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## G = Dic21⋊C4order 336 = 24·3·7

### 3rd semidirect product of Dic21 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — Dic21⋊C4
 Chief series C1 — C7 — C21 — C42 — C2×C42 — C6×Dic7 — Dic21⋊C4
 Lower central C21 — C42 — Dic21⋊C4
 Upper central C1 — C22

Generators and relations for Dic21⋊C4
G = < a,b,c | a42=c4=1, b2=a21, bab-1=a-1, cac-1=a29, cbc-1=a21b >

Smallest permutation representation of Dic21⋊C4
Regular action on 336 points
Generators in S336
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294)(295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336)
(1 267 22 288)(2 266 23 287)(3 265 24 286)(4 264 25 285)(5 263 26 284)(6 262 27 283)(7 261 28 282)(8 260 29 281)(9 259 30 280)(10 258 31 279)(11 257 32 278)(12 256 33 277)(13 255 34 276)(14 254 35 275)(15 253 36 274)(16 294 37 273)(17 293 38 272)(18 292 39 271)(19 291 40 270)(20 290 41 269)(21 289 42 268)(43 299 64 320)(44 298 65 319)(45 297 66 318)(46 296 67 317)(47 295 68 316)(48 336 69 315)(49 335 70 314)(50 334 71 313)(51 333 72 312)(52 332 73 311)(53 331 74 310)(54 330 75 309)(55 329 76 308)(56 328 77 307)(57 327 78 306)(58 326 79 305)(59 325 80 304)(60 324 81 303)(61 323 82 302)(62 322 83 301)(63 321 84 300)(85 213 106 234)(86 212 107 233)(87 211 108 232)(88 252 109 231)(89 251 110 230)(90 250 111 229)(91 249 112 228)(92 248 113 227)(93 247 114 226)(94 246 115 225)(95 245 116 224)(96 244 117 223)(97 243 118 222)(98 242 119 221)(99 241 120 220)(100 240 121 219)(101 239 122 218)(102 238 123 217)(103 237 124 216)(104 236 125 215)(105 235 126 214)(127 199 148 178)(128 198 149 177)(129 197 150 176)(130 196 151 175)(131 195 152 174)(132 194 153 173)(133 193 154 172)(134 192 155 171)(135 191 156 170)(136 190 157 169)(137 189 158 210)(138 188 159 209)(139 187 160 208)(140 186 161 207)(141 185 162 206)(142 184 163 205)(143 183 164 204)(144 182 165 203)(145 181 166 202)(146 180 167 201)(147 179 168 200)
(1 164 68 122)(2 151 69 109)(3 138 70 96)(4 167 71 125)(5 154 72 112)(6 141 73 99)(7 128 74 86)(8 157 75 115)(9 144 76 102)(10 131 77 89)(11 160 78 118)(12 147 79 105)(13 134 80 92)(14 163 81 121)(15 150 82 108)(16 137 83 95)(17 166 84 124)(18 153 43 111)(19 140 44 98)(20 127 45 85)(21 156 46 114)(22 143 47 101)(23 130 48 88)(24 159 49 117)(25 146 50 104)(26 133 51 91)(27 162 52 120)(28 149 53 107)(29 136 54 94)(30 165 55 123)(31 152 56 110)(32 139 57 97)(33 168 58 126)(34 155 59 113)(35 142 60 100)(36 129 61 87)(37 158 62 116)(38 145 63 103)(39 132 64 90)(40 161 65 119)(41 148 66 106)(42 135 67 93)(169 330 225 281)(170 317 226 268)(171 304 227 255)(172 333 228 284)(173 320 229 271)(174 307 230 258)(175 336 231 287)(176 323 232 274)(177 310 233 261)(178 297 234 290)(179 326 235 277)(180 313 236 264)(181 300 237 293)(182 329 238 280)(183 316 239 267)(184 303 240 254)(185 332 241 283)(186 319 242 270)(187 306 243 257)(188 335 244 286)(189 322 245 273)(190 309 246 260)(191 296 247 289)(192 325 248 276)(193 312 249 263)(194 299 250 292)(195 328 251 279)(196 315 252 266)(197 302 211 253)(198 331 212 282)(199 318 213 269)(200 305 214 256)(201 334 215 285)(202 321 216 272)(203 308 217 259)(204 295 218 288)(205 324 219 275)(206 311 220 262)(207 298 221 291)(208 327 222 278)(209 314 223 265)(210 301 224 294)

G:=sub<Sym(336)| (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294)(295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336), (1,267,22,288)(2,266,23,287)(3,265,24,286)(4,264,25,285)(5,263,26,284)(6,262,27,283)(7,261,28,282)(8,260,29,281)(9,259,30,280)(10,258,31,279)(11,257,32,278)(12,256,33,277)(13,255,34,276)(14,254,35,275)(15,253,36,274)(16,294,37,273)(17,293,38,272)(18,292,39,271)(19,291,40,270)(20,290,41,269)(21,289,42,268)(43,299,64,320)(44,298,65,319)(45,297,66,318)(46,296,67,317)(47,295,68,316)(48,336,69,315)(49,335,70,314)(50,334,71,313)(51,333,72,312)(52,332,73,311)(53,331,74,310)(54,330,75,309)(55,329,76,308)(56,328,77,307)(57,327,78,306)(58,326,79,305)(59,325,80,304)(60,324,81,303)(61,323,82,302)(62,322,83,301)(63,321,84,300)(85,213,106,234)(86,212,107,233)(87,211,108,232)(88,252,109,231)(89,251,110,230)(90,250,111,229)(91,249,112,228)(92,248,113,227)(93,247,114,226)(94,246,115,225)(95,245,116,224)(96,244,117,223)(97,243,118,222)(98,242,119,221)(99,241,120,220)(100,240,121,219)(101,239,122,218)(102,238,123,217)(103,237,124,216)(104,236,125,215)(105,235,126,214)(127,199,148,178)(128,198,149,177)(129,197,150,176)(130,196,151,175)(131,195,152,174)(132,194,153,173)(133,193,154,172)(134,192,155,171)(135,191,156,170)(136,190,157,169)(137,189,158,210)(138,188,159,209)(139,187,160,208)(140,186,161,207)(141,185,162,206)(142,184,163,205)(143,183,164,204)(144,182,165,203)(145,181,166,202)(146,180,167,201)(147,179,168,200), (1,164,68,122)(2,151,69,109)(3,138,70,96)(4,167,71,125)(5,154,72,112)(6,141,73,99)(7,128,74,86)(8,157,75,115)(9,144,76,102)(10,131,77,89)(11,160,78,118)(12,147,79,105)(13,134,80,92)(14,163,81,121)(15,150,82,108)(16,137,83,95)(17,166,84,124)(18,153,43,111)(19,140,44,98)(20,127,45,85)(21,156,46,114)(22,143,47,101)(23,130,48,88)(24,159,49,117)(25,146,50,104)(26,133,51,91)(27,162,52,120)(28,149,53,107)(29,136,54,94)(30,165,55,123)(31,152,56,110)(32,139,57,97)(33,168,58,126)(34,155,59,113)(35,142,60,100)(36,129,61,87)(37,158,62,116)(38,145,63,103)(39,132,64,90)(40,161,65,119)(41,148,66,106)(42,135,67,93)(169,330,225,281)(170,317,226,268)(171,304,227,255)(172,333,228,284)(173,320,229,271)(174,307,230,258)(175,336,231,287)(176,323,232,274)(177,310,233,261)(178,297,234,290)(179,326,235,277)(180,313,236,264)(181,300,237,293)(182,329,238,280)(183,316,239,267)(184,303,240,254)(185,332,241,283)(186,319,242,270)(187,306,243,257)(188,335,244,286)(189,322,245,273)(190,309,246,260)(191,296,247,289)(192,325,248,276)(193,312,249,263)(194,299,250,292)(195,328,251,279)(196,315,252,266)(197,302,211,253)(198,331,212,282)(199,318,213,269)(200,305,214,256)(201,334,215,285)(202,321,216,272)(203,308,217,259)(204,295,218,288)(205,324,219,275)(206,311,220,262)(207,298,221,291)(208,327,222,278)(209,314,223,265)(210,301,224,294)>;

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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294)(295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336), (1,267,22,288)(2,266,23,287)(3,265,24,286)(4,264,25,285)(5,263,26,284)(6,262,27,283)(7,261,28,282)(8,260,29,281)(9,259,30,280)(10,258,31,279)(11,257,32,278)(12,256,33,277)(13,255,34,276)(14,254,35,275)(15,253,36,274)(16,294,37,273)(17,293,38,272)(18,292,39,271)(19,291,40,270)(20,290,41,269)(21,289,42,268)(43,299,64,320)(44,298,65,319)(45,297,66,318)(46,296,67,317)(47,295,68,316)(48,336,69,315)(49,335,70,314)(50,334,71,313)(51,333,72,312)(52,332,73,311)(53,331,74,310)(54,330,75,309)(55,329,76,308)(56,328,77,307)(57,327,78,306)(58,326,79,305)(59,325,80,304)(60,324,81,303)(61,323,82,302)(62,322,83,301)(63,321,84,300)(85,213,106,234)(86,212,107,233)(87,211,108,232)(88,252,109,231)(89,251,110,230)(90,250,111,229)(91,249,112,228)(92,248,113,227)(93,247,114,226)(94,246,115,225)(95,245,116,224)(96,244,117,223)(97,243,118,222)(98,242,119,221)(99,241,120,220)(100,240,121,219)(101,239,122,218)(102,238,123,217)(103,237,124,216)(104,236,125,215)(105,235,126,214)(127,199,148,178)(128,198,149,177)(129,197,150,176)(130,196,151,175)(131,195,152,174)(132,194,153,173)(133,193,154,172)(134,192,155,171)(135,191,156,170)(136,190,157,169)(137,189,158,210)(138,188,159,209)(139,187,160,208)(140,186,161,207)(141,185,162,206)(142,184,163,205)(143,183,164,204)(144,182,165,203)(145,181,166,202)(146,180,167,201)(147,179,168,200), (1,164,68,122)(2,151,69,109)(3,138,70,96)(4,167,71,125)(5,154,72,112)(6,141,73,99)(7,128,74,86)(8,157,75,115)(9,144,76,102)(10,131,77,89)(11,160,78,118)(12,147,79,105)(13,134,80,92)(14,163,81,121)(15,150,82,108)(16,137,83,95)(17,166,84,124)(18,153,43,111)(19,140,44,98)(20,127,45,85)(21,156,46,114)(22,143,47,101)(23,130,48,88)(24,159,49,117)(25,146,50,104)(26,133,51,91)(27,162,52,120)(28,149,53,107)(29,136,54,94)(30,165,55,123)(31,152,56,110)(32,139,57,97)(33,168,58,126)(34,155,59,113)(35,142,60,100)(36,129,61,87)(37,158,62,116)(38,145,63,103)(39,132,64,90)(40,161,65,119)(41,148,66,106)(42,135,67,93)(169,330,225,281)(170,317,226,268)(171,304,227,255)(172,333,228,284)(173,320,229,271)(174,307,230,258)(175,336,231,287)(176,323,232,274)(177,310,233,261)(178,297,234,290)(179,326,235,277)(180,313,236,264)(181,300,237,293)(182,329,238,280)(183,316,239,267)(184,303,240,254)(185,332,241,283)(186,319,242,270)(187,306,243,257)(188,335,244,286)(189,322,245,273)(190,309,246,260)(191,296,247,289)(192,325,248,276)(193,312,249,263)(194,299,250,292)(195,328,251,279)(196,315,252,266)(197,302,211,253)(198,331,212,282)(199,318,213,269)(200,305,214,256)(201,334,215,285)(202,321,216,272)(203,308,217,259)(204,295,218,288)(205,324,219,275)(206,311,220,262)(207,298,221,291)(208,327,222,278)(209,314,223,265)(210,301,224,294) );

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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294),(295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336)], [(1,267,22,288),(2,266,23,287),(3,265,24,286),(4,264,25,285),(5,263,26,284),(6,262,27,283),(7,261,28,282),(8,260,29,281),(9,259,30,280),(10,258,31,279),(11,257,32,278),(12,256,33,277),(13,255,34,276),(14,254,35,275),(15,253,36,274),(16,294,37,273),(17,293,38,272),(18,292,39,271),(19,291,40,270),(20,290,41,269),(21,289,42,268),(43,299,64,320),(44,298,65,319),(45,297,66,318),(46,296,67,317),(47,295,68,316),(48,336,69,315),(49,335,70,314),(50,334,71,313),(51,333,72,312),(52,332,73,311),(53,331,74,310),(54,330,75,309),(55,329,76,308),(56,328,77,307),(57,327,78,306),(58,326,79,305),(59,325,80,304),(60,324,81,303),(61,323,82,302),(62,322,83,301),(63,321,84,300),(85,213,106,234),(86,212,107,233),(87,211,108,232),(88,252,109,231),(89,251,110,230),(90,250,111,229),(91,249,112,228),(92,248,113,227),(93,247,114,226),(94,246,115,225),(95,245,116,224),(96,244,117,223),(97,243,118,222),(98,242,119,221),(99,241,120,220),(100,240,121,219),(101,239,122,218),(102,238,123,217),(103,237,124,216),(104,236,125,215),(105,235,126,214),(127,199,148,178),(128,198,149,177),(129,197,150,176),(130,196,151,175),(131,195,152,174),(132,194,153,173),(133,193,154,172),(134,192,155,171),(135,191,156,170),(136,190,157,169),(137,189,158,210),(138,188,159,209),(139,187,160,208),(140,186,161,207),(141,185,162,206),(142,184,163,205),(143,183,164,204),(144,182,165,203),(145,181,166,202),(146,180,167,201),(147,179,168,200)], [(1,164,68,122),(2,151,69,109),(3,138,70,96),(4,167,71,125),(5,154,72,112),(6,141,73,99),(7,128,74,86),(8,157,75,115),(9,144,76,102),(10,131,77,89),(11,160,78,118),(12,147,79,105),(13,134,80,92),(14,163,81,121),(15,150,82,108),(16,137,83,95),(17,166,84,124),(18,153,43,111),(19,140,44,98),(20,127,45,85),(21,156,46,114),(22,143,47,101),(23,130,48,88),(24,159,49,117),(25,146,50,104),(26,133,51,91),(27,162,52,120),(28,149,53,107),(29,136,54,94),(30,165,55,123),(31,152,56,110),(32,139,57,97),(33,168,58,126),(34,155,59,113),(35,142,60,100),(36,129,61,87),(37,158,62,116),(38,145,63,103),(39,132,64,90),(40,161,65,119),(41,148,66,106),(42,135,67,93),(169,330,225,281),(170,317,226,268),(171,304,227,255),(172,333,228,284),(173,320,229,271),(174,307,230,258),(175,336,231,287),(176,323,232,274),(177,310,233,261),(178,297,234,290),(179,326,235,277),(180,313,236,264),(181,300,237,293),(182,329,238,280),(183,316,239,267),(184,303,240,254),(185,332,241,283),(186,319,242,270),(187,306,243,257),(188,335,244,286),(189,322,245,273),(190,309,246,260),(191,296,247,289),(192,325,248,276),(193,312,249,263),(194,299,250,292),(195,328,251,279),(196,315,252,266),(197,302,211,253),(198,331,212,282),(199,318,213,269),(200,305,214,256),(201,334,215,285),(202,321,216,272),(203,308,217,259),(204,295,218,288),(205,324,219,275),(206,311,220,262),(207,298,221,291),(208,327,222,278),(209,314,223,265),(210,301,224,294)]])

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 7A 7B 7C 12A 12B 12C 12D 14A ··· 14I 21A 21B 21C 28A ··· 28L 42A ··· 42I order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 7 7 7 12 12 12 12 14 ··· 14 21 21 21 28 ··· 28 42 ··· 42 size 1 1 1 1 2 6 6 14 14 42 42 2 2 2 2 2 2 14 14 14 14 2 ··· 2 4 4 4 6 ··· 6 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + - + - + + - - image C1 C2 C2 C2 C4 S3 D4 Q8 D6 D7 Dic6 C4×S3 C3⋊D4 D14 Dic14 C4×D7 C7⋊D4 S3×D7 D21⋊C4 C21⋊D4 C21⋊Q8 kernel Dic21⋊C4 C6×Dic7 Dic3×C14 C2×Dic21 Dic21 C2×Dic7 C42 C42 C2×C14 C2×Dic3 C14 C14 C14 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 1 1 1 3 2 2 2 3 6 6 6 3 3 3 3

Matrix representation of Dic21⋊C4 in GL4(𝔽337) generated by

 0 336 0 0 1 110 0 0 0 0 336 1 0 0 336 0
,
 156 150 0 0 177 181 0 0 0 0 150 41 0 0 191 187
,
 243 225 0 0 112 94 0 0 0 0 295 2 0 0 297 42
G:=sub<GL(4,GF(337))| [0,1,0,0,336,110,0,0,0,0,336,336,0,0,1,0],[156,177,0,0,150,181,0,0,0,0,150,191,0,0,41,187],[243,112,0,0,225,94,0,0,0,0,295,297,0,0,2,42] >;

Dic21⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{21}\rtimes C_4
% in TeX

G:=Group("Dic21:C4");
// GroupNames label

G:=SmallGroup(336,46);
// by ID

G=gap.SmallGroup(336,46);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,31,490,10373]);
// Polycyclic

G:=Group<a,b,c|a^42=c^4=1,b^2=a^21,b*a*b^-1=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^21*b>;
// generators/relations

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