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## G = C3×D5⋊C16order 480 = 25·3·5

### Direct product of C3 and D5⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×D5⋊C16
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C3×C5⋊2C8 — C3×C5⋊C16 — C3×D5⋊C16
 Lower central C5 — C3×D5⋊C16
 Upper central C1 — C24

Generators and relations for C3×D5⋊C16
G = < a,b,c,d | a3=b5=c2=d16=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Smallest permutation representation of C3×D5⋊C16
On 240 points
Generators in S240
(1 61 34)(2 62 35)(3 63 36)(4 64 37)(5 49 38)(6 50 39)(7 51 40)(8 52 41)(9 53 42)(10 54 43)(11 55 44)(12 56 45)(13 57 46)(14 58 47)(15 59 48)(16 60 33)(17 82 113)(18 83 114)(19 84 115)(20 85 116)(21 86 117)(22 87 118)(23 88 119)(24 89 120)(25 90 121)(26 91 122)(27 92 123)(28 93 124)(29 94 125)(30 95 126)(31 96 127)(32 81 128)(65 218 168)(66 219 169)(67 220 170)(68 221 171)(69 222 172)(70 223 173)(71 224 174)(72 209 175)(73 210 176)(74 211 161)(75 212 162)(76 213 163)(77 214 164)(78 215 165)(79 216 166)(80 217 167)(97 230 183)(98 231 184)(99 232 185)(100 233 186)(101 234 187)(102 235 188)(103 236 189)(104 237 190)(105 238 191)(106 239 192)(107 240 177)(108 225 178)(109 226 179)(110 227 180)(111 228 181)(112 229 182)(129 194 146)(130 195 147)(131 196 148)(132 197 149)(133 198 150)(134 199 151)(135 200 152)(136 201 153)(137 202 154)(138 203 155)(139 204 156)(140 205 157)(141 206 158)(142 207 159)(143 208 160)(144 193 145)
(1 183 133 115 223)(2 116 184 224 134)(3 209 117 135 185)(4 136 210 186 118)(5 187 137 119 211)(6 120 188 212 138)(7 213 121 139 189)(8 140 214 190 122)(9 191 141 123 215)(10 124 192 216 142)(11 217 125 143 177)(12 144 218 178 126)(13 179 129 127 219)(14 128 180 220 130)(15 221 113 131 181)(16 132 222 182 114)(17 196 111 59 171)(18 60 197 172 112)(19 173 61 97 198)(20 98 174 199 62)(21 200 99 63 175)(22 64 201 176 100)(23 161 49 101 202)(24 102 162 203 50)(25 204 103 51 163)(26 52 205 164 104)(27 165 53 105 206)(28 106 166 207 54)(29 208 107 55 167)(30 56 193 168 108)(31 169 57 109 194)(32 110 170 195 58)(33 149 69 229 83)(34 230 150 84 70)(35 85 231 71 151)(36 72 86 152 232)(37 153 73 233 87)(38 234 154 88 74)(39 89 235 75 155)(40 76 90 156 236)(41 157 77 237 91)(42 238 158 92 78)(43 93 239 79 159)(44 80 94 160 240)(45 145 65 225 95)(46 226 146 96 66)(47 81 227 67 147)(48 68 82 148 228)
(1 223)(2 134)(3 185)(4 118)(5 211)(6 138)(7 189)(8 122)(9 215)(10 142)(11 177)(12 126)(13 219)(14 130)(15 181)(16 114)(18 60)(19 97)(20 174)(22 64)(23 101)(24 162)(26 52)(27 105)(28 166)(30 56)(31 109)(32 170)(33 83)(34 70)(35 151)(36 232)(37 87)(38 74)(39 155)(40 236)(41 91)(42 78)(43 159)(44 240)(45 95)(46 66)(47 147)(48 228)(49 161)(50 203)(51 103)(53 165)(54 207)(55 107)(57 169)(58 195)(59 111)(61 173)(62 199)(63 99)(67 81)(68 148)(71 85)(72 152)(75 89)(76 156)(79 93)(80 160)(84 230)(88 234)(92 238)(96 226)(100 201)(104 205)(108 193)(112 197)(115 183)(116 224)(119 187)(120 212)(123 191)(124 216)(127 179)(128 220)(131 221)(132 182)(135 209)(136 186)(139 213)(140 190)(143 217)(144 178)(145 225)(149 229)(153 233)(157 237)(163 204)(167 208)(171 196)(175 200)
(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)| (1,61,34)(2,62,35)(3,63,36)(4,64,37)(5,49,38)(6,50,39)(7,51,40)(8,52,41)(9,53,42)(10,54,43)(11,55,44)(12,56,45)(13,57,46)(14,58,47)(15,59,48)(16,60,33)(17,82,113)(18,83,114)(19,84,115)(20,85,116)(21,86,117)(22,87,118)(23,88,119)(24,89,120)(25,90,121)(26,91,122)(27,92,123)(28,93,124)(29,94,125)(30,95,126)(31,96,127)(32,81,128)(65,218,168)(66,219,169)(67,220,170)(68,221,171)(69,222,172)(70,223,173)(71,224,174)(72,209,175)(73,210,176)(74,211,161)(75,212,162)(76,213,163)(77,214,164)(78,215,165)(79,216,166)(80,217,167)(97,230,183)(98,231,184)(99,232,185)(100,233,186)(101,234,187)(102,235,188)(103,236,189)(104,237,190)(105,238,191)(106,239,192)(107,240,177)(108,225,178)(109,226,179)(110,227,180)(111,228,181)(112,229,182)(129,194,146)(130,195,147)(131,196,148)(132,197,149)(133,198,150)(134,199,151)(135,200,152)(136,201,153)(137,202,154)(138,203,155)(139,204,156)(140,205,157)(141,206,158)(142,207,159)(143,208,160)(144,193,145), (1,183,133,115,223)(2,116,184,224,134)(3,209,117,135,185)(4,136,210,186,118)(5,187,137,119,211)(6,120,188,212,138)(7,213,121,139,189)(8,140,214,190,122)(9,191,141,123,215)(10,124,192,216,142)(11,217,125,143,177)(12,144,218,178,126)(13,179,129,127,219)(14,128,180,220,130)(15,221,113,131,181)(16,132,222,182,114)(17,196,111,59,171)(18,60,197,172,112)(19,173,61,97,198)(20,98,174,199,62)(21,200,99,63,175)(22,64,201,176,100)(23,161,49,101,202)(24,102,162,203,50)(25,204,103,51,163)(26,52,205,164,104)(27,165,53,105,206)(28,106,166,207,54)(29,208,107,55,167)(30,56,193,168,108)(31,169,57,109,194)(32,110,170,195,58)(33,149,69,229,83)(34,230,150,84,70)(35,85,231,71,151)(36,72,86,152,232)(37,153,73,233,87)(38,234,154,88,74)(39,89,235,75,155)(40,76,90,156,236)(41,157,77,237,91)(42,238,158,92,78)(43,93,239,79,159)(44,80,94,160,240)(45,145,65,225,95)(46,226,146,96,66)(47,81,227,67,147)(48,68,82,148,228), (1,223)(2,134)(3,185)(4,118)(5,211)(6,138)(7,189)(8,122)(9,215)(10,142)(11,177)(12,126)(13,219)(14,130)(15,181)(16,114)(18,60)(19,97)(20,174)(22,64)(23,101)(24,162)(26,52)(27,105)(28,166)(30,56)(31,109)(32,170)(33,83)(34,70)(35,151)(36,232)(37,87)(38,74)(39,155)(40,236)(41,91)(42,78)(43,159)(44,240)(45,95)(46,66)(47,147)(48,228)(49,161)(50,203)(51,103)(53,165)(54,207)(55,107)(57,169)(58,195)(59,111)(61,173)(62,199)(63,99)(67,81)(68,148)(71,85)(72,152)(75,89)(76,156)(79,93)(80,160)(84,230)(88,234)(92,238)(96,226)(100,201)(104,205)(108,193)(112,197)(115,183)(116,224)(119,187)(120,212)(123,191)(124,216)(127,179)(128,220)(131,221)(132,182)(135,209)(136,186)(139,213)(140,190)(143,217)(144,178)(145,225)(149,229)(153,233)(157,237)(163,204)(167,208)(171,196)(175,200), (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( (1,61,34)(2,62,35)(3,63,36)(4,64,37)(5,49,38)(6,50,39)(7,51,40)(8,52,41)(9,53,42)(10,54,43)(11,55,44)(12,56,45)(13,57,46)(14,58,47)(15,59,48)(16,60,33)(17,82,113)(18,83,114)(19,84,115)(20,85,116)(21,86,117)(22,87,118)(23,88,119)(24,89,120)(25,90,121)(26,91,122)(27,92,123)(28,93,124)(29,94,125)(30,95,126)(31,96,127)(32,81,128)(65,218,168)(66,219,169)(67,220,170)(68,221,171)(69,222,172)(70,223,173)(71,224,174)(72,209,175)(73,210,176)(74,211,161)(75,212,162)(76,213,163)(77,214,164)(78,215,165)(79,216,166)(80,217,167)(97,230,183)(98,231,184)(99,232,185)(100,233,186)(101,234,187)(102,235,188)(103,236,189)(104,237,190)(105,238,191)(106,239,192)(107,240,177)(108,225,178)(109,226,179)(110,227,180)(111,228,181)(112,229,182)(129,194,146)(130,195,147)(131,196,148)(132,197,149)(133,198,150)(134,199,151)(135,200,152)(136,201,153)(137,202,154)(138,203,155)(139,204,156)(140,205,157)(141,206,158)(142,207,159)(143,208,160)(144,193,145), (1,183,133,115,223)(2,116,184,224,134)(3,209,117,135,185)(4,136,210,186,118)(5,187,137,119,211)(6,120,188,212,138)(7,213,121,139,189)(8,140,214,190,122)(9,191,141,123,215)(10,124,192,216,142)(11,217,125,143,177)(12,144,218,178,126)(13,179,129,127,219)(14,128,180,220,130)(15,221,113,131,181)(16,132,222,182,114)(17,196,111,59,171)(18,60,197,172,112)(19,173,61,97,198)(20,98,174,199,62)(21,200,99,63,175)(22,64,201,176,100)(23,161,49,101,202)(24,102,162,203,50)(25,204,103,51,163)(26,52,205,164,104)(27,165,53,105,206)(28,106,166,207,54)(29,208,107,55,167)(30,56,193,168,108)(31,169,57,109,194)(32,110,170,195,58)(33,149,69,229,83)(34,230,150,84,70)(35,85,231,71,151)(36,72,86,152,232)(37,153,73,233,87)(38,234,154,88,74)(39,89,235,75,155)(40,76,90,156,236)(41,157,77,237,91)(42,238,158,92,78)(43,93,239,79,159)(44,80,94,160,240)(45,145,65,225,95)(46,226,146,96,66)(47,81,227,67,147)(48,68,82,148,228), (1,223)(2,134)(3,185)(4,118)(5,211)(6,138)(7,189)(8,122)(9,215)(10,142)(11,177)(12,126)(13,219)(14,130)(15,181)(16,114)(18,60)(19,97)(20,174)(22,64)(23,101)(24,162)(26,52)(27,105)(28,166)(30,56)(31,109)(32,170)(33,83)(34,70)(35,151)(36,232)(37,87)(38,74)(39,155)(40,236)(41,91)(42,78)(43,159)(44,240)(45,95)(46,66)(47,147)(48,228)(49,161)(50,203)(51,103)(53,165)(54,207)(55,107)(57,169)(58,195)(59,111)(61,173)(62,199)(63,99)(67,81)(68,148)(71,85)(72,152)(75,89)(76,156)(79,93)(80,160)(84,230)(88,234)(92,238)(96,226)(100,201)(104,205)(108,193)(112,197)(115,183)(116,224)(119,187)(120,212)(123,191)(124,216)(127,179)(128,220)(131,221)(132,182)(135,209)(136,186)(139,213)(140,190)(143,217)(144,178)(145,225)(149,229)(153,233)(157,237)(163,204)(167,208)(171,196)(175,200), (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([[(1,61,34),(2,62,35),(3,63,36),(4,64,37),(5,49,38),(6,50,39),(7,51,40),(8,52,41),(9,53,42),(10,54,43),(11,55,44),(12,56,45),(13,57,46),(14,58,47),(15,59,48),(16,60,33),(17,82,113),(18,83,114),(19,84,115),(20,85,116),(21,86,117),(22,87,118),(23,88,119),(24,89,120),(25,90,121),(26,91,122),(27,92,123),(28,93,124),(29,94,125),(30,95,126),(31,96,127),(32,81,128),(65,218,168),(66,219,169),(67,220,170),(68,221,171),(69,222,172),(70,223,173),(71,224,174),(72,209,175),(73,210,176),(74,211,161),(75,212,162),(76,213,163),(77,214,164),(78,215,165),(79,216,166),(80,217,167),(97,230,183),(98,231,184),(99,232,185),(100,233,186),(101,234,187),(102,235,188),(103,236,189),(104,237,190),(105,238,191),(106,239,192),(107,240,177),(108,225,178),(109,226,179),(110,227,180),(111,228,181),(112,229,182),(129,194,146),(130,195,147),(131,196,148),(132,197,149),(133,198,150),(134,199,151),(135,200,152),(136,201,153),(137,202,154),(138,203,155),(139,204,156),(140,205,157),(141,206,158),(142,207,159),(143,208,160),(144,193,145)], [(1,183,133,115,223),(2,116,184,224,134),(3,209,117,135,185),(4,136,210,186,118),(5,187,137,119,211),(6,120,188,212,138),(7,213,121,139,189),(8,140,214,190,122),(9,191,141,123,215),(10,124,192,216,142),(11,217,125,143,177),(12,144,218,178,126),(13,179,129,127,219),(14,128,180,220,130),(15,221,113,131,181),(16,132,222,182,114),(17,196,111,59,171),(18,60,197,172,112),(19,173,61,97,198),(20,98,174,199,62),(21,200,99,63,175),(22,64,201,176,100),(23,161,49,101,202),(24,102,162,203,50),(25,204,103,51,163),(26,52,205,164,104),(27,165,53,105,206),(28,106,166,207,54),(29,208,107,55,167),(30,56,193,168,108),(31,169,57,109,194),(32,110,170,195,58),(33,149,69,229,83),(34,230,150,84,70),(35,85,231,71,151),(36,72,86,152,232),(37,153,73,233,87),(38,234,154,88,74),(39,89,235,75,155),(40,76,90,156,236),(41,157,77,237,91),(42,238,158,92,78),(43,93,239,79,159),(44,80,94,160,240),(45,145,65,225,95),(46,226,146,96,66),(47,81,227,67,147),(48,68,82,148,228)], [(1,223),(2,134),(3,185),(4,118),(5,211),(6,138),(7,189),(8,122),(9,215),(10,142),(11,177),(12,126),(13,219),(14,130),(15,181),(16,114),(18,60),(19,97),(20,174),(22,64),(23,101),(24,162),(26,52),(27,105),(28,166),(30,56),(31,109),(32,170),(33,83),(34,70),(35,151),(36,232),(37,87),(38,74),(39,155),(40,236),(41,91),(42,78),(43,159),(44,240),(45,95),(46,66),(47,147),(48,228),(49,161),(50,203),(51,103),(53,165),(54,207),(55,107),(57,169),(58,195),(59,111),(61,173),(62,199),(63,99),(67,81),(68,148),(71,85),(72,152),(75,89),(76,156),(79,93),(80,160),(84,230),(88,234),(92,238),(96,226),(100,201),(104,205),(108,193),(112,197),(115,183),(116,224),(119,187),(120,212),(123,191),(124,216),(127,179),(128,220),(131,221),(132,182),(135,209),(136,186),(139,213),(140,190),(143,217),(144,178),(145,225),(149,229),(153,233),(157,237),(163,204),(167,208),(171,196),(175,200)], [(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)]])

120 conjugacy classes

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

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C16 C24 C24 C48 F5 C2×F5 C3×F5 D5⋊C8 C6×F5 D5⋊C16 C3×D5⋊C8 C3×D5⋊C16 kernel C3×D5⋊C16 C3×C5⋊C16 D5×C24 D5⋊C16 C120 D5×C12 C5⋊C16 C8×D5 C3×Dic5 C6×D5 C40 C4×D5 C3×D5 Dic5 D10 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 4 4 16 8 8 32 1 1 2 2 2 4 4 8

Matrix representation of C3×D5⋊C16 in GL5(𝔽241)

 15 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 240 1 0 0 0 240 0 1 0 0 240 0 0 1 0 240 0 0 0
,
 240 0 0 0 0 0 240 0 0 0 0 240 0 0 1 0 240 0 1 0 0 240 1 0 0
,
 1 0 0 0 0 0 44 197 172 0 0 216 197 0 44 0 44 0 197 216 0 0 172 197 44

G:=sub<GL(5,GF(241))| [15,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,240,240,240,240,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0],[240,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0],[1,0,0,0,0,0,44,216,44,0,0,197,197,0,172,0,172,0,197,197,0,0,44,216,44] >;

C3×D5⋊C16 in GAP, Magma, Sage, TeX

C_3\times D_5\rtimes C_{16}
% in TeX

G:=Group("C3xD5:C16");
// GroupNames label

G:=SmallGroup(480,269);
// by ID

G=gap.SmallGroup(480,269);
# by ID

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

G:=Group<a,b,c,d|a^3=b^5=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations

Export

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