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## G = C22×C52⋊6C4order 400 = 24·52

### Direct product of C22 and C52⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C22×C52⋊6C4
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C2×C52⋊6C4 — C22×C52⋊6C4
 Lower central C52 — C22×C52⋊6C4
 Upper central C1 — C23

Generators and relations for C22×C526C4
G = < a,b,c,d,e | a2=b2=c5=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 744 in 216 conjugacy classes, 139 normal (7 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, Dic5, C2×C10, C52, C2×Dic5, C22×C10, C5×C10, C5×C10, C22×Dic5, C526C4, C102, C2×C526C4, C2×C102, C22×C526C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, Dic5, D10, C2×Dic5, C22×D5, C5⋊D5, C22×Dic5, C526C4, C2×C5⋊D5, C2×C526C4, C22×C5⋊D5, C22×C526C4

Smallest permutation representation of C22×C526C4
Regular action on 400 points
Generators in S400
(1 213)(2 214)(3 215)(4 211)(5 212)(6 245)(7 241)(8 242)(9 243)(10 244)(11 240)(12 236)(13 237)(14 238)(15 239)(16 235)(17 231)(18 232)(19 233)(20 234)(21 230)(22 226)(23 227)(24 228)(25 229)(26 216)(27 217)(28 218)(29 219)(30 220)(31 198)(32 199)(33 200)(34 196)(35 197)(36 206)(37 207)(38 208)(39 209)(40 210)(41 201)(42 202)(43 203)(44 204)(45 205)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 181)(57 182)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 194)(70 195)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 169)(95 170)(96 121)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 129)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 145)(221 396)(222 397)(223 398)(224 399)(225 400)(246 371)(247 372)(248 373)(249 374)(250 375)(251 376)(252 377)(253 378)(254 379)(255 380)(256 381)(257 382)(258 383)(259 384)(260 385)(261 386)(262 387)(263 388)(264 389)(265 390)(266 391)(267 392)(268 393)(269 394)(270 395)(271 346)(272 347)(273 348)(274 349)(275 350)(276 351)(277 352)(278 353)(279 354)(280 355)(281 356)(282 357)(283 358)(284 359)(285 360)(286 361)(287 362)(288 363)(289 364)(290 365)(291 366)(292 367)(293 368)(294 369)(295 370)(296 321)(297 322)(298 323)(299 324)(300 325)(301 326)(302 327)(303 328)(304 329)(305 330)(306 331)(307 332)(308 333)(309 334)(310 335)(311 336)(312 337)(313 338)(314 339)(315 340)(316 341)(317 342)(318 343)(319 344)(320 345)
(1 108)(2 109)(3 110)(4 106)(5 107)(6 345)(7 341)(8 342)(9 343)(10 344)(11 340)(12 336)(13 337)(14 338)(15 339)(16 335)(17 331)(18 332)(19 333)(20 334)(21 330)(22 326)(23 327)(24 328)(25 329)(26 101)(27 102)(28 103)(29 104)(30 105)(31 98)(32 99)(33 100)(34 96)(35 97)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 81)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(121 196)(122 197)(123 198)(124 199)(125 200)(126 216)(127 217)(128 218)(129 219)(130 220)(131 211)(132 212)(133 213)(134 214)(135 215)(136 206)(137 207)(138 208)(139 209)(140 210)(141 201)(142 202)(143 203)(144 204)(145 205)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 181)(157 182)(158 183)(159 184)(160 185)(161 186)(162 187)(163 188)(164 189)(165 190)(166 191)(167 192)(168 193)(169 194)(170 195)(221 296)(222 297)(223 298)(224 299)(225 300)(226 301)(227 302)(228 303)(229 304)(230 305)(231 306)(232 307)(233 308)(234 309)(235 310)(236 311)(237 312)(238 313)(239 314)(240 315)(241 316)(242 317)(243 318)(244 319)(245 320)(246 271)(247 272)(248 273)(249 274)(250 275)(251 276)(252 277)(253 278)(254 279)(255 280)(256 281)(257 282)(258 283)(259 284)(260 285)(261 286)(262 287)(263 288)(264 289)(265 290)(266 291)(267 292)(268 293)(269 294)(270 295)(321 396)(322 397)(323 398)(324 399)(325 400)(346 371)(347 372)(348 373)(349 374)(350 375)(351 376)(352 377)(353 378)(354 379)(355 380)(356 381)(357 382)(358 383)(359 384)(360 385)(361 386)(362 387)(363 388)(364 389)(365 390)(366 391)(367 392)(368 393)(369 394)(370 395)
(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)(351 352 353 354 355)(356 357 358 359 360)(361 362 363 364 365)(366 367 368 369 370)(371 372 373 374 375)(376 377 378 379 380)(381 382 383 384 385)(386 387 388 389 390)(391 392 393 394 395)(396 397 398 399 400)
(1 44 28 31 38)(2 45 29 32 39)(3 41 30 33 40)(4 42 26 34 36)(5 43 27 35 37)(6 400 25 16 11)(7 396 21 17 12)(8 397 22 18 13)(9 398 23 19 14)(10 399 24 20 15)(46 61 56 67 51)(47 62 57 68 52)(48 63 58 69 53)(49 64 59 70 54)(50 65 60 66 55)(71 86 81 92 76)(72 87 82 93 77)(73 88 83 94 78)(74 89 84 95 79)(75 90 85 91 80)(96 111 106 117 101)(97 112 107 118 102)(98 113 108 119 103)(99 114 109 120 104)(100 115 110 116 105)(121 136 131 142 126)(122 137 132 143 127)(123 138 133 144 128)(124 139 134 145 129)(125 140 135 141 130)(146 161 156 167 151)(147 162 157 168 152)(148 163 158 169 153)(149 164 159 170 154)(150 165 160 166 155)(171 186 181 192 176)(172 187 182 193 177)(173 188 183 194 178)(174 189 184 195 179)(175 190 185 191 180)(196 206 211 202 216)(197 207 212 203 217)(198 208 213 204 218)(199 209 214 205 219)(200 210 215 201 220)(221 230 231 236 241)(222 226 232 237 242)(223 227 233 238 243)(224 228 234 239 244)(225 229 235 240 245)(246 255 256 261 266)(247 251 257 262 267)(248 252 258 263 268)(249 253 259 264 269)(250 254 260 265 270)(271 280 281 286 291)(272 276 282 287 292)(273 277 283 288 293)(274 278 284 289 294)(275 279 285 290 295)(296 305 306 311 316)(297 301 307 312 317)(298 302 308 313 318)(299 303 309 314 319)(300 304 310 315 320)(321 330 331 336 341)(322 326 332 337 342)(323 327 333 338 343)(324 328 334 339 344)(325 329 335 340 345)(346 355 356 361 366)(347 351 357 362 367)(348 352 358 363 368)(349 353 359 364 369)(350 354 360 365 370)(371 380 381 386 391)(372 376 382 387 392)(373 377 383 388 393)(374 378 384 389 394)(375 379 385 390 395)
(1 307 58 282)(2 306 59 281)(3 310 60 285)(4 309 56 284)(5 308 57 283)(6 150 395 125)(7 149 391 124)(8 148 392 123)(9 147 393 122)(10 146 394 121)(11 165 390 140)(12 164 386 139)(13 163 387 138)(14 162 388 137)(15 161 389 136)(16 160 385 135)(17 159 381 134)(18 158 382 133)(19 157 383 132)(20 156 384 131)(21 170 380 145)(22 169 376 144)(23 168 377 143)(24 167 378 142)(25 166 379 141)(26 299 51 274)(27 298 52 273)(28 297 53 272)(29 296 54 271)(30 300 55 275)(31 317 48 292)(32 316 49 291)(33 320 50 295)(34 319 46 294)(35 318 47 293)(36 314 61 289)(37 313 62 288)(38 312 63 287)(39 311 64 286)(40 315 65 290)(41 304 66 279)(42 303 67 278)(43 302 68 277)(44 301 69 276)(45 305 70 280)(71 269 96 244)(72 268 97 243)(73 267 98 242)(74 266 99 241)(75 270 100 245)(76 249 101 224)(77 248 102 223)(78 247 103 222)(79 246 104 221)(80 250 105 225)(81 259 106 234)(82 258 107 233)(83 257 108 232)(84 256 109 231)(85 260 110 235)(86 264 111 239)(87 263 112 238)(88 262 113 237)(89 261 114 236)(90 265 115 240)(91 254 116 229)(92 253 117 228)(93 252 118 227)(94 251 119 226)(95 255 120 230)(126 399 151 374)(127 398 152 373)(128 397 153 372)(129 396 154 371)(130 400 155 375)(171 369 196 344)(172 368 197 343)(173 367 198 342)(174 366 199 341)(175 370 200 345)(176 349 216 324)(177 348 217 323)(178 347 218 322)(179 346 219 321)(180 350 220 325)(181 359 211 334)(182 358 212 333)(183 357 213 332)(184 356 214 331)(185 360 215 335)(186 364 206 339)(187 363 207 338)(188 362 208 337)(189 361 209 336)(190 365 210 340)(191 354 201 329)(192 353 202 328)(193 352 203 327)(194 351 204 326)(195 355 205 330)

G:=sub<Sym(400)| (1,213)(2,214)(3,215)(4,211)(5,212)(6,245)(7,241)(8,242)(9,243)(10,244)(11,240)(12,236)(13,237)(14,238)(15,239)(16,235)(17,231)(18,232)(19,233)(20,234)(21,230)(22,226)(23,227)(24,228)(25,229)(26,216)(27,217)(28,218)(29,219)(30,220)(31,198)(32,199)(33,200)(34,196)(35,197)(36,206)(37,207)(38,208)(39,209)(40,210)(41,201)(42,202)(43,203)(44,204)(45,205)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(221,396)(222,397)(223,398)(224,399)(225,400)(246,371)(247,372)(248,373)(249,374)(250,375)(251,376)(252,377)(253,378)(254,379)(255,380)(256,381)(257,382)(258,383)(259,384)(260,385)(261,386)(262,387)(263,388)(264,389)(265,390)(266,391)(267,392)(268,393)(269,394)(270,395)(271,346)(272,347)(273,348)(274,349)(275,350)(276,351)(277,352)(278,353)(279,354)(280,355)(281,356)(282,357)(283,358)(284,359)(285,360)(286,361)(287,362)(288,363)(289,364)(290,365)(291,366)(292,367)(293,368)(294,369)(295,370)(296,321)(297,322)(298,323)(299,324)(300,325)(301,326)(302,327)(303,328)(304,329)(305,330)(306,331)(307,332)(308,333)(309,334)(310,335)(311,336)(312,337)(313,338)(314,339)(315,340)(316,341)(317,342)(318,343)(319,344)(320,345), (1,108)(2,109)(3,110)(4,106)(5,107)(6,345)(7,341)(8,342)(9,343)(10,344)(11,340)(12,336)(13,337)(14,338)(15,339)(16,335)(17,331)(18,332)(19,333)(20,334)(21,330)(22,326)(23,327)(24,328)(25,329)(26,101)(27,102)(28,103)(29,104)(30,105)(31,98)(32,99)(33,100)(34,96)(35,97)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(121,196)(122,197)(123,198)(124,199)(125,200)(126,216)(127,217)(128,218)(129,219)(130,220)(131,211)(132,212)(133,213)(134,214)(135,215)(136,206)(137,207)(138,208)(139,209)(140,210)(141,201)(142,202)(143,203)(144,204)(145,205)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181)(157,182)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(221,296)(222,297)(223,298)(224,299)(225,300)(226,301)(227,302)(228,303)(229,304)(230,305)(231,306)(232,307)(233,308)(234,309)(235,310)(236,311)(237,312)(238,313)(239,314)(240,315)(241,316)(242,317)(243,318)(244,319)(245,320)(246,271)(247,272)(248,273)(249,274)(250,275)(251,276)(252,277)(253,278)(254,279)(255,280)(256,281)(257,282)(258,283)(259,284)(260,285)(261,286)(262,287)(263,288)(264,289)(265,290)(266,291)(267,292)(268,293)(269,294)(270,295)(321,396)(322,397)(323,398)(324,399)(325,400)(346,371)(347,372)(348,373)(349,374)(350,375)(351,376)(352,377)(353,378)(354,379)(355,380)(356,381)(357,382)(358,383)(359,384)(360,385)(361,386)(362,387)(363,388)(364,389)(365,390)(366,391)(367,392)(368,393)(369,394)(370,395), (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)(351,352,353,354,355)(356,357,358,359,360)(361,362,363,364,365)(366,367,368,369,370)(371,372,373,374,375)(376,377,378,379,380)(381,382,383,384,385)(386,387,388,389,390)(391,392,393,394,395)(396,397,398,399,400), (1,44,28,31,38)(2,45,29,32,39)(3,41,30,33,40)(4,42,26,34,36)(5,43,27,35,37)(6,400,25,16,11)(7,396,21,17,12)(8,397,22,18,13)(9,398,23,19,14)(10,399,24,20,15)(46,61,56,67,51)(47,62,57,68,52)(48,63,58,69,53)(49,64,59,70,54)(50,65,60,66,55)(71,86,81,92,76)(72,87,82,93,77)(73,88,83,94,78)(74,89,84,95,79)(75,90,85,91,80)(96,111,106,117,101)(97,112,107,118,102)(98,113,108,119,103)(99,114,109,120,104)(100,115,110,116,105)(121,136,131,142,126)(122,137,132,143,127)(123,138,133,144,128)(124,139,134,145,129)(125,140,135,141,130)(146,161,156,167,151)(147,162,157,168,152)(148,163,158,169,153)(149,164,159,170,154)(150,165,160,166,155)(171,186,181,192,176)(172,187,182,193,177)(173,188,183,194,178)(174,189,184,195,179)(175,190,185,191,180)(196,206,211,202,216)(197,207,212,203,217)(198,208,213,204,218)(199,209,214,205,219)(200,210,215,201,220)(221,230,231,236,241)(222,226,232,237,242)(223,227,233,238,243)(224,228,234,239,244)(225,229,235,240,245)(246,255,256,261,266)(247,251,257,262,267)(248,252,258,263,268)(249,253,259,264,269)(250,254,260,265,270)(271,280,281,286,291)(272,276,282,287,292)(273,277,283,288,293)(274,278,284,289,294)(275,279,285,290,295)(296,305,306,311,316)(297,301,307,312,317)(298,302,308,313,318)(299,303,309,314,319)(300,304,310,315,320)(321,330,331,336,341)(322,326,332,337,342)(323,327,333,338,343)(324,328,334,339,344)(325,329,335,340,345)(346,355,356,361,366)(347,351,357,362,367)(348,352,358,363,368)(349,353,359,364,369)(350,354,360,365,370)(371,380,381,386,391)(372,376,382,387,392)(373,377,383,388,393)(374,378,384,389,394)(375,379,385,390,395), (1,307,58,282)(2,306,59,281)(3,310,60,285)(4,309,56,284)(5,308,57,283)(6,150,395,125)(7,149,391,124)(8,148,392,123)(9,147,393,122)(10,146,394,121)(11,165,390,140)(12,164,386,139)(13,163,387,138)(14,162,388,137)(15,161,389,136)(16,160,385,135)(17,159,381,134)(18,158,382,133)(19,157,383,132)(20,156,384,131)(21,170,380,145)(22,169,376,144)(23,168,377,143)(24,167,378,142)(25,166,379,141)(26,299,51,274)(27,298,52,273)(28,297,53,272)(29,296,54,271)(30,300,55,275)(31,317,48,292)(32,316,49,291)(33,320,50,295)(34,319,46,294)(35,318,47,293)(36,314,61,289)(37,313,62,288)(38,312,63,287)(39,311,64,286)(40,315,65,290)(41,304,66,279)(42,303,67,278)(43,302,68,277)(44,301,69,276)(45,305,70,280)(71,269,96,244)(72,268,97,243)(73,267,98,242)(74,266,99,241)(75,270,100,245)(76,249,101,224)(77,248,102,223)(78,247,103,222)(79,246,104,221)(80,250,105,225)(81,259,106,234)(82,258,107,233)(83,257,108,232)(84,256,109,231)(85,260,110,235)(86,264,111,239)(87,263,112,238)(88,262,113,237)(89,261,114,236)(90,265,115,240)(91,254,116,229)(92,253,117,228)(93,252,118,227)(94,251,119,226)(95,255,120,230)(126,399,151,374)(127,398,152,373)(128,397,153,372)(129,396,154,371)(130,400,155,375)(171,369,196,344)(172,368,197,343)(173,367,198,342)(174,366,199,341)(175,370,200,345)(176,349,216,324)(177,348,217,323)(178,347,218,322)(179,346,219,321)(180,350,220,325)(181,359,211,334)(182,358,212,333)(183,357,213,332)(184,356,214,331)(185,360,215,335)(186,364,206,339)(187,363,207,338)(188,362,208,337)(189,361,209,336)(190,365,210,340)(191,354,201,329)(192,353,202,328)(193,352,203,327)(194,351,204,326)(195,355,205,330)>;

G:=Group( (1,213)(2,214)(3,215)(4,211)(5,212)(6,245)(7,241)(8,242)(9,243)(10,244)(11,240)(12,236)(13,237)(14,238)(15,239)(16,235)(17,231)(18,232)(19,233)(20,234)(21,230)(22,226)(23,227)(24,228)(25,229)(26,216)(27,217)(28,218)(29,219)(30,220)(31,198)(32,199)(33,200)(34,196)(35,197)(36,206)(37,207)(38,208)(39,209)(40,210)(41,201)(42,202)(43,203)(44,204)(45,205)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(221,396)(222,397)(223,398)(224,399)(225,400)(246,371)(247,372)(248,373)(249,374)(250,375)(251,376)(252,377)(253,378)(254,379)(255,380)(256,381)(257,382)(258,383)(259,384)(260,385)(261,386)(262,387)(263,388)(264,389)(265,390)(266,391)(267,392)(268,393)(269,394)(270,395)(271,346)(272,347)(273,348)(274,349)(275,350)(276,351)(277,352)(278,353)(279,354)(280,355)(281,356)(282,357)(283,358)(284,359)(285,360)(286,361)(287,362)(288,363)(289,364)(290,365)(291,366)(292,367)(293,368)(294,369)(295,370)(296,321)(297,322)(298,323)(299,324)(300,325)(301,326)(302,327)(303,328)(304,329)(305,330)(306,331)(307,332)(308,333)(309,334)(310,335)(311,336)(312,337)(313,338)(314,339)(315,340)(316,341)(317,342)(318,343)(319,344)(320,345), (1,108)(2,109)(3,110)(4,106)(5,107)(6,345)(7,341)(8,342)(9,343)(10,344)(11,340)(12,336)(13,337)(14,338)(15,339)(16,335)(17,331)(18,332)(19,333)(20,334)(21,330)(22,326)(23,327)(24,328)(25,329)(26,101)(27,102)(28,103)(29,104)(30,105)(31,98)(32,99)(33,100)(34,96)(35,97)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(121,196)(122,197)(123,198)(124,199)(125,200)(126,216)(127,217)(128,218)(129,219)(130,220)(131,211)(132,212)(133,213)(134,214)(135,215)(136,206)(137,207)(138,208)(139,209)(140,210)(141,201)(142,202)(143,203)(144,204)(145,205)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,181)(157,182)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(221,296)(222,297)(223,298)(224,299)(225,300)(226,301)(227,302)(228,303)(229,304)(230,305)(231,306)(232,307)(233,308)(234,309)(235,310)(236,311)(237,312)(238,313)(239,314)(240,315)(241,316)(242,317)(243,318)(244,319)(245,320)(246,271)(247,272)(248,273)(249,274)(250,275)(251,276)(252,277)(253,278)(254,279)(255,280)(256,281)(257,282)(258,283)(259,284)(260,285)(261,286)(262,287)(263,288)(264,289)(265,290)(266,291)(267,292)(268,293)(269,294)(270,295)(321,396)(322,397)(323,398)(324,399)(325,400)(346,371)(347,372)(348,373)(349,374)(350,375)(351,376)(352,377)(353,378)(354,379)(355,380)(356,381)(357,382)(358,383)(359,384)(360,385)(361,386)(362,387)(363,388)(364,389)(365,390)(366,391)(367,392)(368,393)(369,394)(370,395), (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)(351,352,353,354,355)(356,357,358,359,360)(361,362,363,364,365)(366,367,368,369,370)(371,372,373,374,375)(376,377,378,379,380)(381,382,383,384,385)(386,387,388,389,390)(391,392,393,394,395)(396,397,398,399,400), (1,44,28,31,38)(2,45,29,32,39)(3,41,30,33,40)(4,42,26,34,36)(5,43,27,35,37)(6,400,25,16,11)(7,396,21,17,12)(8,397,22,18,13)(9,398,23,19,14)(10,399,24,20,15)(46,61,56,67,51)(47,62,57,68,52)(48,63,58,69,53)(49,64,59,70,54)(50,65,60,66,55)(71,86,81,92,76)(72,87,82,93,77)(73,88,83,94,78)(74,89,84,95,79)(75,90,85,91,80)(96,111,106,117,101)(97,112,107,118,102)(98,113,108,119,103)(99,114,109,120,104)(100,115,110,116,105)(121,136,131,142,126)(122,137,132,143,127)(123,138,133,144,128)(124,139,134,145,129)(125,140,135,141,130)(146,161,156,167,151)(147,162,157,168,152)(148,163,158,169,153)(149,164,159,170,154)(150,165,160,166,155)(171,186,181,192,176)(172,187,182,193,177)(173,188,183,194,178)(174,189,184,195,179)(175,190,185,191,180)(196,206,211,202,216)(197,207,212,203,217)(198,208,213,204,218)(199,209,214,205,219)(200,210,215,201,220)(221,230,231,236,241)(222,226,232,237,242)(223,227,233,238,243)(224,228,234,239,244)(225,229,235,240,245)(246,255,256,261,266)(247,251,257,262,267)(248,252,258,263,268)(249,253,259,264,269)(250,254,260,265,270)(271,280,281,286,291)(272,276,282,287,292)(273,277,283,288,293)(274,278,284,289,294)(275,279,285,290,295)(296,305,306,311,316)(297,301,307,312,317)(298,302,308,313,318)(299,303,309,314,319)(300,304,310,315,320)(321,330,331,336,341)(322,326,332,337,342)(323,327,333,338,343)(324,328,334,339,344)(325,329,335,340,345)(346,355,356,361,366)(347,351,357,362,367)(348,352,358,363,368)(349,353,359,364,369)(350,354,360,365,370)(371,380,381,386,391)(372,376,382,387,392)(373,377,383,388,393)(374,378,384,389,394)(375,379,385,390,395), (1,307,58,282)(2,306,59,281)(3,310,60,285)(4,309,56,284)(5,308,57,283)(6,150,395,125)(7,149,391,124)(8,148,392,123)(9,147,393,122)(10,146,394,121)(11,165,390,140)(12,164,386,139)(13,163,387,138)(14,162,388,137)(15,161,389,136)(16,160,385,135)(17,159,381,134)(18,158,382,133)(19,157,383,132)(20,156,384,131)(21,170,380,145)(22,169,376,144)(23,168,377,143)(24,167,378,142)(25,166,379,141)(26,299,51,274)(27,298,52,273)(28,297,53,272)(29,296,54,271)(30,300,55,275)(31,317,48,292)(32,316,49,291)(33,320,50,295)(34,319,46,294)(35,318,47,293)(36,314,61,289)(37,313,62,288)(38,312,63,287)(39,311,64,286)(40,315,65,290)(41,304,66,279)(42,303,67,278)(43,302,68,277)(44,301,69,276)(45,305,70,280)(71,269,96,244)(72,268,97,243)(73,267,98,242)(74,266,99,241)(75,270,100,245)(76,249,101,224)(77,248,102,223)(78,247,103,222)(79,246,104,221)(80,250,105,225)(81,259,106,234)(82,258,107,233)(83,257,108,232)(84,256,109,231)(85,260,110,235)(86,264,111,239)(87,263,112,238)(88,262,113,237)(89,261,114,236)(90,265,115,240)(91,254,116,229)(92,253,117,228)(93,252,118,227)(94,251,119,226)(95,255,120,230)(126,399,151,374)(127,398,152,373)(128,397,153,372)(129,396,154,371)(130,400,155,375)(171,369,196,344)(172,368,197,343)(173,367,198,342)(174,366,199,341)(175,370,200,345)(176,349,216,324)(177,348,217,323)(178,347,218,322)(179,346,219,321)(180,350,220,325)(181,359,211,334)(182,358,212,333)(183,357,213,332)(184,356,214,331)(185,360,215,335)(186,364,206,339)(187,363,207,338)(188,362,208,337)(189,361,209,336)(190,365,210,340)(191,354,201,329)(192,353,202,328)(193,352,203,327)(194,351,204,326)(195,355,205,330) );

G=PermutationGroup([[(1,213),(2,214),(3,215),(4,211),(5,212),(6,245),(7,241),(8,242),(9,243),(10,244),(11,240),(12,236),(13,237),(14,238),(15,239),(16,235),(17,231),(18,232),(19,233),(20,234),(21,230),(22,226),(23,227),(24,228),(25,229),(26,216),(27,217),(28,218),(29,219),(30,220),(31,198),(32,199),(33,200),(34,196),(35,197),(36,206),(37,207),(38,208),(39,209),(40,210),(41,201),(42,202),(43,203),(44,204),(45,205),(46,171),(47,172),(48,173),(49,174),(50,175),(51,176),(52,177),(53,178),(54,179),(55,180),(56,181),(57,182),(58,183),(59,184),(60,185),(61,186),(62,187),(63,188),(64,189),(65,190),(66,191),(67,192),(68,193),(69,194),(70,195),(71,146),(72,147),(73,148),(74,149),(75,150),(76,151),(77,152),(78,153),(79,154),(80,155),(81,156),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,167),(93,168),(94,169),(95,170),(96,121),(97,122),(98,123),(99,124),(100,125),(101,126),(102,127),(103,128),(104,129),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,145),(221,396),(222,397),(223,398),(224,399),(225,400),(246,371),(247,372),(248,373),(249,374),(250,375),(251,376),(252,377),(253,378),(254,379),(255,380),(256,381),(257,382),(258,383),(259,384),(260,385),(261,386),(262,387),(263,388),(264,389),(265,390),(266,391),(267,392),(268,393),(269,394),(270,395),(271,346),(272,347),(273,348),(274,349),(275,350),(276,351),(277,352),(278,353),(279,354),(280,355),(281,356),(282,357),(283,358),(284,359),(285,360),(286,361),(287,362),(288,363),(289,364),(290,365),(291,366),(292,367),(293,368),(294,369),(295,370),(296,321),(297,322),(298,323),(299,324),(300,325),(301,326),(302,327),(303,328),(304,329),(305,330),(306,331),(307,332),(308,333),(309,334),(310,335),(311,336),(312,337),(313,338),(314,339),(315,340),(316,341),(317,342),(318,343),(319,344),(320,345)], [(1,108),(2,109),(3,110),(4,106),(5,107),(6,345),(7,341),(8,342),(9,343),(10,344),(11,340),(12,336),(13,337),(14,338),(15,339),(16,335),(17,331),(18,332),(19,333),(20,334),(21,330),(22,326),(23,327),(24,328),(25,329),(26,101),(27,102),(28,103),(29,104),(30,105),(31,98),(32,99),(33,100),(34,96),(35,97),(36,111),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,81),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(121,196),(122,197),(123,198),(124,199),(125,200),(126,216),(127,217),(128,218),(129,219),(130,220),(131,211),(132,212),(133,213),(134,214),(135,215),(136,206),(137,207),(138,208),(139,209),(140,210),(141,201),(142,202),(143,203),(144,204),(145,205),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,177),(153,178),(154,179),(155,180),(156,181),(157,182),(158,183),(159,184),(160,185),(161,186),(162,187),(163,188),(164,189),(165,190),(166,191),(167,192),(168,193),(169,194),(170,195),(221,296),(222,297),(223,298),(224,299),(225,300),(226,301),(227,302),(228,303),(229,304),(230,305),(231,306),(232,307),(233,308),(234,309),(235,310),(236,311),(237,312),(238,313),(239,314),(240,315),(241,316),(242,317),(243,318),(244,319),(245,320),(246,271),(247,272),(248,273),(249,274),(250,275),(251,276),(252,277),(253,278),(254,279),(255,280),(256,281),(257,282),(258,283),(259,284),(260,285),(261,286),(262,287),(263,288),(264,289),(265,290),(266,291),(267,292),(268,293),(269,294),(270,295),(321,396),(322,397),(323,398),(324,399),(325,400),(346,371),(347,372),(348,373),(349,374),(350,375),(351,376),(352,377),(353,378),(354,379),(355,380),(356,381),(357,382),(358,383),(359,384),(360,385),(361,386),(362,387),(363,388),(364,389),(365,390),(366,391),(367,392),(368,393),(369,394),(370,395)], [(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),(351,352,353,354,355),(356,357,358,359,360),(361,362,363,364,365),(366,367,368,369,370),(371,372,373,374,375),(376,377,378,379,380),(381,382,383,384,385),(386,387,388,389,390),(391,392,393,394,395),(396,397,398,399,400)], [(1,44,28,31,38),(2,45,29,32,39),(3,41,30,33,40),(4,42,26,34,36),(5,43,27,35,37),(6,400,25,16,11),(7,396,21,17,12),(8,397,22,18,13),(9,398,23,19,14),(10,399,24,20,15),(46,61,56,67,51),(47,62,57,68,52),(48,63,58,69,53),(49,64,59,70,54),(50,65,60,66,55),(71,86,81,92,76),(72,87,82,93,77),(73,88,83,94,78),(74,89,84,95,79),(75,90,85,91,80),(96,111,106,117,101),(97,112,107,118,102),(98,113,108,119,103),(99,114,109,120,104),(100,115,110,116,105),(121,136,131,142,126),(122,137,132,143,127),(123,138,133,144,128),(124,139,134,145,129),(125,140,135,141,130),(146,161,156,167,151),(147,162,157,168,152),(148,163,158,169,153),(149,164,159,170,154),(150,165,160,166,155),(171,186,181,192,176),(172,187,182,193,177),(173,188,183,194,178),(174,189,184,195,179),(175,190,185,191,180),(196,206,211,202,216),(197,207,212,203,217),(198,208,213,204,218),(199,209,214,205,219),(200,210,215,201,220),(221,230,231,236,241),(222,226,232,237,242),(223,227,233,238,243),(224,228,234,239,244),(225,229,235,240,245),(246,255,256,261,266),(247,251,257,262,267),(248,252,258,263,268),(249,253,259,264,269),(250,254,260,265,270),(271,280,281,286,291),(272,276,282,287,292),(273,277,283,288,293),(274,278,284,289,294),(275,279,285,290,295),(296,305,306,311,316),(297,301,307,312,317),(298,302,308,313,318),(299,303,309,314,319),(300,304,310,315,320),(321,330,331,336,341),(322,326,332,337,342),(323,327,333,338,343),(324,328,334,339,344),(325,329,335,340,345),(346,355,356,361,366),(347,351,357,362,367),(348,352,358,363,368),(349,353,359,364,369),(350,354,360,365,370),(371,380,381,386,391),(372,376,382,387,392),(373,377,383,388,393),(374,378,384,389,394),(375,379,385,390,395)], [(1,307,58,282),(2,306,59,281),(3,310,60,285),(4,309,56,284),(5,308,57,283),(6,150,395,125),(7,149,391,124),(8,148,392,123),(9,147,393,122),(10,146,394,121),(11,165,390,140),(12,164,386,139),(13,163,387,138),(14,162,388,137),(15,161,389,136),(16,160,385,135),(17,159,381,134),(18,158,382,133),(19,157,383,132),(20,156,384,131),(21,170,380,145),(22,169,376,144),(23,168,377,143),(24,167,378,142),(25,166,379,141),(26,299,51,274),(27,298,52,273),(28,297,53,272),(29,296,54,271),(30,300,55,275),(31,317,48,292),(32,316,49,291),(33,320,50,295),(34,319,46,294),(35,318,47,293),(36,314,61,289),(37,313,62,288),(38,312,63,287),(39,311,64,286),(40,315,65,290),(41,304,66,279),(42,303,67,278),(43,302,68,277),(44,301,69,276),(45,305,70,280),(71,269,96,244),(72,268,97,243),(73,267,98,242),(74,266,99,241),(75,270,100,245),(76,249,101,224),(77,248,102,223),(78,247,103,222),(79,246,104,221),(80,250,105,225),(81,259,106,234),(82,258,107,233),(83,257,108,232),(84,256,109,231),(85,260,110,235),(86,264,111,239),(87,263,112,238),(88,262,113,237),(89,261,114,236),(90,265,115,240),(91,254,116,229),(92,253,117,228),(93,252,118,227),(94,251,119,226),(95,255,120,230),(126,399,151,374),(127,398,152,373),(128,397,153,372),(129,396,154,371),(130,400,155,375),(171,369,196,344),(172,368,197,343),(173,367,198,342),(174,366,199,341),(175,370,200,345),(176,349,216,324),(177,348,217,323),(178,347,218,322),(179,346,219,321),(180,350,220,325),(181,359,211,334),(182,358,212,333),(183,357,213,332),(184,356,214,331),(185,360,215,335),(186,364,206,339),(187,363,207,338),(188,362,208,337),(189,361,209,336),(190,365,210,340),(191,354,201,329),(192,353,202,328),(193,352,203,327),(194,351,204,326),(195,355,205,330)]])

112 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5A ··· 5L 10A ··· 10CF order 1 2 ··· 2 4 ··· 4 5 ··· 5 10 ··· 10 size 1 1 ··· 1 25 ··· 25 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D5 Dic5 D10 kernel C22×C52⋊6C4 C2×C52⋊6C4 C2×C102 C102 C22×C10 C2×C10 C2×C10 # reps 1 6 1 8 12 48 36

Matrix representation of C22×C526C4 in GL5(𝔽41)

 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 34 0 0 0 7 7 0 0 0 0 0 34 40 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 40 34 0 0 0 0 0 0 1 0 0 0 40 34
,
 1 0 0 0 0 0 3 15 0 0 0 35 38 0 0 0 0 0 40 17 0 0 0 24 1

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,7,0,0,0,34,7,0,0,0,0,0,34,1,0,0,0,40,0],[1,0,0,0,0,0,0,40,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,1,34],[1,0,0,0,0,0,3,35,0,0,0,15,38,0,0,0,0,0,40,24,0,0,0,17,1] >;

C22×C526C4 in GAP, Magma, Sage, TeX

C_2^2\times C_5^2\rtimes_6C_4
% in TeX

G:=Group("C2^2xC5^2:6C4");
// GroupNames label

G:=SmallGroup(400,199);
// by ID

G=gap.SmallGroup(400,199);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,1924,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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