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## G = C20.48SD16order 320 = 26·5

### 3rd non-split extension by C20 of SD16 acting via SD16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20.48SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C20⋊2Q8 — C20.48SD16
 Lower central C5 — C10 — C2×C20 — C20.48SD16
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for C20.48SD16
G = < a,b,c | a20=b8=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a10b3 >

Subgroups: 310 in 96 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, Dic5, C20, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C52C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8⋊Q8, C2×C52C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, Q8×C10, C203C8, C20.Q8, Q8⋊Dic5, C202Q8, Q8×C20, C20.48SD16
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8.C22, Dic10, C5⋊D4, C22×D5, Q8⋊Q8, Q8⋊D5, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, C2×Q8⋊D5, D4.9D10, C20.48SD16

Smallest permutation representation of C20.48SD16
Regular action on 320 points
Generators in S320
(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)
(1 257 262 69 50 231 285 187)(2 256 263 68 51 230 286 186)(3 255 264 67 52 229 287 185)(4 254 265 66 53 228 288 184)(5 253 266 65 54 227 289 183)(6 252 267 64 55 226 290 182)(7 251 268 63 56 225 291 181)(8 250 269 62 57 224 292 200)(9 249 270 61 58 223 293 199)(10 248 271 80 59 222 294 198)(11 247 272 79 60 221 295 197)(12 246 273 78 41 240 296 196)(13 245 274 77 42 239 297 195)(14 244 275 76 43 238 298 194)(15 243 276 75 44 237 299 193)(16 242 277 74 45 236 300 192)(17 241 278 73 46 235 281 191)(18 260 279 72 47 234 282 190)(19 259 280 71 48 233 283 189)(20 258 261 70 49 232 284 188)(21 312 95 148 161 118 214 127)(22 311 96 147 162 117 215 126)(23 310 97 146 163 116 216 125)(24 309 98 145 164 115 217 124)(25 308 99 144 165 114 218 123)(26 307 100 143 166 113 219 122)(27 306 81 142 167 112 220 121)(28 305 82 141 168 111 201 140)(29 304 83 160 169 110 202 139)(30 303 84 159 170 109 203 138)(31 302 85 158 171 108 204 137)(32 301 86 157 172 107 205 136)(33 320 87 156 173 106 206 135)(34 319 88 155 174 105 207 134)(35 318 89 154 175 104 208 133)(36 317 90 153 176 103 209 132)(37 316 91 152 177 102 210 131)(38 315 92 151 178 101 211 130)(39 314 93 150 179 120 212 129)(40 313 94 149 180 119 213 128)
(1 313 11 303)(2 312 12 302)(3 311 13 301)(4 310 14 320)(5 309 15 319)(6 308 16 318)(7 307 17 317)(8 306 18 316)(9 305 19 315)(10 304 20 314)(21 230 31 240)(22 229 32 239)(23 228 33 238)(24 227 34 237)(25 226 35 236)(26 225 36 235)(27 224 37 234)(28 223 38 233)(29 222 39 232)(30 221 40 231)(41 108 51 118)(42 107 52 117)(43 106 53 116)(44 105 54 115)(45 104 55 114)(46 103 56 113)(47 102 57 112)(48 101 58 111)(49 120 59 110)(50 119 60 109)(61 92 71 82)(62 91 72 81)(63 90 73 100)(64 89 74 99)(65 88 75 98)(66 87 76 97)(67 86 77 96)(68 85 78 95)(69 84 79 94)(70 83 80 93)(121 279 131 269)(122 278 132 268)(123 277 133 267)(124 276 134 266)(125 275 135 265)(126 274 136 264)(127 273 137 263)(128 272 138 262)(129 271 139 261)(130 270 140 280)(141 283 151 293)(142 282 152 292)(143 281 153 291)(144 300 154 290)(145 299 155 289)(146 298 156 288)(147 297 157 287)(148 296 158 286)(149 295 159 285)(150 294 160 284)(161 256 171 246)(162 255 172 245)(163 254 173 244)(164 253 174 243)(165 252 175 242)(166 251 176 241)(167 250 177 260)(168 249 178 259)(169 248 179 258)(170 247 180 257)(181 209 191 219)(182 208 192 218)(183 207 193 217)(184 206 194 216)(185 205 195 215)(186 204 196 214)(187 203 197 213)(188 202 198 212)(189 201 199 211)(190 220 200 210)

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4I 4J 4K 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 4 4 4 4 4 ··· 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 4 ··· 4 40 40 2 2 20 20 20 20 2 ··· 2 2 ··· 2 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C2 D4 Q8 D5 SD16 C4○D4 D10 D10 D10 C5⋊D4 Dic10 C4○D20 C8.C22 Q8⋊D5 D4.9D10 kernel C20.48SD16 C20⋊3C8 C20.Q8 Q8⋊Dic5 C20⋊2Q8 Q8×C20 C2×C20 C5×Q8 C4×Q8 C20 C20 C42 C4⋊C4 C2×Q8 C2×C4 Q8 C4 C10 C4 C2 # reps 1 1 2 2 1 1 2 2 2 4 2 2 2 2 8 8 8 1 4 4

Matrix representation of C20.48SD16 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 16 11 0 0 14 2
,
 15 15 0 0 26 15 0 0 0 0 28 22 0 0 37 13
,
 24 9 0 0 9 17 0 0 0 0 28 22 0 0 37 13
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,14,0,0,11,2],[15,26,0,0,15,15,0,0,0,0,28,37,0,0,22,13],[24,9,0,0,9,17,0,0,0,0,28,37,0,0,22,13] >;

C20.48SD16 in GAP, Magma, Sage, TeX

C_{20}._{48}{\rm SD}_{16}
% in TeX

G:=Group("C20.48SD16");
// GroupNames label

G:=SmallGroup(320,647);
// by ID

G=gap.SmallGroup(320,647);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,184,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^10*b^3>;
// generators/relations

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