direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19×C3⋊C8, C3⋊C152, C57⋊3C8, 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
| C3 — C19×C3⋊C8 |
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)
(1 392 225)(2 393 226)(3 394 227)(4 395 228)(5 396 210)(6 397 211)(7 398 212)(8 399 213)(9 381 214)(10 382 215)(11 383 216)(12 384 217)(13 385 218)(14 386 219)(15 387 220)(16 388 221)(17 389 222)(18 390 223)(19 391 224)(20 245 299)(21 246 300)(22 247 301)(23 229 302)(24 230 303)(25 231 304)(26 232 286)(27 233 287)(28 234 288)(29 235 289)(30 236 290)(31 237 291)(32 238 292)(33 239 293)(34 240 294)(35 241 295)(36 242 296)(37 243 297)(38 244 298)(39 150 372)(40 151 373)(41 152 374)(42 134 375)(43 135 376)(44 136 377)(45 137 378)(46 138 379)(47 139 380)(48 140 362)(49 141 363)(50 142 364)(51 143 365)(52 144 366)(53 145 367)(54 146 368)(55 147 369)(56 148 370)(57 149 371)(58 133 179)(59 115 180)(60 116 181)(61 117 182)(62 118 183)(63 119 184)(64 120 185)(65 121 186)(66 122 187)(67 123 188)(68 124 189)(69 125 190)(70 126 172)(71 127 173)(72 128 174)(73 129 175)(74 130 176)(75 131 177)(76 132 178)(77 336 249)(78 337 250)(79 338 251)(80 339 252)(81 340 253)(82 341 254)(83 342 255)(84 324 256)(85 325 257)(86 326 258)(87 327 259)(88 328 260)(89 329 261)(90 330 262)(91 331 263)(92 332 264)(93 333 265)(94 334 266)(95 335 248)(96 453 432)(97 454 433)(98 455 434)(99 456 435)(100 438 436)(101 439 437)(102 440 419)(103 441 420)(104 442 421)(105 443 422)(106 444 423)(107 445 424)(108 446 425)(109 447 426)(110 448 427)(111 449 428)(112 450 429)(113 451 430)(114 452 431)(153 268 196)(154 269 197)(155 270 198)(156 271 199)(157 272 200)(158 273 201)(159 274 202)(160 275 203)(161 276 204)(162 277 205)(163 278 206)(164 279 207)(165 280 208)(166 281 209)(167 282 191)(168 283 192)(169 284 193)(170 285 194)(171 267 195)(305 404 355)(306 405 356)(307 406 357)(308 407 358)(309 408 359)(310 409 360)(311 410 361)(312 411 343)(313 412 344)(314 413 345)(315 414 346)(316 415 347)(317 416 348)(318 417 349)(319 418 350)(320 400 351)(321 401 352)(322 402 353)(323 403 354)
(1 63 361 276 299 428 253 40)(2 64 343 277 300 429 254 41)(3 65 344 278 301 430 255 42)(4 66 345 279 302 431 256 43)(5 67 346 280 303 432 257 44)(6 68 347 281 304 433 258 45)(7 69 348 282 286 434 259 46)(8 70 349 283 287 435 260 47)(9 71 350 284 288 436 261 48)(10 72 351 285 289 437 262 49)(11 73 352 267 290 419 263 50)(12 74 353 268 291 420 264 51)(13 75 354 269 292 421 265 52)(14 76 355 270 293 422 266 53)(15 58 356 271 294 423 248 54)(16 59 357 272 295 424 249 55)(17 60 358 273 296 425 250 56)(18 61 359 274 297 426 251 57)(19 62 360 275 298 427 252 39)(20 449 81 373 392 184 311 161)(21 450 82 374 393 185 312 162)(22 451 83 375 394 186 313 163)(23 452 84 376 395 187 314 164)(24 453 85 377 396 188 315 165)(25 454 86 378 397 189 316 166)(26 455 87 379 398 190 317 167)(27 456 88 380 399 172 318 168)(28 438 89 362 381 173 319 169)(29 439 90 363 382 174 320 170)(30 440 91 364 383 175 321 171)(31 441 92 365 384 176 322 153)(32 442 93 366 385 177 323 154)(33 443 94 367 386 178 305 155)(34 444 95 368 387 179 306 156)(35 445 77 369 388 180 307 157)(36 446 78 370 389 181 308 158)(37 447 79 371 390 182 309 159)(38 448 80 372 391 183 310 160)(96 325 136 210 123 414 208 230)(97 326 137 211 124 415 209 231)(98 327 138 212 125 416 191 232)(99 328 139 213 126 417 192 233)(100 329 140 214 127 418 193 234)(101 330 141 215 128 400 194 235)(102 331 142 216 129 401 195 236)(103 332 143 217 130 402 196 237)(104 333 144 218 131 403 197 238)(105 334 145 219 132 404 198 239)(106 335 146 220 133 405 199 240)(107 336 147 221 115 406 200 241)(108 337 148 222 116 407 201 242)(109 338 149 223 117 408 202 243)(110 339 150 224 118 409 203 244)(111 340 151 225 119 410 204 245)(112 341 152 226 120 411 205 246)(113 342 134 227 121 412 206 247)(114 324 135 228 122 413 207 229)
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), (1,392,225)(2,393,226)(3,394,227)(4,395,228)(5,396,210)(6,397,211)(7,398,212)(8,399,213)(9,381,214)(10,382,215)(11,383,216)(12,384,217)(13,385,218)(14,386,219)(15,387,220)(16,388,221)(17,389,222)(18,390,223)(19,391,224)(20,245,299)(21,246,300)(22,247,301)(23,229,302)(24,230,303)(25,231,304)(26,232,286)(27,233,287)(28,234,288)(29,235,289)(30,236,290)(31,237,291)(32,238,292)(33,239,293)(34,240,294)(35,241,295)(36,242,296)(37,243,297)(38,244,298)(39,150,372)(40,151,373)(41,152,374)(42,134,375)(43,135,376)(44,136,377)(45,137,378)(46,138,379)(47,139,380)(48,140,362)(49,141,363)(50,142,364)(51,143,365)(52,144,366)(53,145,367)(54,146,368)(55,147,369)(56,148,370)(57,149,371)(58,133,179)(59,115,180)(60,116,181)(61,117,182)(62,118,183)(63,119,184)(64,120,185)(65,121,186)(66,122,187)(67,123,188)(68,124,189)(69,125,190)(70,126,172)(71,127,173)(72,128,174)(73,129,175)(74,130,176)(75,131,177)(76,132,178)(77,336,249)(78,337,250)(79,338,251)(80,339,252)(81,340,253)(82,341,254)(83,342,255)(84,324,256)(85,325,257)(86,326,258)(87,327,259)(88,328,260)(89,329,261)(90,330,262)(91,331,263)(92,332,264)(93,333,265)(94,334,266)(95,335,248)(96,453,432)(97,454,433)(98,455,434)(99,456,435)(100,438,436)(101,439,437)(102,440,419)(103,441,420)(104,442,421)(105,443,422)(106,444,423)(107,445,424)(108,446,425)(109,447,426)(110,448,427)(111,449,428)(112,450,429)(113,451,430)(114,452,431)(153,268,196)(154,269,197)(155,270,198)(156,271,199)(157,272,200)(158,273,201)(159,274,202)(160,275,203)(161,276,204)(162,277,205)(163,278,206)(164,279,207)(165,280,208)(166,281,209)(167,282,191)(168,283,192)(169,284,193)(170,285,194)(171,267,195)(305,404,355)(306,405,356)(307,406,357)(308,407,358)(309,408,359)(310,409,360)(311,410,361)(312,411,343)(313,412,344)(314,413,345)(315,414,346)(316,415,347)(317,416,348)(318,417,349)(319,418,350)(320,400,351)(321,401,352)(322,402,353)(323,403,354), (1,63,361,276,299,428,253,40)(2,64,343,277,300,429,254,41)(3,65,344,278,301,430,255,42)(4,66,345,279,302,431,256,43)(5,67,346,280,303,432,257,44)(6,68,347,281,304,433,258,45)(7,69,348,282,286,434,259,46)(8,70,349,283,287,435,260,47)(9,71,350,284,288,436,261,48)(10,72,351,285,289,437,262,49)(11,73,352,267,290,419,263,50)(12,74,353,268,291,420,264,51)(13,75,354,269,292,421,265,52)(14,76,355,270,293,422,266,53)(15,58,356,271,294,423,248,54)(16,59,357,272,295,424,249,55)(17,60,358,273,296,425,250,56)(18,61,359,274,297,426,251,57)(19,62,360,275,298,427,252,39)(20,449,81,373,392,184,311,161)(21,450,82,374,393,185,312,162)(22,451,83,375,394,186,313,163)(23,452,84,376,395,187,314,164)(24,453,85,377,396,188,315,165)(25,454,86,378,397,189,316,166)(26,455,87,379,398,190,317,167)(27,456,88,380,399,172,318,168)(28,438,89,362,381,173,319,169)(29,439,90,363,382,174,320,170)(30,440,91,364,383,175,321,171)(31,441,92,365,384,176,322,153)(32,442,93,366,385,177,323,154)(33,443,94,367,386,178,305,155)(34,444,95,368,387,179,306,156)(35,445,77,369,388,180,307,157)(36,446,78,370,389,181,308,158)(37,447,79,371,390,182,309,159)(38,448,80,372,391,183,310,160)(96,325,136,210,123,414,208,230)(97,326,137,211,124,415,209,231)(98,327,138,212,125,416,191,232)(99,328,139,213,126,417,192,233)(100,329,140,214,127,418,193,234)(101,330,141,215,128,400,194,235)(102,331,142,216,129,401,195,236)(103,332,143,217,130,402,196,237)(104,333,144,218,131,403,197,238)(105,334,145,219,132,404,198,239)(106,335,146,220,133,405,199,240)(107,336,147,221,115,406,200,241)(108,337,148,222,116,407,201,242)(109,338,149,223,117,408,202,243)(110,339,150,224,118,409,203,244)(111,340,151,225,119,410,204,245)(112,341,152,226,120,411,205,246)(113,342,134,227,121,412,206,247)(114,324,135,228,122,413,207,229)>;
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), (1,392,225)(2,393,226)(3,394,227)(4,395,228)(5,396,210)(6,397,211)(7,398,212)(8,399,213)(9,381,214)(10,382,215)(11,383,216)(12,384,217)(13,385,218)(14,386,219)(15,387,220)(16,388,221)(17,389,222)(18,390,223)(19,391,224)(20,245,299)(21,246,300)(22,247,301)(23,229,302)(24,230,303)(25,231,304)(26,232,286)(27,233,287)(28,234,288)(29,235,289)(30,236,290)(31,237,291)(32,238,292)(33,239,293)(34,240,294)(35,241,295)(36,242,296)(37,243,297)(38,244,298)(39,150,372)(40,151,373)(41,152,374)(42,134,375)(43,135,376)(44,136,377)(45,137,378)(46,138,379)(47,139,380)(48,140,362)(49,141,363)(50,142,364)(51,143,365)(52,144,366)(53,145,367)(54,146,368)(55,147,369)(56,148,370)(57,149,371)(58,133,179)(59,115,180)(60,116,181)(61,117,182)(62,118,183)(63,119,184)(64,120,185)(65,121,186)(66,122,187)(67,123,188)(68,124,189)(69,125,190)(70,126,172)(71,127,173)(72,128,174)(73,129,175)(74,130,176)(75,131,177)(76,132,178)(77,336,249)(78,337,250)(79,338,251)(80,339,252)(81,340,253)(82,341,254)(83,342,255)(84,324,256)(85,325,257)(86,326,258)(87,327,259)(88,328,260)(89,329,261)(90,330,262)(91,331,263)(92,332,264)(93,333,265)(94,334,266)(95,335,248)(96,453,432)(97,454,433)(98,455,434)(99,456,435)(100,438,436)(101,439,437)(102,440,419)(103,441,420)(104,442,421)(105,443,422)(106,444,423)(107,445,424)(108,446,425)(109,447,426)(110,448,427)(111,449,428)(112,450,429)(113,451,430)(114,452,431)(153,268,196)(154,269,197)(155,270,198)(156,271,199)(157,272,200)(158,273,201)(159,274,202)(160,275,203)(161,276,204)(162,277,205)(163,278,206)(164,279,207)(165,280,208)(166,281,209)(167,282,191)(168,283,192)(169,284,193)(170,285,194)(171,267,195)(305,404,355)(306,405,356)(307,406,357)(308,407,358)(309,408,359)(310,409,360)(311,410,361)(312,411,343)(313,412,344)(314,413,345)(315,414,346)(316,415,347)(317,416,348)(318,417,349)(319,418,350)(320,400,351)(321,401,352)(322,402,353)(323,403,354), (1,63,361,276,299,428,253,40)(2,64,343,277,300,429,254,41)(3,65,344,278,301,430,255,42)(4,66,345,279,302,431,256,43)(5,67,346,280,303,432,257,44)(6,68,347,281,304,433,258,45)(7,69,348,282,286,434,259,46)(8,70,349,283,287,435,260,47)(9,71,350,284,288,436,261,48)(10,72,351,285,289,437,262,49)(11,73,352,267,290,419,263,50)(12,74,353,268,291,420,264,51)(13,75,354,269,292,421,265,52)(14,76,355,270,293,422,266,53)(15,58,356,271,294,423,248,54)(16,59,357,272,295,424,249,55)(17,60,358,273,296,425,250,56)(18,61,359,274,297,426,251,57)(19,62,360,275,298,427,252,39)(20,449,81,373,392,184,311,161)(21,450,82,374,393,185,312,162)(22,451,83,375,394,186,313,163)(23,452,84,376,395,187,314,164)(24,453,85,377,396,188,315,165)(25,454,86,378,397,189,316,166)(26,455,87,379,398,190,317,167)(27,456,88,380,399,172,318,168)(28,438,89,362,381,173,319,169)(29,439,90,363,382,174,320,170)(30,440,91,364,383,175,321,171)(31,441,92,365,384,176,322,153)(32,442,93,366,385,177,323,154)(33,443,94,367,386,178,305,155)(34,444,95,368,387,179,306,156)(35,445,77,369,388,180,307,157)(36,446,78,370,389,181,308,158)(37,447,79,371,390,182,309,159)(38,448,80,372,391,183,310,160)(96,325,136,210,123,414,208,230)(97,326,137,211,124,415,209,231)(98,327,138,212,125,416,191,232)(99,328,139,213,126,417,192,233)(100,329,140,214,127,418,193,234)(101,330,141,215,128,400,194,235)(102,331,142,216,129,401,195,236)(103,332,143,217,130,402,196,237)(104,333,144,218,131,403,197,238)(105,334,145,219,132,404,198,239)(106,335,146,220,133,405,199,240)(107,336,147,221,115,406,200,241)(108,337,148,222,116,407,201,242)(109,338,149,223,117,408,202,243)(110,339,150,224,118,409,203,244)(111,340,151,225,119,410,204,245)(112,341,152,226,120,411,205,246)(113,342,134,227,121,412,206,247)(114,324,135,228,122,413,207,229) );
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)], [(1,392,225),(2,393,226),(3,394,227),(4,395,228),(5,396,210),(6,397,211),(7,398,212),(8,399,213),(9,381,214),(10,382,215),(11,383,216),(12,384,217),(13,385,218),(14,386,219),(15,387,220),(16,388,221),(17,389,222),(18,390,223),(19,391,224),(20,245,299),(21,246,300),(22,247,301),(23,229,302),(24,230,303),(25,231,304),(26,232,286),(27,233,287),(28,234,288),(29,235,289),(30,236,290),(31,237,291),(32,238,292),(33,239,293),(34,240,294),(35,241,295),(36,242,296),(37,243,297),(38,244,298),(39,150,372),(40,151,373),(41,152,374),(42,134,375),(43,135,376),(44,136,377),(45,137,378),(46,138,379),(47,139,380),(48,140,362),(49,141,363),(50,142,364),(51,143,365),(52,144,366),(53,145,367),(54,146,368),(55,147,369),(56,148,370),(57,149,371),(58,133,179),(59,115,180),(60,116,181),(61,117,182),(62,118,183),(63,119,184),(64,120,185),(65,121,186),(66,122,187),(67,123,188),(68,124,189),(69,125,190),(70,126,172),(71,127,173),(72,128,174),(73,129,175),(74,130,176),(75,131,177),(76,132,178),(77,336,249),(78,337,250),(79,338,251),(80,339,252),(81,340,253),(82,341,254),(83,342,255),(84,324,256),(85,325,257),(86,326,258),(87,327,259),(88,328,260),(89,329,261),(90,330,262),(91,331,263),(92,332,264),(93,333,265),(94,334,266),(95,335,248),(96,453,432),(97,454,433),(98,455,434),(99,456,435),(100,438,436),(101,439,437),(102,440,419),(103,441,420),(104,442,421),(105,443,422),(106,444,423),(107,445,424),(108,446,425),(109,447,426),(110,448,427),(111,449,428),(112,450,429),(113,451,430),(114,452,431),(153,268,196),(154,269,197),(155,270,198),(156,271,199),(157,272,200),(158,273,201),(159,274,202),(160,275,203),(161,276,204),(162,277,205),(163,278,206),(164,279,207),(165,280,208),(166,281,209),(167,282,191),(168,283,192),(169,284,193),(170,285,194),(171,267,195),(305,404,355),(306,405,356),(307,406,357),(308,407,358),(309,408,359),(310,409,360),(311,410,361),(312,411,343),(313,412,344),(314,413,345),(315,414,346),(316,415,347),(317,416,348),(318,417,349),(319,418,350),(320,400,351),(321,401,352),(322,402,353),(323,403,354)], [(1,63,361,276,299,428,253,40),(2,64,343,277,300,429,254,41),(3,65,344,278,301,430,255,42),(4,66,345,279,302,431,256,43),(5,67,346,280,303,432,257,44),(6,68,347,281,304,433,258,45),(7,69,348,282,286,434,259,46),(8,70,349,283,287,435,260,47),(9,71,350,284,288,436,261,48),(10,72,351,285,289,437,262,49),(11,73,352,267,290,419,263,50),(12,74,353,268,291,420,264,51),(13,75,354,269,292,421,265,52),(14,76,355,270,293,422,266,53),(15,58,356,271,294,423,248,54),(16,59,357,272,295,424,249,55),(17,60,358,273,296,425,250,56),(18,61,359,274,297,426,251,57),(19,62,360,275,298,427,252,39),(20,449,81,373,392,184,311,161),(21,450,82,374,393,185,312,162),(22,451,83,375,394,186,313,163),(23,452,84,376,395,187,314,164),(24,453,85,377,396,188,315,165),(25,454,86,378,397,189,316,166),(26,455,87,379,398,190,317,167),(27,456,88,380,399,172,318,168),(28,438,89,362,381,173,319,169),(29,439,90,363,382,174,320,170),(30,440,91,364,383,175,321,171),(31,441,92,365,384,176,322,153),(32,442,93,366,385,177,323,154),(33,443,94,367,386,178,305,155),(34,444,95,368,387,179,306,156),(35,445,77,369,388,180,307,157),(36,446,78,370,389,181,308,158),(37,447,79,371,390,182,309,159),(38,448,80,372,391,183,310,160),(96,325,136,210,123,414,208,230),(97,326,137,211,124,415,209,231),(98,327,138,212,125,416,191,232),(99,328,139,213,126,417,192,233),(100,329,140,214,127,418,193,234),(101,330,141,215,128,400,194,235),(102,331,142,216,129,401,195,236),(103,332,143,217,130,402,196,237),(104,333,144,218,131,403,197,238),(105,334,145,219,132,404,198,239),(106,335,146,220,133,405,199,240),(107,336,147,221,115,406,200,241),(108,337,148,222,116,407,201,242),(109,338,149,223,117,408,202,243),(110,339,150,224,118,409,203,244),(111,340,151,225,119,410,204,245),(112,341,152,226,120,411,205,246),(113,342,134,227,121,412,206,247),(114,324,135,228,122,413,207,229)]])
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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