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G = C3×Dic29order 348 = 22·3·29

Direct product of C3 and Dic29

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×Dic29, C874C4, C58.C6, C292C12, C6.2D29, C174.2C2, C2.(C3×D29), SmallGroup(348,2)

Series: Derived Chief Lower central Upper central

C1C29 — C3×Dic29
C1C29C58C174 — C3×Dic29
C29 — C3×Dic29
C1C6

Generators and relations for C3×Dic29
 G = < a,b,c | a3=b58=1, c2=b29, ab=ba, ac=ca, cbc-1=b-1 >

29C4
29C12

Smallest permutation representation of C3×Dic29
Regular action on 348 points
Generators in S348
(1 153 71)(2 154 72)(3 155 73)(4 156 74)(5 157 75)(6 158 76)(7 159 77)(8 160 78)(9 161 79)(10 162 80)(11 163 81)(12 164 82)(13 165 83)(14 166 84)(15 167 85)(16 168 86)(17 169 87)(18 170 88)(19 171 89)(20 172 90)(21 173 91)(22 174 92)(23 117 93)(24 118 94)(25 119 95)(26 120 96)(27 121 97)(28 122 98)(29 123 99)(30 124 100)(31 125 101)(32 126 102)(33 127 103)(34 128 104)(35 129 105)(36 130 106)(37 131 107)(38 132 108)(39 133 109)(40 134 110)(41 135 111)(42 136 112)(43 137 113)(44 138 114)(45 139 115)(46 140 116)(47 141 59)(48 142 60)(49 143 61)(50 144 62)(51 145 63)(52 146 64)(53 147 65)(54 148 66)(55 149 67)(56 150 68)(57 151 69)(58 152 70)(175 291 233)(176 292 234)(177 293 235)(178 294 236)(179 295 237)(180 296 238)(181 297 239)(182 298 240)(183 299 241)(184 300 242)(185 301 243)(186 302 244)(187 303 245)(188 304 246)(189 305 247)(190 306 248)(191 307 249)(192 308 250)(193 309 251)(194 310 252)(195 311 253)(196 312 254)(197 313 255)(198 314 256)(199 315 257)(200 316 258)(201 317 259)(202 318 260)(203 319 261)(204 320 262)(205 321 263)(206 322 264)(207 323 265)(208 324 266)(209 325 267)(210 326 268)(211 327 269)(212 328 270)(213 329 271)(214 330 272)(215 331 273)(216 332 274)(217 333 275)(218 334 276)(219 335 277)(220 336 278)(221 337 279)(222 338 280)(223 339 281)(224 340 282)(225 341 283)(226 342 284)(227 343 285)(228 344 286)(229 345 287)(230 346 288)(231 347 289)(232 348 290)
(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)
(1 175 30 204)(2 232 31 203)(3 231 32 202)(4 230 33 201)(5 229 34 200)(6 228 35 199)(7 227 36 198)(8 226 37 197)(9 225 38 196)(10 224 39 195)(11 223 40 194)(12 222 41 193)(13 221 42 192)(14 220 43 191)(15 219 44 190)(16 218 45 189)(17 217 46 188)(18 216 47 187)(19 215 48 186)(20 214 49 185)(21 213 50 184)(22 212 51 183)(23 211 52 182)(24 210 53 181)(25 209 54 180)(26 208 55 179)(27 207 56 178)(28 206 57 177)(29 205 58 176)(59 245 88 274)(60 244 89 273)(61 243 90 272)(62 242 91 271)(63 241 92 270)(64 240 93 269)(65 239 94 268)(66 238 95 267)(67 237 96 266)(68 236 97 265)(69 235 98 264)(70 234 99 263)(71 233 100 262)(72 290 101 261)(73 289 102 260)(74 288 103 259)(75 287 104 258)(76 286 105 257)(77 285 106 256)(78 284 107 255)(79 283 108 254)(80 282 109 253)(81 281 110 252)(82 280 111 251)(83 279 112 250)(84 278 113 249)(85 277 114 248)(86 276 115 247)(87 275 116 246)(117 327 146 298)(118 326 147 297)(119 325 148 296)(120 324 149 295)(121 323 150 294)(122 322 151 293)(123 321 152 292)(124 320 153 291)(125 319 154 348)(126 318 155 347)(127 317 156 346)(128 316 157 345)(129 315 158 344)(130 314 159 343)(131 313 160 342)(132 312 161 341)(133 311 162 340)(134 310 163 339)(135 309 164 338)(136 308 165 337)(137 307 166 336)(138 306 167 335)(139 305 168 334)(140 304 169 333)(141 303 170 332)(142 302 171 331)(143 301 172 330)(144 300 173 329)(145 299 174 328)

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

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

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

96 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D29A···29N58A···58N87A···87AB174A···174AB
order123344661212121229···2958···5887···87174···174
size1111292911292929292···22···22···22···2

96 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12D29Dic29C3×D29C3×Dic29
kernelC3×Dic29C174Dic29C87C58C29C6C3C2C1
# reps11222414142828

Matrix representation of C3×Dic29 in GL4(𝔽349) generated by

1000
022600
0010
0001
,
348000
0100
0001
0034818
,
213000
034800
0030190
0027348
G:=sub<GL(4,GF(349))| [1,0,0,0,0,226,0,0,0,0,1,0,0,0,0,1],[348,0,0,0,0,1,0,0,0,0,0,348,0,0,1,18],[213,0,0,0,0,348,0,0,0,0,301,273,0,0,90,48] >;

C3×Dic29 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{29}
% in TeX

G:=Group("C3xDic29");
// GroupNames label

G:=SmallGroup(348,2);
// by ID

G=gap.SmallGroup(348,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-29,24,5379]);
// Polycyclic

G:=Group<a,b,c|a^3=b^58=1,c^2=b^29,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic29 in TeX

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