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G = C3×C19⋊C8order 456 = 23·3·19

Direct product of C3 and C19⋊C8

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

Aliases: C3×C19⋊C8, C572C8, C193C24, 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

C1C19 — C3×C19⋊C8
C1C19C38C76C228 — C3×C19⋊C8
C19 — C3×C19⋊C8
C1C12

Generators and relations for C3×C19⋊C8
 G = < a,b,c | a3=b19=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

19C8
19C24

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

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

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

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

132 conjugacy classes

class 1  2 3A3B4A4B6A6B8A8B8C8D12A12B12C12D19A···19I24A···24H38A···38I57A···57R76A···76R114A···114R228A···228AJ
order1233446688881212121219···1924···2438···3857···5776···76114···114228···228
size111111111919191911112···219···192···22···22···22···22···2

132 irreducible representations

dim11111111222222
type+++-
imageC1C2C3C4C6C8C12C24D19Dic19C3×D19C19⋊C8C3×Dic19C3×C19⋊C8
kernelC3×C19⋊C8C228C19⋊C8C114C76C57C38C19C12C6C4C3C2C1
# reps112224489918181836

Matrix representation of C3×C19⋊C8 in GL2(𝔽37) generated by

260
026
,
2912
2227
,
08
100
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

Export

Subgroup lattice of C3×C19⋊C8 in TeX

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