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## G = C19×C3⋊C8order 456 = 23·3·19

### Direct product of C19 and C3⋊C8

Aliases: C19×C3⋊C8, C3⋊C152, C573C8, C6.C76, C76.4S3, C228.6C2, C12.2C38, C114.3C4, C38.2Dic3, C4.2(S3×C19), C2.(Dic3×C19), SmallGroup(456,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C19×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C228 — C19×C3⋊C8
 Lower central C3 — C19×C3⋊C8
 Upper central C1 — C76

Generators and relations for C19×C3⋊C8
G = < a,b,c | a19=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C19×C3⋊C8
Regular action on 456 points
Generators in S456
(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)(400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418)(419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437)(438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456)
(1 422 207)(2 423 208)(3 424 209)(4 425 191)(5 426 192)(6 427 193)(7 428 194)(8 429 195)(9 430 196)(10 431 197)(11 432 198)(12 433 199)(13 434 200)(14 435 201)(15 436 202)(16 437 203)(17 419 204)(18 420 205)(19 421 206)(20 168 304)(21 169 286)(22 170 287)(23 171 288)(24 153 289)(25 154 290)(26 155 291)(27 156 292)(28 157 293)(29 158 294)(30 159 295)(31 160 296)(32 161 297)(33 162 298)(34 163 299)(35 164 300)(36 165 301)(37 166 302)(38 167 303)(39 330 221)(40 331 222)(41 332 223)(42 333 224)(43 334 225)(44 335 226)(45 336 227)(46 337 228)(47 338 210)(48 339 211)(49 340 212)(50 341 213)(51 342 214)(52 324 215)(53 325 216)(54 326 217)(55 327 218)(56 328 219)(57 329 220)(58 377 381)(59 378 382)(60 379 383)(61 380 384)(62 362 385)(63 363 386)(64 364 387)(65 365 388)(66 366 389)(67 367 390)(68 368 391)(69 369 392)(70 370 393)(71 371 394)(72 372 395)(73 373 396)(74 374 397)(75 375 398)(76 376 399)(77 278 242)(78 279 243)(79 280 244)(80 281 245)(81 282 246)(82 283 247)(83 284 229)(84 285 230)(85 267 231)(86 268 232)(87 269 233)(88 270 234)(89 271 235)(90 272 236)(91 273 237)(92 274 238)(93 275 239)(94 276 240)(95 277 241)(96 177 320)(97 178 321)(98 179 322)(99 180 323)(100 181 305)(101 182 306)(102 183 307)(103 184 308)(104 185 309)(105 186 310)(106 187 311)(107 188 312)(108 189 313)(109 190 314)(110 172 315)(111 173 316)(112 174 317)(113 175 318)(114 176 319)(115 151 406)(116 152 407)(117 134 408)(118 135 409)(119 136 410)(120 137 411)(121 138 412)(122 139 413)(123 140 414)(124 141 415)(125 142 416)(126 143 417)(127 144 418)(128 145 400)(129 146 401)(130 147 402)(131 148 403)(132 149 404)(133 150 405)(248 439 343)(249 440 344)(250 441 345)(251 442 346)(252 443 347)(253 444 348)(254 445 349)(255 446 350)(256 447 351)(257 448 352)(258 449 353)(259 450 354)(260 451 355)(261 452 356)(262 453 357)(263 454 358)(264 455 359)(265 456 360)(266 438 361)
(1 445 414 38 80 227 306 365)(2 446 415 20 81 228 307 366)(3 447 416 21 82 210 308 367)(4 448 417 22 83 211 309 368)(5 449 418 23 84 212 310 369)(6 450 400 24 85 213 311 370)(7 451 401 25 86 214 312 371)(8 452 402 26 87 215 313 372)(9 453 403 27 88 216 314 373)(10 454 404 28 89 217 315 374)(11 455 405 29 90 218 316 375)(12 456 406 30 91 219 317 376)(13 438 407 31 92 220 318 377)(14 439 408 32 93 221 319 378)(15 440 409 33 94 222 320 379)(16 441 410 34 95 223 321 380)(17 442 411 35 77 224 322 362)(18 443 412 36 78 225 323 363)(19 444 413 37 79 226 305 364)(39 176 382 201 343 134 161 239)(40 177 383 202 344 135 162 240)(41 178 384 203 345 136 163 241)(42 179 385 204 346 137 164 242)(43 180 386 205 347 138 165 243)(44 181 387 206 348 139 166 244)(45 182 388 207 349 140 167 245)(46 183 389 208 350 141 168 246)(47 184 390 209 351 142 169 247)(48 185 391 191 352 143 170 229)(49 186 392 192 353 144 171 230)(50 187 393 193 354 145 153 231)(51 188 394 194 355 146 154 232)(52 189 395 195 356 147 155 233)(53 190 396 196 357 148 156 234)(54 172 397 197 358 149 157 235)(55 173 398 198 359 150 158 236)(56 174 399 199 360 151 159 237)(57 175 381 200 361 152 160 238)(58 434 266 116 296 274 329 113)(59 435 248 117 297 275 330 114)(60 436 249 118 298 276 331 96)(61 437 250 119 299 277 332 97)(62 419 251 120 300 278 333 98)(63 420 252 121 301 279 334 99)(64 421 253 122 302 280 335 100)(65 422 254 123 303 281 336 101)(66 423 255 124 304 282 337 102)(67 424 256 125 286 283 338 103)(68 425 257 126 287 284 339 104)(69 426 258 127 288 285 340 105)(70 427 259 128 289 267 341 106)(71 428 260 129 290 268 342 107)(72 429 261 130 291 269 324 108)(73 430 262 131 292 270 325 109)(74 431 263 132 293 271 326 110)(75 432 264 133 294 272 327 111)(76 433 265 115 295 273 328 112)

G:=sub<Sym(456)| (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)(400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418)(419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437)(438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456), (1,422,207)(2,423,208)(3,424,209)(4,425,191)(5,426,192)(6,427,193)(7,428,194)(8,429,195)(9,430,196)(10,431,197)(11,432,198)(12,433,199)(13,434,200)(14,435,201)(15,436,202)(16,437,203)(17,419,204)(18,420,205)(19,421,206)(20,168,304)(21,169,286)(22,170,287)(23,171,288)(24,153,289)(25,154,290)(26,155,291)(27,156,292)(28,157,293)(29,158,294)(30,159,295)(31,160,296)(32,161,297)(33,162,298)(34,163,299)(35,164,300)(36,165,301)(37,166,302)(38,167,303)(39,330,221)(40,331,222)(41,332,223)(42,333,224)(43,334,225)(44,335,226)(45,336,227)(46,337,228)(47,338,210)(48,339,211)(49,340,212)(50,341,213)(51,342,214)(52,324,215)(53,325,216)(54,326,217)(55,327,218)(56,328,219)(57,329,220)(58,377,381)(59,378,382)(60,379,383)(61,380,384)(62,362,385)(63,363,386)(64,364,387)(65,365,388)(66,366,389)(67,367,390)(68,368,391)(69,369,392)(70,370,393)(71,371,394)(72,372,395)(73,373,396)(74,374,397)(75,375,398)(76,376,399)(77,278,242)(78,279,243)(79,280,244)(80,281,245)(81,282,246)(82,283,247)(83,284,229)(84,285,230)(85,267,231)(86,268,232)(87,269,233)(88,270,234)(89,271,235)(90,272,236)(91,273,237)(92,274,238)(93,275,239)(94,276,240)(95,277,241)(96,177,320)(97,178,321)(98,179,322)(99,180,323)(100,181,305)(101,182,306)(102,183,307)(103,184,308)(104,185,309)(105,186,310)(106,187,311)(107,188,312)(108,189,313)(109,190,314)(110,172,315)(111,173,316)(112,174,317)(113,175,318)(114,176,319)(115,151,406)(116,152,407)(117,134,408)(118,135,409)(119,136,410)(120,137,411)(121,138,412)(122,139,413)(123,140,414)(124,141,415)(125,142,416)(126,143,417)(127,144,418)(128,145,400)(129,146,401)(130,147,402)(131,148,403)(132,149,404)(133,150,405)(248,439,343)(249,440,344)(250,441,345)(251,442,346)(252,443,347)(253,444,348)(254,445,349)(255,446,350)(256,447,351)(257,448,352)(258,449,353)(259,450,354)(260,451,355)(261,452,356)(262,453,357)(263,454,358)(264,455,359)(265,456,360)(266,438,361), (1,445,414,38,80,227,306,365)(2,446,415,20,81,228,307,366)(3,447,416,21,82,210,308,367)(4,448,417,22,83,211,309,368)(5,449,418,23,84,212,310,369)(6,450,400,24,85,213,311,370)(7,451,401,25,86,214,312,371)(8,452,402,26,87,215,313,372)(9,453,403,27,88,216,314,373)(10,454,404,28,89,217,315,374)(11,455,405,29,90,218,316,375)(12,456,406,30,91,219,317,376)(13,438,407,31,92,220,318,377)(14,439,408,32,93,221,319,378)(15,440,409,33,94,222,320,379)(16,441,410,34,95,223,321,380)(17,442,411,35,77,224,322,362)(18,443,412,36,78,225,323,363)(19,444,413,37,79,226,305,364)(39,176,382,201,343,134,161,239)(40,177,383,202,344,135,162,240)(41,178,384,203,345,136,163,241)(42,179,385,204,346,137,164,242)(43,180,386,205,347,138,165,243)(44,181,387,206,348,139,166,244)(45,182,388,207,349,140,167,245)(46,183,389,208,350,141,168,246)(47,184,390,209,351,142,169,247)(48,185,391,191,352,143,170,229)(49,186,392,192,353,144,171,230)(50,187,393,193,354,145,153,231)(51,188,394,194,355,146,154,232)(52,189,395,195,356,147,155,233)(53,190,396,196,357,148,156,234)(54,172,397,197,358,149,157,235)(55,173,398,198,359,150,158,236)(56,174,399,199,360,151,159,237)(57,175,381,200,361,152,160,238)(58,434,266,116,296,274,329,113)(59,435,248,117,297,275,330,114)(60,436,249,118,298,276,331,96)(61,437,250,119,299,277,332,97)(62,419,251,120,300,278,333,98)(63,420,252,121,301,279,334,99)(64,421,253,122,302,280,335,100)(65,422,254,123,303,281,336,101)(66,423,255,124,304,282,337,102)(67,424,256,125,286,283,338,103)(68,425,257,126,287,284,339,104)(69,426,258,127,288,285,340,105)(70,427,259,128,289,267,341,106)(71,428,260,129,290,268,342,107)(72,429,261,130,291,269,324,108)(73,430,262,131,292,270,325,109)(74,431,263,132,293,271,326,110)(75,432,264,133,294,272,327,111)(76,433,265,115,295,273,328,112)>;

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)(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342)(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)(400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418)(419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437)(438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456), (1,422,207)(2,423,208)(3,424,209)(4,425,191)(5,426,192)(6,427,193)(7,428,194)(8,429,195)(9,430,196)(10,431,197)(11,432,198)(12,433,199)(13,434,200)(14,435,201)(15,436,202)(16,437,203)(17,419,204)(18,420,205)(19,421,206)(20,168,304)(21,169,286)(22,170,287)(23,171,288)(24,153,289)(25,154,290)(26,155,291)(27,156,292)(28,157,293)(29,158,294)(30,159,295)(31,160,296)(32,161,297)(33,162,298)(34,163,299)(35,164,300)(36,165,301)(37,166,302)(38,167,303)(39,330,221)(40,331,222)(41,332,223)(42,333,224)(43,334,225)(44,335,226)(45,336,227)(46,337,228)(47,338,210)(48,339,211)(49,340,212)(50,341,213)(51,342,214)(52,324,215)(53,325,216)(54,326,217)(55,327,218)(56,328,219)(57,329,220)(58,377,381)(59,378,382)(60,379,383)(61,380,384)(62,362,385)(63,363,386)(64,364,387)(65,365,388)(66,366,389)(67,367,390)(68,368,391)(69,369,392)(70,370,393)(71,371,394)(72,372,395)(73,373,396)(74,374,397)(75,375,398)(76,376,399)(77,278,242)(78,279,243)(79,280,244)(80,281,245)(81,282,246)(82,283,247)(83,284,229)(84,285,230)(85,267,231)(86,268,232)(87,269,233)(88,270,234)(89,271,235)(90,272,236)(91,273,237)(92,274,238)(93,275,239)(94,276,240)(95,277,241)(96,177,320)(97,178,321)(98,179,322)(99,180,323)(100,181,305)(101,182,306)(102,183,307)(103,184,308)(104,185,309)(105,186,310)(106,187,311)(107,188,312)(108,189,313)(109,190,314)(110,172,315)(111,173,316)(112,174,317)(113,175,318)(114,176,319)(115,151,406)(116,152,407)(117,134,408)(118,135,409)(119,136,410)(120,137,411)(121,138,412)(122,139,413)(123,140,414)(124,141,415)(125,142,416)(126,143,417)(127,144,418)(128,145,400)(129,146,401)(130,147,402)(131,148,403)(132,149,404)(133,150,405)(248,439,343)(249,440,344)(250,441,345)(251,442,346)(252,443,347)(253,444,348)(254,445,349)(255,446,350)(256,447,351)(257,448,352)(258,449,353)(259,450,354)(260,451,355)(261,452,356)(262,453,357)(263,454,358)(264,455,359)(265,456,360)(266,438,361), (1,445,414,38,80,227,306,365)(2,446,415,20,81,228,307,366)(3,447,416,21,82,210,308,367)(4,448,417,22,83,211,309,368)(5,449,418,23,84,212,310,369)(6,450,400,24,85,213,311,370)(7,451,401,25,86,214,312,371)(8,452,402,26,87,215,313,372)(9,453,403,27,88,216,314,373)(10,454,404,28,89,217,315,374)(11,455,405,29,90,218,316,375)(12,456,406,30,91,219,317,376)(13,438,407,31,92,220,318,377)(14,439,408,32,93,221,319,378)(15,440,409,33,94,222,320,379)(16,441,410,34,95,223,321,380)(17,442,411,35,77,224,322,362)(18,443,412,36,78,225,323,363)(19,444,413,37,79,226,305,364)(39,176,382,201,343,134,161,239)(40,177,383,202,344,135,162,240)(41,178,384,203,345,136,163,241)(42,179,385,204,346,137,164,242)(43,180,386,205,347,138,165,243)(44,181,387,206,348,139,166,244)(45,182,388,207,349,140,167,245)(46,183,389,208,350,141,168,246)(47,184,390,209,351,142,169,247)(48,185,391,191,352,143,170,229)(49,186,392,192,353,144,171,230)(50,187,393,193,354,145,153,231)(51,188,394,194,355,146,154,232)(52,189,395,195,356,147,155,233)(53,190,396,196,357,148,156,234)(54,172,397,197,358,149,157,235)(55,173,398,198,359,150,158,236)(56,174,399,199,360,151,159,237)(57,175,381,200,361,152,160,238)(58,434,266,116,296,274,329,113)(59,435,248,117,297,275,330,114)(60,436,249,118,298,276,331,96)(61,437,250,119,299,277,332,97)(62,419,251,120,300,278,333,98)(63,420,252,121,301,279,334,99)(64,421,253,122,302,280,335,100)(65,422,254,123,303,281,336,101)(66,423,255,124,304,282,337,102)(67,424,256,125,286,283,338,103)(68,425,257,126,287,284,339,104)(69,426,258,127,288,285,340,105)(70,427,259,128,289,267,341,106)(71,428,260,129,290,268,342,107)(72,429,261,130,291,269,324,108)(73,430,262,131,292,270,325,109)(74,431,263,132,293,271,326,110)(75,432,264,133,294,272,327,111)(76,433,265,115,295,273,328,112) );

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),(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342),(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),(400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418),(419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437),(438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456)], [(1,422,207),(2,423,208),(3,424,209),(4,425,191),(5,426,192),(6,427,193),(7,428,194),(8,429,195),(9,430,196),(10,431,197),(11,432,198),(12,433,199),(13,434,200),(14,435,201),(15,436,202),(16,437,203),(17,419,204),(18,420,205),(19,421,206),(20,168,304),(21,169,286),(22,170,287),(23,171,288),(24,153,289),(25,154,290),(26,155,291),(27,156,292),(28,157,293),(29,158,294),(30,159,295),(31,160,296),(32,161,297),(33,162,298),(34,163,299),(35,164,300),(36,165,301),(37,166,302),(38,167,303),(39,330,221),(40,331,222),(41,332,223),(42,333,224),(43,334,225),(44,335,226),(45,336,227),(46,337,228),(47,338,210),(48,339,211),(49,340,212),(50,341,213),(51,342,214),(52,324,215),(53,325,216),(54,326,217),(55,327,218),(56,328,219),(57,329,220),(58,377,381),(59,378,382),(60,379,383),(61,380,384),(62,362,385),(63,363,386),(64,364,387),(65,365,388),(66,366,389),(67,367,390),(68,368,391),(69,369,392),(70,370,393),(71,371,394),(72,372,395),(73,373,396),(74,374,397),(75,375,398),(76,376,399),(77,278,242),(78,279,243),(79,280,244),(80,281,245),(81,282,246),(82,283,247),(83,284,229),(84,285,230),(85,267,231),(86,268,232),(87,269,233),(88,270,234),(89,271,235),(90,272,236),(91,273,237),(92,274,238),(93,275,239),(94,276,240),(95,277,241),(96,177,320),(97,178,321),(98,179,322),(99,180,323),(100,181,305),(101,182,306),(102,183,307),(103,184,308),(104,185,309),(105,186,310),(106,187,311),(107,188,312),(108,189,313),(109,190,314),(110,172,315),(111,173,316),(112,174,317),(113,175,318),(114,176,319),(115,151,406),(116,152,407),(117,134,408),(118,135,409),(119,136,410),(120,137,411),(121,138,412),(122,139,413),(123,140,414),(124,141,415),(125,142,416),(126,143,417),(127,144,418),(128,145,400),(129,146,401),(130,147,402),(131,148,403),(132,149,404),(133,150,405),(248,439,343),(249,440,344),(250,441,345),(251,442,346),(252,443,347),(253,444,348),(254,445,349),(255,446,350),(256,447,351),(257,448,352),(258,449,353),(259,450,354),(260,451,355),(261,452,356),(262,453,357),(263,454,358),(264,455,359),(265,456,360),(266,438,361)], [(1,445,414,38,80,227,306,365),(2,446,415,20,81,228,307,366),(3,447,416,21,82,210,308,367),(4,448,417,22,83,211,309,368),(5,449,418,23,84,212,310,369),(6,450,400,24,85,213,311,370),(7,451,401,25,86,214,312,371),(8,452,402,26,87,215,313,372),(9,453,403,27,88,216,314,373),(10,454,404,28,89,217,315,374),(11,455,405,29,90,218,316,375),(12,456,406,30,91,219,317,376),(13,438,407,31,92,220,318,377),(14,439,408,32,93,221,319,378),(15,440,409,33,94,222,320,379),(16,441,410,34,95,223,321,380),(17,442,411,35,77,224,322,362),(18,443,412,36,78,225,323,363),(19,444,413,37,79,226,305,364),(39,176,382,201,343,134,161,239),(40,177,383,202,344,135,162,240),(41,178,384,203,345,136,163,241),(42,179,385,204,346,137,164,242),(43,180,386,205,347,138,165,243),(44,181,387,206,348,139,166,244),(45,182,388,207,349,140,167,245),(46,183,389,208,350,141,168,246),(47,184,390,209,351,142,169,247),(48,185,391,191,352,143,170,229),(49,186,392,192,353,144,171,230),(50,187,393,193,354,145,153,231),(51,188,394,194,355,146,154,232),(52,189,395,195,356,147,155,233),(53,190,396,196,357,148,156,234),(54,172,397,197,358,149,157,235),(55,173,398,198,359,150,158,236),(56,174,399,199,360,151,159,237),(57,175,381,200,361,152,160,238),(58,434,266,116,296,274,329,113),(59,435,248,117,297,275,330,114),(60,436,249,118,298,276,331,96),(61,437,250,119,299,277,332,97),(62,419,251,120,300,278,333,98),(63,420,252,121,301,279,334,99),(64,421,253,122,302,280,335,100),(65,422,254,123,303,281,336,101),(66,423,255,124,304,282,337,102),(67,424,256,125,286,283,338,103),(68,425,257,126,287,284,339,104),(69,426,258,127,288,285,340,105),(70,427,259,128,289,267,341,106),(71,428,260,129,290,268,342,107),(72,429,261,130,291,269,324,108),(73,430,262,131,292,270,325,109),(74,431,263,132,293,271,326,110),(75,432,264,133,294,272,327,111),(76,433,265,115,295,273,328,112)])

228 conjugacy classes

 class 1 2 3 4A 4B 6 8A 8B 8C 8D 12A 12B 19A ··· 19R 38A ··· 38R 57A ··· 57R 76A ··· 76AJ 114A ··· 114R 152A ··· 152BT 228A ··· 228AJ order 1 2 3 4 4 6 8 8 8 8 12 12 19 ··· 19 38 ··· 38 57 ··· 57 76 ··· 76 114 ··· 114 152 ··· 152 228 ··· 228 size 1 1 2 1 1 2 3 3 3 3 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

228 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C8 C19 C38 C76 C152 S3 Dic3 C3⋊C8 S3×C19 Dic3×C19 C19×C3⋊C8 kernel C19×C3⋊C8 C228 C114 C57 C3⋊C8 C12 C6 C3 C76 C38 C19 C4 C2 C1 # reps 1 1 2 4 18 18 36 72 1 1 2 18 18 36

Matrix representation of C19×C3⋊C8 in GL2(𝔽457) generated by

 185 0 0 185
,
 456 456 1 0
,
 222 246 24 235
G:=sub<GL(2,GF(457))| [185,0,0,185],[456,1,456,0],[222,24,246,235] >;

C19×C3⋊C8 in GAP, Magma, Sage, TeX

C_{19}\times C_3\rtimes C_8
% in TeX

G:=Group("C19xC3:C8");
// GroupNames label

G:=SmallGroup(456,3);
// by ID

G=gap.SmallGroup(456,3);
# by ID

G:=PCGroup([5,-2,-19,-2,-2,-3,190,42,7604]);
// Polycyclic

G:=Group<a,b,c|a^19=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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