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## G = C4×C11⋊C8order 352 = 25·11

### Direct product of C4 and C11⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C4×C11⋊C8
 Chief series C1 — C11 — C22 — C44 — C2×C44 — C2×C11⋊C8 — C4×C11⋊C8
 Lower central C11 — C4×C11⋊C8
 Upper central C1 — C42

Generators and relations for C4×C11⋊C8
G = < a,b,c | a4=b11=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C4×C11⋊C8
Regular action on 352 points
Generators in S352
(1 166 34 133)(2 167 35 134)(3 168 36 135)(4 169 37 136)(5 170 38 137)(6 171 39 138)(7 172 40 139)(8 173 41 140)(9 174 42 141)(10 175 43 142)(11 176 44 143)(12 155 23 144)(13 156 24 145)(14 157 25 146)(15 158 26 147)(16 159 27 148)(17 160 28 149)(18 161 29 150)(19 162 30 151)(20 163 31 152)(21 164 32 153)(22 165 33 154)(45 111 78 100)(46 112 79 101)(47 113 80 102)(48 114 81 103)(49 115 82 104)(50 116 83 105)(51 117 84 106)(52 118 85 107)(53 119 86 108)(54 120 87 109)(55 121 88 110)(56 122 67 89)(57 123 68 90)(58 124 69 91)(59 125 70 92)(60 126 71 93)(61 127 72 94)(62 128 73 95)(63 129 74 96)(64 130 75 97)(65 131 76 98)(66 132 77 99)(177 342 210 309)(178 343 211 310)(179 344 212 311)(180 345 213 312)(181 346 214 313)(182 347 215 314)(183 348 216 315)(184 349 217 316)(185 350 218 317)(186 351 219 318)(187 352 220 319)(188 331 199 320)(189 332 200 321)(190 333 201 322)(191 334 202 323)(192 335 203 324)(193 336 204 325)(194 337 205 326)(195 338 206 327)(196 339 207 328)(197 340 208 329)(198 341 209 330)(221 287 254 276)(222 288 255 277)(223 289 256 278)(224 290 257 279)(225 291 258 280)(226 292 259 281)(227 293 260 282)(228 294 261 283)(229 295 262 284)(230 296 263 285)(231 297 264 286)(232 298 243 265)(233 299 244 266)(234 300 245 267)(235 301 246 268)(236 302 247 269)(237 303 248 270)(238 304 249 271)(239 305 250 272)(240 306 251 273)(241 307 252 274)(242 308 253 275)
(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)(331 332 333 334 335 336 337 338 339 340 341)(342 343 344 345 346 347 348 349 350 351 352)
(1 241 56 197 12 230 45 186)(2 240 57 196 13 229 46 185)(3 239 58 195 14 228 47 184)(4 238 59 194 15 227 48 183)(5 237 60 193 16 226 49 182)(6 236 61 192 17 225 50 181)(7 235 62 191 18 224 51 180)(8 234 63 190 19 223 52 179)(9 233 64 189 20 222 53 178)(10 232 65 188 21 221 54 177)(11 242 66 198 22 231 55 187)(23 263 78 219 34 252 67 208)(24 262 79 218 35 251 68 207)(25 261 80 217 36 250 69 206)(26 260 81 216 37 249 70 205)(27 259 82 215 38 248 71 204)(28 258 83 214 39 247 72 203)(29 257 84 213 40 246 73 202)(30 256 85 212 41 245 74 201)(31 255 86 211 42 244 75 200)(32 254 87 210 43 243 76 199)(33 264 88 220 44 253 77 209)(89 329 144 285 100 318 133 274)(90 328 145 284 101 317 134 273)(91 327 146 283 102 316 135 272)(92 326 147 282 103 315 136 271)(93 325 148 281 104 314 137 270)(94 324 149 280 105 313 138 269)(95 323 150 279 106 312 139 268)(96 322 151 278 107 311 140 267)(97 321 152 277 108 310 141 266)(98 320 153 276 109 309 142 265)(99 330 154 286 110 319 143 275)(111 351 166 307 122 340 155 296)(112 350 167 306 123 339 156 295)(113 349 168 305 124 338 157 294)(114 348 169 304 125 337 158 293)(115 347 170 303 126 336 159 292)(116 346 171 302 127 335 160 291)(117 345 172 301 128 334 161 290)(118 344 173 300 129 333 162 289)(119 343 174 299 130 332 163 288)(120 342 175 298 131 331 164 287)(121 352 176 308 132 341 165 297)

G:=sub<Sym(352)| (1,166,34,133)(2,167,35,134)(3,168,36,135)(4,169,37,136)(5,170,38,137)(6,171,39,138)(7,172,40,139)(8,173,41,140)(9,174,42,141)(10,175,43,142)(11,176,44,143)(12,155,23,144)(13,156,24,145)(14,157,25,146)(15,158,26,147)(16,159,27,148)(17,160,28,149)(18,161,29,150)(19,162,30,151)(20,163,31,152)(21,164,32,153)(22,165,33,154)(45,111,78,100)(46,112,79,101)(47,113,80,102)(48,114,81,103)(49,115,82,104)(50,116,83,105)(51,117,84,106)(52,118,85,107)(53,119,86,108)(54,120,87,109)(55,121,88,110)(56,122,67,89)(57,123,68,90)(58,124,69,91)(59,125,70,92)(60,126,71,93)(61,127,72,94)(62,128,73,95)(63,129,74,96)(64,130,75,97)(65,131,76,98)(66,132,77,99)(177,342,210,309)(178,343,211,310)(179,344,212,311)(180,345,213,312)(181,346,214,313)(182,347,215,314)(183,348,216,315)(184,349,217,316)(185,350,218,317)(186,351,219,318)(187,352,220,319)(188,331,199,320)(189,332,200,321)(190,333,201,322)(191,334,202,323)(192,335,203,324)(193,336,204,325)(194,337,205,326)(195,338,206,327)(196,339,207,328)(197,340,208,329)(198,341,209,330)(221,287,254,276)(222,288,255,277)(223,289,256,278)(224,290,257,279)(225,291,258,280)(226,292,259,281)(227,293,260,282)(228,294,261,283)(229,295,262,284)(230,296,263,285)(231,297,264,286)(232,298,243,265)(233,299,244,266)(234,300,245,267)(235,301,246,268)(236,302,247,269)(237,303,248,270)(238,304,249,271)(239,305,250,272)(240,306,251,273)(241,307,252,274)(242,308,253,275), (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)(331,332,333,334,335,336,337,338,339,340,341)(342,343,344,345,346,347,348,349,350,351,352), (1,241,56,197,12,230,45,186)(2,240,57,196,13,229,46,185)(3,239,58,195,14,228,47,184)(4,238,59,194,15,227,48,183)(5,237,60,193,16,226,49,182)(6,236,61,192,17,225,50,181)(7,235,62,191,18,224,51,180)(8,234,63,190,19,223,52,179)(9,233,64,189,20,222,53,178)(10,232,65,188,21,221,54,177)(11,242,66,198,22,231,55,187)(23,263,78,219,34,252,67,208)(24,262,79,218,35,251,68,207)(25,261,80,217,36,250,69,206)(26,260,81,216,37,249,70,205)(27,259,82,215,38,248,71,204)(28,258,83,214,39,247,72,203)(29,257,84,213,40,246,73,202)(30,256,85,212,41,245,74,201)(31,255,86,211,42,244,75,200)(32,254,87,210,43,243,76,199)(33,264,88,220,44,253,77,209)(89,329,144,285,100,318,133,274)(90,328,145,284,101,317,134,273)(91,327,146,283,102,316,135,272)(92,326,147,282,103,315,136,271)(93,325,148,281,104,314,137,270)(94,324,149,280,105,313,138,269)(95,323,150,279,106,312,139,268)(96,322,151,278,107,311,140,267)(97,321,152,277,108,310,141,266)(98,320,153,276,109,309,142,265)(99,330,154,286,110,319,143,275)(111,351,166,307,122,340,155,296)(112,350,167,306,123,339,156,295)(113,349,168,305,124,338,157,294)(114,348,169,304,125,337,158,293)(115,347,170,303,126,336,159,292)(116,346,171,302,127,335,160,291)(117,345,172,301,128,334,161,290)(118,344,173,300,129,333,162,289)(119,343,174,299,130,332,163,288)(120,342,175,298,131,331,164,287)(121,352,176,308,132,341,165,297)>;

G:=Group( (1,166,34,133)(2,167,35,134)(3,168,36,135)(4,169,37,136)(5,170,38,137)(6,171,39,138)(7,172,40,139)(8,173,41,140)(9,174,42,141)(10,175,43,142)(11,176,44,143)(12,155,23,144)(13,156,24,145)(14,157,25,146)(15,158,26,147)(16,159,27,148)(17,160,28,149)(18,161,29,150)(19,162,30,151)(20,163,31,152)(21,164,32,153)(22,165,33,154)(45,111,78,100)(46,112,79,101)(47,113,80,102)(48,114,81,103)(49,115,82,104)(50,116,83,105)(51,117,84,106)(52,118,85,107)(53,119,86,108)(54,120,87,109)(55,121,88,110)(56,122,67,89)(57,123,68,90)(58,124,69,91)(59,125,70,92)(60,126,71,93)(61,127,72,94)(62,128,73,95)(63,129,74,96)(64,130,75,97)(65,131,76,98)(66,132,77,99)(177,342,210,309)(178,343,211,310)(179,344,212,311)(180,345,213,312)(181,346,214,313)(182,347,215,314)(183,348,216,315)(184,349,217,316)(185,350,218,317)(186,351,219,318)(187,352,220,319)(188,331,199,320)(189,332,200,321)(190,333,201,322)(191,334,202,323)(192,335,203,324)(193,336,204,325)(194,337,205,326)(195,338,206,327)(196,339,207,328)(197,340,208,329)(198,341,209,330)(221,287,254,276)(222,288,255,277)(223,289,256,278)(224,290,257,279)(225,291,258,280)(226,292,259,281)(227,293,260,282)(228,294,261,283)(229,295,262,284)(230,296,263,285)(231,297,264,286)(232,298,243,265)(233,299,244,266)(234,300,245,267)(235,301,246,268)(236,302,247,269)(237,303,248,270)(238,304,249,271)(239,305,250,272)(240,306,251,273)(241,307,252,274)(242,308,253,275), (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)(331,332,333,334,335,336,337,338,339,340,341)(342,343,344,345,346,347,348,349,350,351,352), (1,241,56,197,12,230,45,186)(2,240,57,196,13,229,46,185)(3,239,58,195,14,228,47,184)(4,238,59,194,15,227,48,183)(5,237,60,193,16,226,49,182)(6,236,61,192,17,225,50,181)(7,235,62,191,18,224,51,180)(8,234,63,190,19,223,52,179)(9,233,64,189,20,222,53,178)(10,232,65,188,21,221,54,177)(11,242,66,198,22,231,55,187)(23,263,78,219,34,252,67,208)(24,262,79,218,35,251,68,207)(25,261,80,217,36,250,69,206)(26,260,81,216,37,249,70,205)(27,259,82,215,38,248,71,204)(28,258,83,214,39,247,72,203)(29,257,84,213,40,246,73,202)(30,256,85,212,41,245,74,201)(31,255,86,211,42,244,75,200)(32,254,87,210,43,243,76,199)(33,264,88,220,44,253,77,209)(89,329,144,285,100,318,133,274)(90,328,145,284,101,317,134,273)(91,327,146,283,102,316,135,272)(92,326,147,282,103,315,136,271)(93,325,148,281,104,314,137,270)(94,324,149,280,105,313,138,269)(95,323,150,279,106,312,139,268)(96,322,151,278,107,311,140,267)(97,321,152,277,108,310,141,266)(98,320,153,276,109,309,142,265)(99,330,154,286,110,319,143,275)(111,351,166,307,122,340,155,296)(112,350,167,306,123,339,156,295)(113,349,168,305,124,338,157,294)(114,348,169,304,125,337,158,293)(115,347,170,303,126,336,159,292)(116,346,171,302,127,335,160,291)(117,345,172,301,128,334,161,290)(118,344,173,300,129,333,162,289)(119,343,174,299,130,332,163,288)(120,342,175,298,131,331,164,287)(121,352,176,308,132,341,165,297) );

G=PermutationGroup([[(1,166,34,133),(2,167,35,134),(3,168,36,135),(4,169,37,136),(5,170,38,137),(6,171,39,138),(7,172,40,139),(8,173,41,140),(9,174,42,141),(10,175,43,142),(11,176,44,143),(12,155,23,144),(13,156,24,145),(14,157,25,146),(15,158,26,147),(16,159,27,148),(17,160,28,149),(18,161,29,150),(19,162,30,151),(20,163,31,152),(21,164,32,153),(22,165,33,154),(45,111,78,100),(46,112,79,101),(47,113,80,102),(48,114,81,103),(49,115,82,104),(50,116,83,105),(51,117,84,106),(52,118,85,107),(53,119,86,108),(54,120,87,109),(55,121,88,110),(56,122,67,89),(57,123,68,90),(58,124,69,91),(59,125,70,92),(60,126,71,93),(61,127,72,94),(62,128,73,95),(63,129,74,96),(64,130,75,97),(65,131,76,98),(66,132,77,99),(177,342,210,309),(178,343,211,310),(179,344,212,311),(180,345,213,312),(181,346,214,313),(182,347,215,314),(183,348,216,315),(184,349,217,316),(185,350,218,317),(186,351,219,318),(187,352,220,319),(188,331,199,320),(189,332,200,321),(190,333,201,322),(191,334,202,323),(192,335,203,324),(193,336,204,325),(194,337,205,326),(195,338,206,327),(196,339,207,328),(197,340,208,329),(198,341,209,330),(221,287,254,276),(222,288,255,277),(223,289,256,278),(224,290,257,279),(225,291,258,280),(226,292,259,281),(227,293,260,282),(228,294,261,283),(229,295,262,284),(230,296,263,285),(231,297,264,286),(232,298,243,265),(233,299,244,266),(234,300,245,267),(235,301,246,268),(236,302,247,269),(237,303,248,270),(238,304,249,271),(239,305,250,272),(240,306,251,273),(241,307,252,274),(242,308,253,275)], [(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),(331,332,333,334,335,336,337,338,339,340,341),(342,343,344,345,346,347,348,349,350,351,352)], [(1,241,56,197,12,230,45,186),(2,240,57,196,13,229,46,185),(3,239,58,195,14,228,47,184),(4,238,59,194,15,227,48,183),(5,237,60,193,16,226,49,182),(6,236,61,192,17,225,50,181),(7,235,62,191,18,224,51,180),(8,234,63,190,19,223,52,179),(9,233,64,189,20,222,53,178),(10,232,65,188,21,221,54,177),(11,242,66,198,22,231,55,187),(23,263,78,219,34,252,67,208),(24,262,79,218,35,251,68,207),(25,261,80,217,36,250,69,206),(26,260,81,216,37,249,70,205),(27,259,82,215,38,248,71,204),(28,258,83,214,39,247,72,203),(29,257,84,213,40,246,73,202),(30,256,85,212,41,245,74,201),(31,255,86,211,42,244,75,200),(32,254,87,210,43,243,76,199),(33,264,88,220,44,253,77,209),(89,329,144,285,100,318,133,274),(90,328,145,284,101,317,134,273),(91,327,146,283,102,316,135,272),(92,326,147,282,103,315,136,271),(93,325,148,281,104,314,137,270),(94,324,149,280,105,313,138,269),(95,323,150,279,106,312,139,268),(96,322,151,278,107,311,140,267),(97,321,152,277,108,310,141,266),(98,320,153,276,109,309,142,265),(99,330,154,286,110,319,143,275),(111,351,166,307,122,340,155,296),(112,350,167,306,123,339,156,295),(113,349,168,305,124,338,157,294),(114,348,169,304,125,337,158,293),(115,347,170,303,126,336,159,292),(116,346,171,302,127,335,160,291),(117,345,172,301,128,334,161,290),(118,344,173,300,129,333,162,289),(119,343,174,299,130,332,163,288),(120,342,175,298,131,331,164,287),(121,352,176,308,132,341,165,297)]])

112 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 8A ··· 8P 11A ··· 11E 22A ··· 22O 44A ··· 44BH order 1 2 2 2 4 ··· 4 8 ··· 8 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 1 ··· 1 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C8 D11 Dic11 D22 C11⋊C8 C4×D11 kernel C4×C11⋊C8 C2×C11⋊C8 C4×C44 C11⋊C8 C2×C44 C44 C42 C2×C4 C2×C4 C4 C4 # reps 1 2 1 8 4 16 5 10 5 40 20

Matrix representation of C4×C11⋊C8 in GL3(𝔽89) generated by

 55 0 0 0 55 0 0 0 55
,
 1 0 0 0 55 88 0 1 0
,
 12 0 0 0 6 45 0 19 83
G:=sub<GL(3,GF(89))| [55,0,0,0,55,0,0,0,55],[1,0,0,0,55,1,0,88,0],[12,0,0,0,6,19,0,45,83] >;

C4×C11⋊C8 in GAP, Magma, Sage, TeX

C_4\times C_{11}\rtimes C_8
% in TeX

G:=Group("C4xC11:C8");
// GroupNames label

G:=SmallGroup(352,8);
// by ID

G=gap.SmallGroup(352,8);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,55,86,11525]);
// Polycyclic

G:=Group<a,b,c|a^4=b^11=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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