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## G = Q16×C21order 336 = 24·3·7

### Direct product of C21 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: Q16×C21, C8.C42, C56.7C6, C24.2C14, C168.6C2, C42.56D4, Q8.2C42, C84.79C22, C4.3(C2×C42), C6.16(C7×D4), C2.5(D4×C21), (C7×Q8).8C6, C28.42(C2×C6), C14.32(C3×D4), (Q8×C21).4C2, (C3×Q8).2C14, C12.19(C2×C14), SmallGroup(336,113)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — Q16×C21
 Chief series C1 — C2 — C4 — C28 — C84 — Q8×C21 — Q16×C21
 Lower central C1 — C2 — C4 — Q16×C21
 Upper central C1 — C42 — C84 — Q16×C21

Generators and relations for Q16×C21
G = < a,b,c | a21=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Q16×C21
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 39 82 309 63 231 172)(2 268 40 83 310 43 211 173)(3 269 41 84 311 44 212 174)(4 270 42 64 312 45 213 175)(5 271 22 65 313 46 214 176)(6 272 23 66 314 47 215 177)(7 273 24 67 315 48 216 178)(8 253 25 68 295 49 217 179)(9 254 26 69 296 50 218 180)(10 255 27 70 297 51 219 181)(11 256 28 71 298 52 220 182)(12 257 29 72 299 53 221 183)(13 258 30 73 300 54 222 184)(14 259 31 74 301 55 223 185)(15 260 32 75 302 56 224 186)(16 261 33 76 303 57 225 187)(17 262 34 77 304 58 226 188)(18 263 35 78 305 59 227 189)(19 264 36 79 306 60 228 169)(20 265 37 80 307 61 229 170)(21 266 38 81 308 62 230 171)(85 163 290 201 147 233 319 113)(86 164 291 202 127 234 320 114)(87 165 292 203 128 235 321 115)(88 166 293 204 129 236 322 116)(89 167 294 205 130 237 323 117)(90 168 274 206 131 238 324 118)(91 148 275 207 132 239 325 119)(92 149 276 208 133 240 326 120)(93 150 277 209 134 241 327 121)(94 151 278 210 135 242 328 122)(95 152 279 190 136 243 329 123)(96 153 280 191 137 244 330 124)(97 154 281 192 138 245 331 125)(98 155 282 193 139 246 332 126)(99 156 283 194 140 247 333 106)(100 157 284 195 141 248 334 107)(101 158 285 196 142 249 335 108)(102 159 286 197 143 250 336 109)(103 160 287 198 144 251 316 110)(104 161 288 199 145 252 317 111)(105 162 289 200 146 232 318 112)
(1 137 309 96)(2 138 310 97)(3 139 311 98)(4 140 312 99)(5 141 313 100)(6 142 314 101)(7 143 315 102)(8 144 295 103)(9 145 296 104)(10 146 297 105)(11 147 298 85)(12 127 299 86)(13 128 300 87)(14 129 301 88)(15 130 302 89)(16 131 303 90)(17 132 304 91)(18 133 305 92)(19 134 306 93)(20 135 307 94)(21 136 308 95)(22 284 214 334)(23 285 215 335)(24 286 216 336)(25 287 217 316)(26 288 218 317)(27 289 219 318)(28 290 220 319)(29 291 221 320)(30 292 222 321)(31 293 223 322)(32 294 224 323)(33 274 225 324)(34 275 226 325)(35 276 227 326)(36 277 228 327)(37 278 229 328)(38 279 230 329)(39 280 231 330)(40 281 211 331)(41 282 212 332)(42 283 213 333)(43 125 268 192)(44 126 269 193)(45 106 270 194)(46 107 271 195)(47 108 272 196)(48 109 273 197)(49 110 253 198)(50 111 254 199)(51 112 255 200)(52 113 256 201)(53 114 257 202)(54 115 258 203)(55 116 259 204)(56 117 260 205)(57 118 261 206)(58 119 262 207)(59 120 263 208)(60 121 264 209)(61 122 265 210)(62 123 266 190)(63 124 267 191)(64 156 175 247)(65 157 176 248)(66 158 177 249)(67 159 178 250)(68 160 179 251)(69 161 180 252)(70 162 181 232)(71 163 182 233)(72 164 183 234)(73 165 184 235)(74 166 185 236)(75 167 186 237)(76 168 187 238)(77 148 188 239)(78 149 189 240)(79 150 169 241)(80 151 170 242)(81 152 171 243)(82 153 172 244)(83 154 173 245)(84 155 174 246)

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

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

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

147 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 6A 6B 7A ··· 7F 8A 8B 12A 12B 12C 12D 12E 12F 14A ··· 14F 21A ··· 21L 24A 24B 24C 24D 28A ··· 28F 28G ··· 28R 42A ··· 42L 56A ··· 56L 84A ··· 84L 84M ··· 84AJ 168A ··· 168X order 1 2 3 3 4 4 4 6 6 7 ··· 7 8 8 12 12 12 12 12 12 14 ··· 14 21 ··· 21 24 24 24 24 28 ··· 28 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 84 ··· 84 168 ··· 168 size 1 1 1 1 2 4 4 1 1 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

147 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C3 C6 C6 C7 C14 C14 C21 C42 C42 D4 Q16 C3×D4 C3×Q16 C7×D4 C7×Q16 D4×C21 Q16×C21 kernel Q16×C21 C168 Q8×C21 C7×Q16 C56 C7×Q8 C3×Q16 C24 C3×Q8 Q16 C8 Q8 C42 C21 C14 C7 C6 C3 C2 C1 # reps 1 1 2 2 2 4 6 6 12 12 12 24 1 2 2 4 6 12 12 24

Matrix representation of Q16×C21 in GL2(𝔽337) generated by

 26 0 0 26
,
 324 13 324 324
,
 244 198 198 93
G:=sub<GL(2,GF(337))| [26,0,0,26],[324,324,13,324],[244,198,198,93] >;

Q16×C21 in GAP, Magma, Sage, TeX

Q_{16}\times C_{21}
% in TeX

G:=Group("Q16xC21");
// GroupNames label

G:=SmallGroup(336,113);
// by ID

G=gap.SmallGroup(336,113);
# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,-2,1008,1033,1015,7564,3790,88]);
// Polycyclic

G:=Group<a,b,c|a^21=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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