181 220 214 120 187 226 208)
(1 180 10 177 7 174 4 171)(2 173 11 170 8 179 5 176)(3 178 12 175 9 172 6 169)(13 233 22 230 19 239 16 236)(14 238 23 235 20 232 17 229)(15 231 24 240 21 237 18 234)(25 101 34 98 31 107 28 104)(26 106 35 103 32 100 29 97)(27 99 36 108 33 105 30 102)(37 76 46 73 43 82 40 79)(38 81 47 78 44 75 41 84)(39 74 48 83 45 80 42 77)(49 197 58 194 55 203 52 200)(50 202 59 199 56 196 53 193)(51 195 60 204 57 201 54 198)(61 141 70 138 67 135 64 144)(62 134 71 143 68 140 65 137)(63 139 72 136 69 133 66 142)(85 122 94 131 91 128 88 125)(86 127 95 124 92 121 89 130)(87 132 96 129 93 126 90 123)(109 216 118 213 115 210 112 207)(110 209 119 206 116 215 113 212)(111 214 120 211 117 208 114 205)(145 168 154 165 151 162 148 159)(146 161 155 158 152 167 149 164)(147 166 156 163 153 160 150 157)(181 220 190 217 187 226 184 223)(182 225 191 222 188 219 185 228)(183 218 192 227 189 224 186 221)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A ··· 6G 8A ··· 8H 10A ··· 10L 10M ··· 10T 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20X 30A ··· 30AB 40A ··· 40AF 60A ··· 60AF order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 ··· 6 8 ··· 8 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 2 2 1 1 1 1 2 2 1 1 1 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 S3 D4 Dic3 D6 Dic3 M4(2) C3⋊D4 C3⋊C8 C5×S3 C5×D4 C4.Dic3 C5×Dic3 S3×C10 C5×Dic3 C5×M4(2) C5×C3⋊D4 C5×C3⋊C8 C5×C4.Dic3 kernel C5×C12.55D4 C10×C3⋊C8 C22×C60 C2×C60 C22×C30 C12.55D4 C2×C30 C2×C3⋊C8 C22×C12 C2×C12 C22×C6 C2×C6 C22×C20 C60 C2×C20 C2×C20 C22×C10 C30 C20 C2×C10 C22×C4 C12 C10 C2×C4 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 2 4 8 8 4 8 8 32 1 2 1 1 1 2 4 4 4 8 4 4 4 4 8 16 16 16

Matrix representation of C5×C12.55D4 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 91 0 0 0 0 91
,
 240 0 0 0 0 240 0 0 0 0 237 0 0 0 0 181
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 177 0
,
 0 1 0 0 240 0 0 0 0 0 0 1 0 0 64 0
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,91,0,0,0,0,91],[240,0,0,0,0,240,0,0,0,0,237,0,0,0,0,181],[0,1,0,0,1,0,0,0,0,0,0,177,0,0,1,0],[0,240,0,0,1,0,0,0,0,0,0,64,0,0,1,0] >;

C5×C12.55D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}._{55}D_4
% in TeX

G:=Group("C5xC12.55D4");
// GroupNames label

G:=SmallGroup(480,149);
// by ID

G=gap.SmallGroup(480,149);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,136,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations

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