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## G = C5×C24.C4order 480 = 25·3·5

### Direct product of C5 and C24.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C24.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C5×C4.Dic3 — C5×C24.C4
 Lower central C3 — C6 — C12 — C5×C24.C4
 Upper central C1 — C20 — C2×C20 — C2×C40

Generators and relations for C5×C24.C4
G = < a,b,c,d | a5=b8=1, c6=b4, d2=b4c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 100 in 60 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4 [×2], C22, C5, C6, C6, C8 [×2], C8 [×2], C2×C4, C10, C10, C12 [×2], C2×C6, C15, C2×C8, M4(2) [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C24 [×2], C2×C12, C30, C30, C8.C4, C40 [×2], C40 [×2], C2×C20, C4.Dic3 [×2], C2×C24, C60 [×2], C2×C30, C2×C40, C5×M4(2) [×2], C24.C4, C5×C3⋊C8 [×2], C120 [×2], C2×C60, C5×C8.C4, C5×C4.Dic3 [×2], C2×C120, C5×C24.C4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4, Q8, C10 [×3], Dic3 [×2], D6, C4⋊C4, C20 [×2], C2×C10, Dic6, D12, C2×Dic3, C5×S3, C8.C4, C2×C20, C5×D4, C5×Q8, C4⋊Dic3, C5×Dic3 [×2], S3×C10, C5×C4⋊C4, C24.C4, C5×Dic6, C5×D12, C10×Dic3, C5×C8.C4, C5×C4⋊Dic3, C5×C24.C4

Smallest permutation representation of C5×C24.C4
On 240 points
Generators in S240
(1 49 37 25 13)(2 50 38 26 14)(3 51 39 27 15)(4 52 40 28 16)(5 53 41 29 17)(6 54 42 30 18)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(121 169 157 145 133)(122 170 158 146 134)(123 171 159 147 135)(124 172 160 148 136)(125 173 161 149 137)(126 174 162 150 138)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(181 229 217 205 193)(182 230 218 206 194)(183 231 219 207 195)(184 232 220 208 196)(185 233 221 209 197)(186 234 222 210 198)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(1 61 10 70 7 67 4 64)(2 62 11 71 8 68 5 65)(3 63 12 72 9 69 6 66)(13 73 22 82 19 79 16 76)(14 74 23 83 20 80 17 77)(15 75 24 84 21 81 18 78)(25 85 34 94 31 91 28 88)(26 86 35 95 32 92 29 89)(27 87 36 96 33 93 30 90)(37 97 46 106 43 103 40 100)(38 98 47 107 44 104 41 101)(39 99 48 108 45 105 42 102)(49 109 58 118 55 115 52 112)(50 110 59 119 56 116 53 113)(51 111 60 120 57 117 54 114)(121 184 124 187 127 190 130 181)(122 185 125 188 128 191 131 182)(123 186 126 189 129 192 132 183)(133 196 136 199 139 202 142 193)(134 197 137 200 140 203 143 194)(135 198 138 201 141 204 144 195)(145 208 148 211 151 214 154 205)(146 209 149 212 152 215 155 206)(147 210 150 213 153 216 156 207)(157 220 160 223 163 226 166 217)(158 221 161 224 164 227 167 218)(159 222 162 225 165 228 168 219)(169 232 172 235 175 238 178 229)(170 233 173 236 176 239 179 230)(171 234 174 237 177 240 180 231)
(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)
(1 122 10 131 7 128 4 125)(2 127 11 124 8 121 5 130)(3 132 12 129 9 126 6 123)(13 134 22 143 19 140 16 137)(14 139 23 136 20 133 17 142)(15 144 24 141 21 138 18 135)(25 146 34 155 31 152 28 149)(26 151 35 148 32 145 29 154)(27 156 36 153 33 150 30 147)(37 158 46 167 43 164 40 161)(38 163 47 160 44 157 41 166)(39 168 48 165 45 162 42 159)(49 170 58 179 55 176 52 173)(50 175 59 172 56 169 53 178)(51 180 60 177 57 174 54 171)(61 182 70 191 67 188 64 185)(62 187 71 184 68 181 65 190)(63 192 72 189 69 186 66 183)(73 194 82 203 79 200 76 197)(74 199 83 196 80 193 77 202)(75 204 84 201 81 198 78 195)(85 206 94 215 91 212 88 209)(86 211 95 208 92 205 89 214)(87 216 96 213 93 210 90 207)(97 218 106 227 103 224 100 221)(98 223 107 220 104 217 101 226)(99 228 108 225 105 222 102 219)(109 230 118 239 115 236 112 233)(110 235 119 232 116 229 113 238)(111 240 120 237 117 234 114 231)

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

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

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

150 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 24A ··· 24H 30A ··· 30L 40A ··· 40P 40Q ··· 40AF 60A ··· 60P 120A ··· 120AF order 1 2 2 3 4 4 4 5 5 5 5 6 6 6 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 24 ··· 24 30 ··· 30 40 ··· 40 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 2 2 1 1 2 1 1 1 1 2 2 2 2 2 2 2 12 12 12 12 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

150 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 2 2 type + + + + + - - + + - image C1 C2 C2 C4 C5 C10 C10 C20 S3 D4 Q8 Dic3 D6 D12 Dic6 C5×S3 C8.C4 C5×D4 C5×Q8 C5×Dic3 S3×C10 C24.C4 C5×D12 C5×Dic6 C5×C8.C4 C5×C24.C4 kernel C5×C24.C4 C5×C4.Dic3 C2×C120 C120 C24.C4 C4.Dic3 C2×C24 C24 C2×C40 C60 C2×C30 C40 C2×C20 C20 C2×C10 C2×C8 C15 C12 C2×C6 C8 C2×C4 C5 C4 C22 C3 C1 # reps 1 2 1 4 4 8 4 16 1 1 1 2 1 2 2 4 4 4 4 8 4 8 8 8 16 32

Matrix representation of C5×C24.C4 in GL2(𝔽241) generated by

 205 0 0 205
,
 211 0 103 8
,
 60 0 178 4
,
 58 239 27 183
G:=sub<GL(2,GF(241))| [205,0,0,205],[211,103,0,8],[60,178,0,4],[58,27,239,183] >;

C5×C24.C4 in GAP, Magma, Sage, TeX

C_5\times C_{24}.C_4
% in TeX

G:=Group("C5xC24.C4");
// GroupNames label

G:=SmallGroup(480,138);
// by ID

G=gap.SmallGroup(480,138);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,288,136,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=1,c^6=b^4,d^2=b^4*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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