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G = C15×C22⋊C8order 480 = 25·3·5

Direct product of C15 and C22⋊C8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C15×C22⋊C8, C222C120, C60.250D4, C23.3C60, C30.44M4(2), (C2×C6)⋊1C40, (C2×C30)⋊5C8, (C2×C40)⋊3C6, (C2×C8)⋊1C30, (C2×C10)⋊6C24, (C2×C120)⋊7C2, (C2×C24)⋊3C10, (C2×C4).3C60, C30.73(C2×C8), (C2×C60).36C4, C2.1(C2×C120), C6.11(C2×C40), (C2×C12).7C20, C12.65(C5×D4), C20.65(C3×D4), C4.16(D4×C15), C10.20(C2×C24), (C2×C20).18C12, (C22×C4).4C30, C22.8(C2×C60), (C22×C60).5C2, (C22×C6).3C20, (C22×C20).7C6, C6.8(C5×M4(2)), (C22×C30).13C4, (C22×C12).3C10, C2.2(C15×M4(2)), (C2×C60).588C22, (C22×C10).10C12, C10.13(C3×M4(2)), C30.124(C22⋊C4), (C2×C4).32(C2×C30), (C2×C6).38(C2×C20), C2.2(C15×C22⋊C4), C6.20(C5×C22⋊C4), (C2×C30).206(C2×C4), (C2×C20).134(C2×C6), (C2×C10).58(C2×C12), C10.31(C3×C22⋊C4), (C2×C12).135(C2×C10), SmallGroup(480,201)

Series: Derived Chief Lower central Upper central

C1C2 — C15×C22⋊C8
C1C2C4C2×C4C2×C20C2×C60C2×C120 — C15×C22⋊C8
C1C2 — C15×C22⋊C8
C1C2×C60 — C15×C22⋊C8

Generators and relations for C15×C22⋊C8
 G = < a,b,c,d | a15=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 136 in 100 conjugacy classes, 64 normal (40 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C10 [×3], C10 [×2], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×C8 [×2], C22×C4, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C24 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊C8, C40 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C2×C24 [×2], C22×C12, C60 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C40 [×2], C22×C20, C3×C22⋊C8, C120 [×2], C2×C60 [×2], C2×C60 [×2], C22×C30, C5×C22⋊C8, C2×C120 [×2], C22×C60, C15×C22⋊C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, D4 [×2], C10 [×3], C12 [×2], C2×C6, C15, C22⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, C24 [×2], C2×C12, C3×D4 [×2], C30 [×3], C22⋊C8, C40 [×2], C2×C20, C5×D4 [×2], C3×C22⋊C4, C2×C24, C3×M4(2), C60 [×2], C2×C30, C5×C22⋊C4, C2×C40, C5×M4(2), C3×C22⋊C8, C120 [×2], C2×C60, D4×C15 [×2], C5×C22⋊C8, C15×C22⋊C4, C2×C120, C15×M4(2), C15×C22⋊C8

Smallest permutation representation of C15×C22⋊C8
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)
(16 230)(17 231)(18 232)(19 233)(20 234)(21 235)(22 236)(23 237)(24 238)(25 239)(26 240)(27 226)(28 227)(29 228)(30 229)(31 74)(32 75)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 73)(106 214)(107 215)(108 216)(109 217)(110 218)(111 219)(112 220)(113 221)(114 222)(115 223)(116 224)(117 225)(118 211)(119 212)(120 213)(136 190)(137 191)(138 192)(139 193)(140 194)(141 195)(142 181)(143 182)(144 183)(145 184)(146 185)(147 186)(148 187)(149 188)(150 189)
(1 60)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 230)(17 231)(18 232)(19 233)(20 234)(21 235)(22 236)(23 237)(24 238)(25 239)(26 240)(27 226)(28 227)(29 228)(30 229)(31 74)(32 75)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 73)(76 159)(77 160)(78 161)(79 162)(80 163)(81 164)(82 165)(83 151)(84 152)(85 153)(86 154)(87 155)(88 156)(89 157)(90 158)(91 208)(92 209)(93 210)(94 196)(95 197)(96 198)(97 199)(98 200)(99 201)(100 202)(101 203)(102 204)(103 205)(104 206)(105 207)(106 214)(107 215)(108 216)(109 217)(110 218)(111 219)(112 220)(113 221)(114 222)(115 223)(116 224)(117 225)(118 211)(119 212)(120 213)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 166)(134 167)(135 168)(136 190)(137 191)(138 192)(139 193)(140 194)(141 195)(142 181)(143 182)(144 183)(145 184)(146 185)(147 186)(148 187)(149 188)(150 189)
(1 69 157 185 172 20 105 118)(2 70 158 186 173 21 91 119)(3 71 159 187 174 22 92 120)(4 72 160 188 175 23 93 106)(5 73 161 189 176 24 94 107)(6 74 162 190 177 25 95 108)(7 75 163 191 178 26 96 109)(8 61 164 192 179 27 97 110)(9 62 165 193 180 28 98 111)(10 63 151 194 166 29 99 112)(11 64 152 195 167 30 100 113)(12 65 153 181 168 16 101 114)(13 66 154 182 169 17 102 115)(14 67 155 183 170 18 103 116)(15 68 156 184 171 19 104 117)(31 79 136 129 239 197 216 50)(32 80 137 130 240 198 217 51)(33 81 138 131 226 199 218 52)(34 82 139 132 227 200 219 53)(35 83 140 133 228 201 220 54)(36 84 141 134 229 202 221 55)(37 85 142 135 230 203 222 56)(38 86 143 121 231 204 223 57)(39 87 144 122 232 205 224 58)(40 88 145 123 233 206 225 59)(41 89 146 124 234 207 211 60)(42 90 147 125 235 208 212 46)(43 76 148 126 236 209 213 47)(44 77 149 127 237 210 214 48)(45 78 150 128 238 196 215 49)

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

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

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

300 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F5A5B5C5D6A···6F6G6H6I6J8A···8H10A···10L10M···10T12A···12H12I12J12K12L15A···15H20A···20P20Q···20X24A···24P30A···30X30Y···30AN40A···40AF60A···60AF60AG···60AV120A···120BL
order1222223344444455556···666668···810···1010···1012···121212121215···1520···2020···2024···2430···3030···3040···4060···6060···60120···120
size1111221111112211111···122222···21···12···21···122221···11···12···22···21···12···22···21···12···22···2

300 irreducible representations

dim11111111111111111111111122222222
type++++
imageC1C2C2C3C4C4C5C6C6C8C10C10C12C12C15C20C20C24C30C30C40C60C60C120D4M4(2)C3×D4C5×D4C3×M4(2)C5×M4(2)D4×C15C15×M4(2)
kernelC15×C22⋊C8C2×C120C22×C60C5×C22⋊C8C2×C60C22×C30C3×C22⋊C8C2×C40C22×C20C2×C30C2×C24C22×C12C2×C20C22×C10C22⋊C8C2×C12C22×C6C2×C10C2×C8C22×C4C2×C6C2×C4C23C22C60C30C20C12C10C6C4C2
# reps1212224428844488816168321616642248481616

Matrix representation of C15×C22⋊C8 in GL4(𝔽241) generated by

87000
0100
00150
00015
,
240000
024000
0010
000240
,
1000
0100
002400
000240
,
240000
03000
0001
002400
G:=sub<GL(4,GF(241))| [87,0,0,0,0,1,0,0,0,0,15,0,0,0,0,15],[240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[240,0,0,0,0,30,0,0,0,0,0,240,0,0,1,0] >;

C15×C22⋊C8 in GAP, Magma, Sage, TeX

C_{15}\times C_2^2\rtimes C_8
% in TeX

G:=Group("C15xC2^2:C8");
// GroupNames label

G:=SmallGroup(480,201);
// by ID

G=gap.SmallGroup(480,201);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,124]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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