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G = C19×Dic6order 456 = 23·3·19

Direct product of C19 and Dic6

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

Aliases: C19×Dic6, C574Q8, C76.3S3, C12.1C38, C228.5C2, C38.13D6, Dic3.C38, C114.18C22, C3⋊(Q8×C19), C4.(S3×C19), C2.3(S3×C38), C6.1(C2×C38), (Dic3×C19).2C2, SmallGroup(456,29)

Series: Derived Chief Lower central Upper central

C1C6 — C19×Dic6
C1C3C6C114Dic3×C19 — C19×Dic6
C3C6 — C19×Dic6
C1C38C76

Generators and relations for C19×Dic6
 G = < a,b,c | a19=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3Q8
3C76
3C76
3Q8×C19

Smallest permutation representation of C19×Dic6
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)
(1 143 227 62 260 372 328 402 44 129 169 281)(2 144 228 63 261 373 329 403 45 130 170 282)(3 145 210 64 262 374 330 404 46 131 171 283)(4 146 211 65 263 375 331 405 47 132 153 284)(5 147 212 66 264 376 332 406 48 133 154 285)(6 148 213 67 265 377 333 407 49 115 155 267)(7 149 214 68 266 378 334 408 50 116 156 268)(8 150 215 69 248 379 335 409 51 117 157 269)(9 151 216 70 249 380 336 410 52 118 158 270)(10 152 217 71 250 362 337 411 53 119 159 271)(11 134 218 72 251 363 338 412 54 120 160 272)(12 135 219 73 252 364 339 413 55 121 161 273)(13 136 220 74 253 365 340 414 56 122 162 274)(14 137 221 75 254 366 341 415 57 123 163 275)(15 138 222 76 255 367 342 416 39 124 164 276)(16 139 223 58 256 368 324 417 40 125 165 277)(17 140 224 59 257 369 325 418 41 126 166 278)(18 141 225 60 258 370 326 400 42 127 167 279)(19 142 226 61 259 371 327 401 43 128 168 280)(20 349 174 236 423 390 305 456 291 108 197 86)(21 350 175 237 424 391 306 438 292 109 198 87)(22 351 176 238 425 392 307 439 293 110 199 88)(23 352 177 239 426 393 308 440 294 111 200 89)(24 353 178 240 427 394 309 441 295 112 201 90)(25 354 179 241 428 395 310 442 296 113 202 91)(26 355 180 242 429 396 311 443 297 114 203 92)(27 356 181 243 430 397 312 444 298 96 204 93)(28 357 182 244 431 398 313 445 299 97 205 94)(29 358 183 245 432 399 314 446 300 98 206 95)(30 359 184 246 433 381 315 447 301 99 207 77)(31 360 185 247 434 382 316 448 302 100 208 78)(32 361 186 229 435 383 317 449 303 101 209 79)(33 343 187 230 436 384 318 450 304 102 191 80)(34 344 188 231 437 385 319 451 286 103 192 81)(35 345 189 232 419 386 320 452 287 104 193 82)(36 346 190 233 420 387 321 453 288 105 194 83)(37 347 172 234 421 388 322 454 289 106 195 84)(38 348 173 235 422 389 323 455 290 107 196 85)
(1 237 328 109)(2 238 329 110)(3 239 330 111)(4 240 331 112)(5 241 332 113)(6 242 333 114)(7 243 334 96)(8 244 335 97)(9 245 336 98)(10 246 337 99)(11 247 338 100)(12 229 339 101)(13 230 340 102)(14 231 341 103)(15 232 342 104)(16 233 324 105)(17 234 325 106)(18 235 326 107)(19 236 327 108)(20 128 305 61)(21 129 306 62)(22 130 307 63)(23 131 308 64)(24 132 309 65)(25 133 310 66)(26 115 311 67)(27 116 312 68)(28 117 313 69)(29 118 314 70)(30 119 315 71)(31 120 316 72)(32 121 317 73)(33 122 318 74)(34 123 319 75)(35 124 320 76)(36 125 321 58)(37 126 322 59)(38 127 323 60)(39 452 222 345)(40 453 223 346)(41 454 224 347)(42 455 225 348)(43 456 226 349)(44 438 227 350)(45 439 228 351)(46 440 210 352)(47 441 211 353)(48 442 212 354)(49 443 213 355)(50 444 214 356)(51 445 215 357)(52 446 216 358)(53 447 217 359)(54 448 218 360)(55 449 219 361)(56 450 220 343)(57 451 221 344)(77 159 381 250)(78 160 382 251)(79 161 383 252)(80 162 384 253)(81 163 385 254)(82 164 386 255)(83 165 387 256)(84 166 388 257)(85 167 389 258)(86 168 390 259)(87 169 391 260)(88 170 392 261)(89 171 393 262)(90 153 394 263)(91 154 395 264)(92 155 396 265)(93 156 397 266)(94 157 398 248)(95 158 399 249)(134 185 412 302)(135 186 413 303)(136 187 414 304)(137 188 415 286)(138 189 416 287)(139 190 417 288)(140 172 418 289)(141 173 400 290)(142 174 401 291)(143 175 402 292)(144 176 403 293)(145 177 404 294)(146 178 405 295)(147 179 406 296)(148 180 407 297)(149 181 408 298)(150 182 409 299)(151 183 410 300)(152 184 411 301)(191 274 436 365)(192 275 437 366)(193 276 419 367)(194 277 420 368)(195 278 421 369)(196 279 422 370)(197 280 423 371)(198 281 424 372)(199 282 425 373)(200 283 426 374)(201 284 427 375)(202 285 428 376)(203 267 429 377)(204 268 430 378)(205 269 431 379)(206 270 432 380)(207 271 433 362)(208 272 434 363)(209 273 435 364)

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

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

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

171 conjugacy classes

class 1  2  3 4A4B4C 6 12A12B19A···19R38A···38R57A···57R76A···76R76S···76BB114A···114R228A···228AJ
order1234446121219···1938···3857···5776···7676···76114···114228···228
size1122662221···11···12···22···26···62···22···2

171 irreducible representations

dim11111122222222
type++++-+-
imageC1C2C2C19C38C38S3Q8D6Dic6S3×C19Q8×C19S3×C38C19×Dic6
kernelC19×Dic6Dic3×C19C228Dic6Dic3C12C76C57C38C19C4C3C2C1
# reps121183618111218181836

Matrix representation of C19×Dic6 in GL2(𝔽229) generated by

600
060
,
29129
100129
,
0107
1070
G:=sub<GL(2,GF(229))| [60,0,0,60],[29,100,129,129],[0,107,107,0] >;

C19×Dic6 in GAP, Magma, Sage, TeX

C_{19}\times {\rm Dic}_6
% in TeX

G:=Group("C19xDic6");
// GroupNames label

G:=SmallGroup(456,29);
// by ID

G=gap.SmallGroup(456,29);
# by ID

G:=PCGroup([5,-2,-2,-19,-2,-3,380,781,386,7604]);
// Polycyclic

G:=Group<a,b,c|a^19=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C19×Dic6 in TeX

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