direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C19⋊C8, C57⋊2C8, C19⋊3C24, C76.6C6, C228.4C2, C114.2C4, C38.3C12, C12.4D19, C6.2Dic19, C4.2(C3×D19), C2.(C3×Dic19), SmallGroup(456,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C3×C19⋊C8 |
Generators and relations for C3×C19⋊C8
G = < a,b,c | a3=b19=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C19⋊C8
►Regular action on 456 points
Generators in S456
(1 153 77)(2 154 78)(3 155 79)(4 156 80)(5 157 81)(6 158 82)(7 159 83)(8 160 84)(9 161 85)(10 162 86)(11 163 87)(12 164 88)(13 165 89)(14 166 90)(15 167 91)(16 168 92)(17 169 93)(18 170 94)(19 171 95)(20 172 96)(21 173 97)(22 174 98)(23 175 99)(24 176 100)(25 177 101)(26 178 102)(27 179 103)(28 180 104)(29 181 105)(30 182 106)(31 183 107)(32 184 108)(33 185 109)(34 186 110)(35 187 111)(36 188 112)(37 189 113)(38 190 114)(39 191 115)(40 192 116)(41 193 117)(42 194 118)(43 195 119)(44 196 120)(45 197 121)(46 198 122)(47 199 123)(48 200 124)(49 201 125)(50 202 126)(51 203 127)(52 204 128)(53 205 129)(54 206 130)(55 207 131)(56 208 132)(57 209 133)(58 210 134)(59 211 135)(60 212 136)(61 213 137)(62 214 138)(63 215 139)(64 216 140)(65 217 141)(66 218 142)(67 219 143)(68 220 144)(69 221 145)(70 222 146)(71 223 147)(72 224 148)(73 225 149)(74 226 150)(75 227 151)(76 228 152)(229 385 322)(230 386 323)(231 387 305)(232 388 306)(233 389 307)(234 390 308)(235 391 309)(236 392 310)(237 393 311)(238 394 312)(239 395 313)(240 396 314)(241 397 315)(242 398 316)(243 399 317)(244 381 318)(245 382 319)(246 383 320)(247 384 321)(248 413 334)(249 414 335)(250 415 336)(251 416 337)(252 417 338)(253 418 339)(254 400 340)(255 401 341)(256 402 342)(257 403 324)(258 404 325)(259 405 326)(260 406 327)(261 407 328)(262 408 329)(263 409 330)(264 410 331)(265 411 332)(266 412 333)(267 435 359)(268 436 360)(269 437 361)(270 419 343)(271 420 344)(272 421 345)(273 422 346)(274 423 347)(275 424 348)(276 425 349)(277 426 350)(278 427 351)(279 428 352)(280 429 353)(281 430 354)(282 431 355)(283 432 356)(284 433 357)(285 434 358)(286 453 378)(287 454 379)(288 455 380)(289 456 362)(290 438 363)(291 439 364)(292 440 365)(293 441 366)(294 442 367)(295 443 368)(296 444 369)(297 445 370)(298 446 371)(299 447 372)(300 448 373)(301 449 374)(302 450 375)(303 451 376)(304 452 377)
(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)
(1 294 58 255 20 268 39 246)(2 293 59 254 21 267 40 245)(3 292 60 253 22 285 41 244)(4 291 61 252 23 284 42 243)(5 290 62 251 24 283 43 242)(6 289 63 250 25 282 44 241)(7 288 64 249 26 281 45 240)(8 287 65 248 27 280 46 239)(9 286 66 266 28 279 47 238)(10 304 67 265 29 278 48 237)(11 303 68 264 30 277 49 236)(12 302 69 263 31 276 50 235)(13 301 70 262 32 275 51 234)(14 300 71 261 33 274 52 233)(15 299 72 260 34 273 53 232)(16 298 73 259 35 272 54 231)(17 297 74 258 36 271 55 230)(18 296 75 257 37 270 56 229)(19 295 76 256 38 269 57 247)(77 367 134 341 96 360 115 320)(78 366 135 340 97 359 116 319)(79 365 136 339 98 358 117 318)(80 364 137 338 99 357 118 317)(81 363 138 337 100 356 119 316)(82 362 139 336 101 355 120 315)(83 380 140 335 102 354 121 314)(84 379 141 334 103 353 122 313)(85 378 142 333 104 352 123 312)(86 377 143 332 105 351 124 311)(87 376 144 331 106 350 125 310)(88 375 145 330 107 349 126 309)(89 374 146 329 108 348 127 308)(90 373 147 328 109 347 128 307)(91 372 148 327 110 346 129 306)(92 371 149 326 111 345 130 305)(93 370 150 325 112 344 131 323)(94 369 151 324 113 343 132 322)(95 368 152 342 114 361 133 321)(153 442 210 401 172 436 191 383)(154 441 211 400 173 435 192 382)(155 440 212 418 174 434 193 381)(156 439 213 417 175 433 194 399)(157 438 214 416 176 432 195 398)(158 456 215 415 177 431 196 397)(159 455 216 414 178 430 197 396)(160 454 217 413 179 429 198 395)(161 453 218 412 180 428 199 394)(162 452 219 411 181 427 200 393)(163 451 220 410 182 426 201 392)(164 450 221 409 183 425 202 391)(165 449 222 408 184 424 203 390)(166 448 223 407 185 423 204 389)(167 447 224 406 186 422 205 388)(168 446 225 405 187 421 206 387)(169 445 226 404 188 420 207 386)(170 444 227 403 189 419 208 385)(171 443 228 402 190 437 209 384)
G:=sub<Sym(456)| (1,153,77)(2,154,78)(3,155,79)(4,156,80)(5,157,81)(6,158,82)(7,159,83)(8,160,84)(9,161,85)(10,162,86)(11,163,87)(12,164,88)(13,165,89)(14,166,90)(15,167,91)(16,168,92)(17,169,93)(18,170,94)(19,171,95)(20,172,96)(21,173,97)(22,174,98)(23,175,99)(24,176,100)(25,177,101)(26,178,102)(27,179,103)(28,180,104)(29,181,105)(30,182,106)(31,183,107)(32,184,108)(33,185,109)(34,186,110)(35,187,111)(36,188,112)(37,189,113)(38,190,114)(39,191,115)(40,192,116)(41,193,117)(42,194,118)(43,195,119)(44,196,120)(45,197,121)(46,198,122)(47,199,123)(48,200,124)(49,201,125)(50,202,126)(51,203,127)(52,204,128)(53,205,129)(54,206,130)(55,207,131)(56,208,132)(57,209,133)(58,210,134)(59,211,135)(60,212,136)(61,213,137)(62,214,138)(63,215,139)(64,216,140)(65,217,141)(66,218,142)(67,219,143)(68,220,144)(69,221,145)(70,222,146)(71,223,147)(72,224,148)(73,225,149)(74,226,150)(75,227,151)(76,228,152)(229,385,322)(230,386,323)(231,387,305)(232,388,306)(233,389,307)(234,390,308)(235,391,309)(236,392,310)(237,393,311)(238,394,312)(239,395,313)(240,396,314)(241,397,315)(242,398,316)(243,399,317)(244,381,318)(245,382,319)(246,383,320)(247,384,321)(248,413,334)(249,414,335)(250,415,336)(251,416,337)(252,417,338)(253,418,339)(254,400,340)(255,401,341)(256,402,342)(257,403,324)(258,404,325)(259,405,326)(260,406,327)(261,407,328)(262,408,329)(263,409,330)(264,410,331)(265,411,332)(266,412,333)(267,435,359)(268,436,360)(269,437,361)(270,419,343)(271,420,344)(272,421,345)(273,422,346)(274,423,347)(275,424,348)(276,425,349)(277,426,350)(278,427,351)(279,428,352)(280,429,353)(281,430,354)(282,431,355)(283,432,356)(284,433,357)(285,434,358)(286,453,378)(287,454,379)(288,455,380)(289,456,362)(290,438,363)(291,439,364)(292,440,365)(293,441,366)(294,442,367)(295,443,368)(296,444,369)(297,445,370)(298,446,371)(299,447,372)(300,448,373)(301,449,374)(302,450,375)(303,451,376)(304,452,377), (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), (1,294,58,255,20,268,39,246)(2,293,59,254,21,267,40,245)(3,292,60,253,22,285,41,244)(4,291,61,252,23,284,42,243)(5,290,62,251,24,283,43,242)(6,289,63,250,25,282,44,241)(7,288,64,249,26,281,45,240)(8,287,65,248,27,280,46,239)(9,286,66,266,28,279,47,238)(10,304,67,265,29,278,48,237)(11,303,68,264,30,277,49,236)(12,302,69,263,31,276,50,235)(13,301,70,262,32,275,51,234)(14,300,71,261,33,274,52,233)(15,299,72,260,34,273,53,232)(16,298,73,259,35,272,54,231)(17,297,74,258,36,271,55,230)(18,296,75,257,37,270,56,229)(19,295,76,256,38,269,57,247)(77,367,134,341,96,360,115,320)(78,366,135,340,97,359,116,319)(79,365,136,339,98,358,117,318)(80,364,137,338,99,357,118,317)(81,363,138,337,100,356,119,316)(82,362,139,336,101,355,120,315)(83,380,140,335,102,354,121,314)(84,379,141,334,103,353,122,313)(85,378,142,333,104,352,123,312)(86,377,143,332,105,351,124,311)(87,376,144,331,106,350,125,310)(88,375,145,330,107,349,126,309)(89,374,146,329,108,348,127,308)(90,373,147,328,109,347,128,307)(91,372,148,327,110,346,129,306)(92,371,149,326,111,345,130,305)(93,370,150,325,112,344,131,323)(94,369,151,324,113,343,132,322)(95,368,152,342,114,361,133,321)(153,442,210,401,172,436,191,383)(154,441,211,400,173,435,192,382)(155,440,212,418,174,434,193,381)(156,439,213,417,175,433,194,399)(157,438,214,416,176,432,195,398)(158,456,215,415,177,431,196,397)(159,455,216,414,178,430,197,396)(160,454,217,413,179,429,198,395)(161,453,218,412,180,428,199,394)(162,452,219,411,181,427,200,393)(163,451,220,410,182,426,201,392)(164,450,221,409,183,425,202,391)(165,449,222,408,184,424,203,390)(166,448,223,407,185,423,204,389)(167,447,224,406,186,422,205,388)(168,446,225,405,187,421,206,387)(169,445,226,404,188,420,207,386)(170,444,227,403,189,419,208,385)(171,443,228,402,190,437,209,384)>;
G:=Group( (1,153,77)(2,154,78)(3,155,79)(4,156,80)(5,157,81)(6,158,82)(7,159,83)(8,160,84)(9,161,85)(10,162,86)(11,163,87)(12,164,88)(13,165,89)(14,166,90)(15,167,91)(16,168,92)(17,169,93)(18,170,94)(19,171,95)(20,172,96)(21,173,97)(22,174,98)(23,175,99)(24,176,100)(25,177,101)(26,178,102)(27,179,103)(28,180,104)(29,181,105)(30,182,106)(31,183,107)(32,184,108)(33,185,109)(34,186,110)(35,187,111)(36,188,112)(37,189,113)(38,190,114)(39,191,115)(40,192,116)(41,193,117)(42,194,118)(43,195,119)(44,196,120)(45,197,121)(46,198,122)(47,199,123)(48,200,124)(49,201,125)(50,202,126)(51,203,127)(52,204,128)(53,205,129)(54,206,130)(55,207,131)(56,208,132)(57,209,133)(58,210,134)(59,211,135)(60,212,136)(61,213,137)(62,214,138)(63,215,139)(64,216,140)(65,217,141)(66,218,142)(67,219,143)(68,220,144)(69,221,145)(70,222,146)(71,223,147)(72,224,148)(73,225,149)(74,226,150)(75,227,151)(76,228,152)(229,385,322)(230,386,323)(231,387,305)(232,388,306)(233,389,307)(234,390,308)(235,391,309)(236,392,310)(237,393,311)(238,394,312)(239,395,313)(240,396,314)(241,397,315)(242,398,316)(243,399,317)(244,381,318)(245,382,319)(246,383,320)(247,384,321)(248,413,334)(249,414,335)(250,415,336)(251,416,337)(252,417,338)(253,418,339)(254,400,340)(255,401,341)(256,402,342)(257,403,324)(258,404,325)(259,405,326)(260,406,327)(261,407,328)(262,408,329)(263,409,330)(264,410,331)(265,411,332)(266,412,333)(267,435,359)(268,436,360)(269,437,361)(270,419,343)(271,420,344)(272,421,345)(273,422,346)(274,423,347)(275,424,348)(276,425,349)(277,426,350)(278,427,351)(279,428,352)(280,429,353)(281,430,354)(282,431,355)(283,432,356)(284,433,357)(285,434,358)(286,453,378)(287,454,379)(288,455,380)(289,456,362)(290,438,363)(291,439,364)(292,440,365)(293,441,366)(294,442,367)(295,443,368)(296,444,369)(297,445,370)(298,446,371)(299,447,372)(300,448,373)(301,449,374)(302,450,375)(303,451,376)(304,452,377), (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), (1,294,58,255,20,268,39,246)(2,293,59,254,21,267,40,245)(3,292,60,253,22,285,41,244)(4,291,61,252,23,284,42,243)(5,290,62,251,24,283,43,242)(6,289,63,250,25,282,44,241)(7,288,64,249,26,281,45,240)(8,287,65,248,27,280,46,239)(9,286,66,266,28,279,47,238)(10,304,67,265,29,278,48,237)(11,303,68,264,30,277,49,236)(12,302,69,263,31,276,50,235)(13,301,70,262,32,275,51,234)(14,300,71,261,33,274,52,233)(15,299,72,260,34,273,53,232)(16,298,73,259,35,272,54,231)(17,297,74,258,36,271,55,230)(18,296,75,257,37,270,56,229)(19,295,76,256,38,269,57,247)(77,367,134,341,96,360,115,320)(78,366,135,340,97,359,116,319)(79,365,136,339,98,358,117,318)(80,364,137,338,99,357,118,317)(81,363,138,337,100,356,119,316)(82,362,139,336,101,355,120,315)(83,380,140,335,102,354,121,314)(84,379,141,334,103,353,122,313)(85,378,142,333,104,352,123,312)(86,377,143,332,105,351,124,311)(87,376,144,331,106,350,125,310)(88,375,145,330,107,349,126,309)(89,374,146,329,108,348,127,308)(90,373,147,328,109,347,128,307)(91,372,148,327,110,346,129,306)(92,371,149,326,111,345,130,305)(93,370,150,325,112,344,131,323)(94,369,151,324,113,343,132,322)(95,368,152,342,114,361,133,321)(153,442,210,401,172,436,191,383)(154,441,211,400,173,435,192,382)(155,440,212,418,174,434,193,381)(156,439,213,417,175,433,194,399)(157,438,214,416,176,432,195,398)(158,456,215,415,177,431,196,397)(159,455,216,414,178,430,197,396)(160,454,217,413,179,429,198,395)(161,453,218,412,180,428,199,394)(162,452,219,411,181,427,200,393)(163,451,220,410,182,426,201,392)(164,450,221,409,183,425,202,391)(165,449,222,408,184,424,203,390)(166,448,223,407,185,423,204,389)(167,447,224,406,186,422,205,388)(168,446,225,405,187,421,206,387)(169,445,226,404,188,420,207,386)(170,444,227,403,189,419,208,385)(171,443,228,402,190,437,209,384) );
G=PermutationGroup([[(1,153,77),(2,154,78),(3,155,79),(4,156,80),(5,157,81),(6,158,82),(7,159,83),(8,160,84),(9,161,85),(10,162,86),(11,163,87),(12,164,88),(13,165,89),(14,166,90),(15,167,91),(16,168,92),(17,169,93),(18,170,94),(19,171,95),(20,172,96),(21,173,97),(22,174,98),(23,175,99),(24,176,100),(25,177,101),(26,178,102),(27,179,103),(28,180,104),(29,181,105),(30,182,106),(31,183,107),(32,184,108),(33,185,109),(34,186,110),(35,187,111),(36,188,112),(37,189,113),(38,190,114),(39,191,115),(40,192,116),(41,193,117),(42,194,118),(43,195,119),(44,196,120),(45,197,121),(46,198,122),(47,199,123),(48,200,124),(49,201,125),(50,202,126),(51,203,127),(52,204,128),(53,205,129),(54,206,130),(55,207,131),(56,208,132),(57,209,133),(58,210,134),(59,211,135),(60,212,136),(61,213,137),(62,214,138),(63,215,139),(64,216,140),(65,217,141),(66,218,142),(67,219,143),(68,220,144),(69,221,145),(70,222,146),(71,223,147),(72,224,148),(73,225,149),(74,226,150),(75,227,151),(76,228,152),(229,385,322),(230,386,323),(231,387,305),(232,388,306),(233,389,307),(234,390,308),(235,391,309),(236,392,310),(237,393,311),(238,394,312),(239,395,313),(240,396,314),(241,397,315),(242,398,316),(243,399,317),(244,381,318),(245,382,319),(246,383,320),(247,384,321),(248,413,334),(249,414,335),(250,415,336),(251,416,337),(252,417,338),(253,418,339),(254,400,340),(255,401,341),(256,402,342),(257,403,324),(258,404,325),(259,405,326),(260,406,327),(261,407,328),(262,408,329),(263,409,330),(264,410,331),(265,411,332),(266,412,333),(267,435,359),(268,436,360),(269,437,361),(270,419,343),(271,420,344),(272,421,345),(273,422,346),(274,423,347),(275,424,348),(276,425,349),(277,426,350),(278,427,351),(279,428,352),(280,429,353),(281,430,354),(282,431,355),(283,432,356),(284,433,357),(285,434,358),(286,453,378),(287,454,379),(288,455,380),(289,456,362),(290,438,363),(291,439,364),(292,440,365),(293,441,366),(294,442,367),(295,443,368),(296,444,369),(297,445,370),(298,446,371),(299,447,372),(300,448,373),(301,449,374),(302,450,375),(303,451,376),(304,452,377)], [(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)], [(1,294,58,255,20,268,39,246),(2,293,59,254,21,267,40,245),(3,292,60,253,22,285,41,244),(4,291,61,252,23,284,42,243),(5,290,62,251,24,283,43,242),(6,289,63,250,25,282,44,241),(7,288,64,249,26,281,45,240),(8,287,65,248,27,280,46,239),(9,286,66,266,28,279,47,238),(10,304,67,265,29,278,48,237),(11,303,68,264,30,277,49,236),(12,302,69,263,31,276,50,235),(13,301,70,262,32,275,51,234),(14,300,71,261,33,274,52,233),(15,299,72,260,34,273,53,232),(16,298,73,259,35,272,54,231),(17,297,74,258,36,271,55,230),(18,296,75,257,37,270,56,229),(19,295,76,256,38,269,57,247),(77,367,134,341,96,360,115,320),(78,366,135,340,97,359,116,319),(79,365,136,339,98,358,117,318),(80,364,137,338,99,357,118,317),(81,363,138,337,100,356,119,316),(82,362,139,336,101,355,120,315),(83,380,140,335,102,354,121,314),(84,379,141,334,103,353,122,313),(85,378,142,333,104,352,123,312),(86,377,143,332,105,351,124,311),(87,376,144,331,106,350,125,310),(88,375,145,330,107,349,126,309),(89,374,146,329,108,348,127,308),(90,373,147,328,109,347,128,307),(91,372,148,327,110,346,129,306),(92,371,149,326,111,345,130,305),(93,370,150,325,112,344,131,323),(94,369,151,324,113,343,132,322),(95,368,152,342,114,361,133,321),(153,442,210,401,172,436,191,383),(154,441,211,400,173,435,192,382),(155,440,212,418,174,434,193,381),(156,439,213,417,175,433,194,399),(157,438,214,416,176,432,195,398),(158,456,215,415,177,431,196,397),(159,455,216,414,178,430,197,396),(160,454,217,413,179,429,198,395),(161,453,218,412,180,428,199,394),(162,452,219,411,181,427,200,393),(163,451,220,410,182,426,201,392),(164,450,221,409,183,425,202,391),(165,449,222,408,184,424,203,390),(166,448,223,407,185,423,204,389),(167,447,224,406,186,422,205,388),(168,446,225,405,187,421,206,387),(169,445,226,404,188,420,207,386),(170,444,227,403,189,419,208,385),(171,443,228,402,190,437,209,384)]])
132 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 19A | ··· | 19I | 24A | ··· | 24H | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 19 | ··· | 19 | 24 | ··· | 24 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | D19 | Dic19 | C3×D19 | C19⋊C8 | C3×Dic19 | C3×C19⋊C8 |
| kernel | C3×C19⋊C8 | C228 | C19⋊C8 | C114 | C76 | C57 | C38 | C19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C3×C19⋊C8 ►in GL2(𝔽37) generated by
| 26 | 0 |
| 0 | 26 |
| 29 | 12 |
| 22 | 27 |
| 0 | 8 |
| 10 | 0 |
G:=sub<GL(2,GF(37))| [26,0,0,26],[29,22,12,27],[0,10,8,0] >;
C3×C19⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_{19}\rtimes C_8 % in TeX
G:=Group("C3xC19:C8"); // GroupNames label
G:=SmallGroup(456,4);
// by ID
G=gap.SmallGroup(456,4);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-19,30,42,10804]);
// Polycyclic
G:=Group<a,b,c|a^3=b^19=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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