direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C19×Dic6, C57⋊4Q8, 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
Generators and relations for C19×Dic6
G = < a,b,c | a19=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
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)
(1 341 255 51 419 72 23 219 157 272 456 301)(2 342 256 52 420 73 24 220 158 273 438 302)(3 324 257 53 421 74 25 221 159 274 439 303)(4 325 258 54 422 75 26 222 160 275 440 304)(5 326 259 55 423 76 27 223 161 276 441 286)(6 327 260 56 424 58 28 224 162 277 442 287)(7 328 261 57 425 59 29 225 163 278 443 288)(8 329 262 39 426 60 30 226 164 279 444 289)(9 330 263 40 427 61 31 227 165 280 445 290)(10 331 264 41 428 62 32 228 166 281 446 291)(11 332 265 42 429 63 33 210 167 282 447 292)(12 333 266 43 430 64 34 211 168 283 448 293)(13 334 248 44 431 65 35 212 169 284 449 294)(14 335 249 45 432 66 36 213 170 285 450 295)(15 336 250 46 433 67 37 214 171 267 451 296)(16 337 251 47 434 68 38 215 153 268 452 297)(17 338 252 48 435 69 20 216 154 269 453 298)(18 339 253 49 436 70 21 217 155 270 454 299)(19 340 254 50 437 71 22 218 156 271 455 300)(77 186 137 360 240 376 411 388 319 209 104 119)(78 187 138 361 241 377 412 389 320 191 105 120)(79 188 139 343 242 378 413 390 321 192 106 121)(80 189 140 344 243 379 414 391 322 193 107 122)(81 190 141 345 244 380 415 392 323 194 108 123)(82 172 142 346 245 362 416 393 305 195 109 124)(83 173 143 347 246 363 417 394 306 196 110 125)(84 174 144 348 247 364 418 395 307 197 111 126)(85 175 145 349 229 365 400 396 308 198 112 127)(86 176 146 350 230 366 401 397 309 199 113 128)(87 177 147 351 231 367 402 398 310 200 114 129)(88 178 148 352 232 368 403 399 311 201 96 130)(89 179 149 353 233 369 404 381 312 202 97 131)(90 180 150 354 234 370 405 382 313 203 98 132)(91 181 151 355 235 371 406 383 314 204 99 133)(92 182 152 356 236 372 407 384 315 205 100 115)(93 183 134 357 237 373 408 385 316 206 101 116)(94 184 135 358 238 374 409 386 317 207 102 117)(95 185 136 359 239 375 410 387 318 208 103 118)
(1 82 23 416)(2 83 24 417)(3 84 25 418)(4 85 26 400)(5 86 27 401)(6 87 28 402)(7 88 29 403)(8 89 30 404)(9 90 31 405)(10 91 32 406)(11 92 33 407)(12 93 34 408)(13 94 35 409)(14 95 36 410)(15 77 37 411)(16 78 38 412)(17 79 20 413)(18 80 21 414)(19 81 22 415)(39 202 279 353)(40 203 280 354)(41 204 281 355)(42 205 282 356)(43 206 283 357)(44 207 284 358)(45 208 285 359)(46 209 267 360)(47 191 268 361)(48 192 269 343)(49 193 270 344)(50 194 271 345)(51 195 272 346)(52 196 273 347)(53 197 274 348)(54 198 275 349)(55 199 276 350)(56 200 277 351)(57 201 278 352)(58 398 287 177)(59 399 288 178)(60 381 289 179)(61 382 290 180)(62 383 291 181)(63 384 292 182)(64 385 293 183)(65 386 294 184)(66 387 295 185)(67 388 296 186)(68 389 297 187)(69 390 298 188)(70 391 299 189)(71 392 300 190)(72 393 301 172)(73 394 302 173)(74 395 303 174)(75 396 304 175)(76 397 286 176)(96 163 232 261)(97 164 233 262)(98 165 234 263)(99 166 235 264)(100 167 236 265)(101 168 237 266)(102 169 238 248)(103 170 239 249)(104 171 240 250)(105 153 241 251)(106 154 242 252)(107 155 243 253)(108 156 244 254)(109 157 245 255)(110 158 246 256)(111 159 247 257)(112 160 229 258)(113 161 230 259)(114 162 231 260)(115 210 372 332)(116 211 373 333)(117 212 374 334)(118 213 375 335)(119 214 376 336)(120 215 377 337)(121 216 378 338)(122 217 379 339)(123 218 380 340)(124 219 362 341)(125 220 363 342)(126 221 364 324)(127 222 365 325)(128 223 366 326)(129 224 367 327)(130 225 368 328)(131 226 369 329)(132 227 370 330)(133 228 371 331)(134 430 316 448)(135 431 317 449)(136 432 318 450)(137 433 319 451)(138 434 320 452)(139 435 321 453)(140 436 322 454)(141 437 323 455)(142 419 305 456)(143 420 306 438)(144 421 307 439)(145 422 308 440)(146 423 309 441)(147 424 310 442)(148 425 311 443)(149 426 312 444)(150 427 313 445)(151 428 314 446)(152 429 315 447)
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), (1,341,255,51,419,72,23,219,157,272,456,301)(2,342,256,52,420,73,24,220,158,273,438,302)(3,324,257,53,421,74,25,221,159,274,439,303)(4,325,258,54,422,75,26,222,160,275,440,304)(5,326,259,55,423,76,27,223,161,276,441,286)(6,327,260,56,424,58,28,224,162,277,442,287)(7,328,261,57,425,59,29,225,163,278,443,288)(8,329,262,39,426,60,30,226,164,279,444,289)(9,330,263,40,427,61,31,227,165,280,445,290)(10,331,264,41,428,62,32,228,166,281,446,291)(11,332,265,42,429,63,33,210,167,282,447,292)(12,333,266,43,430,64,34,211,168,283,448,293)(13,334,248,44,431,65,35,212,169,284,449,294)(14,335,249,45,432,66,36,213,170,285,450,295)(15,336,250,46,433,67,37,214,171,267,451,296)(16,337,251,47,434,68,38,215,153,268,452,297)(17,338,252,48,435,69,20,216,154,269,453,298)(18,339,253,49,436,70,21,217,155,270,454,299)(19,340,254,50,437,71,22,218,156,271,455,300)(77,186,137,360,240,376,411,388,319,209,104,119)(78,187,138,361,241,377,412,389,320,191,105,120)(79,188,139,343,242,378,413,390,321,192,106,121)(80,189,140,344,243,379,414,391,322,193,107,122)(81,190,141,345,244,380,415,392,323,194,108,123)(82,172,142,346,245,362,416,393,305,195,109,124)(83,173,143,347,246,363,417,394,306,196,110,125)(84,174,144,348,247,364,418,395,307,197,111,126)(85,175,145,349,229,365,400,396,308,198,112,127)(86,176,146,350,230,366,401,397,309,199,113,128)(87,177,147,351,231,367,402,398,310,200,114,129)(88,178,148,352,232,368,403,399,311,201,96,130)(89,179,149,353,233,369,404,381,312,202,97,131)(90,180,150,354,234,370,405,382,313,203,98,132)(91,181,151,355,235,371,406,383,314,204,99,133)(92,182,152,356,236,372,407,384,315,205,100,115)(93,183,134,357,237,373,408,385,316,206,101,116)(94,184,135,358,238,374,409,386,317,207,102,117)(95,185,136,359,239,375,410,387,318,208,103,118), (1,82,23,416)(2,83,24,417)(3,84,25,418)(4,85,26,400)(5,86,27,401)(6,87,28,402)(7,88,29,403)(8,89,30,404)(9,90,31,405)(10,91,32,406)(11,92,33,407)(12,93,34,408)(13,94,35,409)(14,95,36,410)(15,77,37,411)(16,78,38,412)(17,79,20,413)(18,80,21,414)(19,81,22,415)(39,202,279,353)(40,203,280,354)(41,204,281,355)(42,205,282,356)(43,206,283,357)(44,207,284,358)(45,208,285,359)(46,209,267,360)(47,191,268,361)(48,192,269,343)(49,193,270,344)(50,194,271,345)(51,195,272,346)(52,196,273,347)(53,197,274,348)(54,198,275,349)(55,199,276,350)(56,200,277,351)(57,201,278,352)(58,398,287,177)(59,399,288,178)(60,381,289,179)(61,382,290,180)(62,383,291,181)(63,384,292,182)(64,385,293,183)(65,386,294,184)(66,387,295,185)(67,388,296,186)(68,389,297,187)(69,390,298,188)(70,391,299,189)(71,392,300,190)(72,393,301,172)(73,394,302,173)(74,395,303,174)(75,396,304,175)(76,397,286,176)(96,163,232,261)(97,164,233,262)(98,165,234,263)(99,166,235,264)(100,167,236,265)(101,168,237,266)(102,169,238,248)(103,170,239,249)(104,171,240,250)(105,153,241,251)(106,154,242,252)(107,155,243,253)(108,156,244,254)(109,157,245,255)(110,158,246,256)(111,159,247,257)(112,160,229,258)(113,161,230,259)(114,162,231,260)(115,210,372,332)(116,211,373,333)(117,212,374,334)(118,213,375,335)(119,214,376,336)(120,215,377,337)(121,216,378,338)(122,217,379,339)(123,218,380,340)(124,219,362,341)(125,220,363,342)(126,221,364,324)(127,222,365,325)(128,223,366,326)(129,224,367,327)(130,225,368,328)(131,226,369,329)(132,227,370,330)(133,228,371,331)(134,430,316,448)(135,431,317,449)(136,432,318,450)(137,433,319,451)(138,434,320,452)(139,435,321,453)(140,436,322,454)(141,437,323,455)(142,419,305,456)(143,420,306,438)(144,421,307,439)(145,422,308,440)(146,423,309,441)(147,424,310,442)(148,425,311,443)(149,426,312,444)(150,427,313,445)(151,428,314,446)(152,429,315,447)>;
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), (1,341,255,51,419,72,23,219,157,272,456,301)(2,342,256,52,420,73,24,220,158,273,438,302)(3,324,257,53,421,74,25,221,159,274,439,303)(4,325,258,54,422,75,26,222,160,275,440,304)(5,326,259,55,423,76,27,223,161,276,441,286)(6,327,260,56,424,58,28,224,162,277,442,287)(7,328,261,57,425,59,29,225,163,278,443,288)(8,329,262,39,426,60,30,226,164,279,444,289)(9,330,263,40,427,61,31,227,165,280,445,290)(10,331,264,41,428,62,32,228,166,281,446,291)(11,332,265,42,429,63,33,210,167,282,447,292)(12,333,266,43,430,64,34,211,168,283,448,293)(13,334,248,44,431,65,35,212,169,284,449,294)(14,335,249,45,432,66,36,213,170,285,450,295)(15,336,250,46,433,67,37,214,171,267,451,296)(16,337,251,47,434,68,38,215,153,268,452,297)(17,338,252,48,435,69,20,216,154,269,453,298)(18,339,253,49,436,70,21,217,155,270,454,299)(19,340,254,50,437,71,22,218,156,271,455,300)(77,186,137,360,240,376,411,388,319,209,104,119)(78,187,138,361,241,377,412,389,320,191,105,120)(79,188,139,343,242,378,413,390,321,192,106,121)(80,189,140,344,243,379,414,391,322,193,107,122)(81,190,141,345,244,380,415,392,323,194,108,123)(82,172,142,346,245,362,416,393,305,195,109,124)(83,173,143,347,246,363,417,394,306,196,110,125)(84,174,144,348,247,364,418,395,307,197,111,126)(85,175,145,349,229,365,400,396,308,198,112,127)(86,176,146,350,230,366,401,397,309,199,113,128)(87,177,147,351,231,367,402,398,310,200,114,129)(88,178,148,352,232,368,403,399,311,201,96,130)(89,179,149,353,233,369,404,381,312,202,97,131)(90,180,150,354,234,370,405,382,313,203,98,132)(91,181,151,355,235,371,406,383,314,204,99,133)(92,182,152,356,236,372,407,384,315,205,100,115)(93,183,134,357,237,373,408,385,316,206,101,116)(94,184,135,358,238,374,409,386,317,207,102,117)(95,185,136,359,239,375,410,387,318,208,103,118), (1,82,23,416)(2,83,24,417)(3,84,25,418)(4,85,26,400)(5,86,27,401)(6,87,28,402)(7,88,29,403)(8,89,30,404)(9,90,31,405)(10,91,32,406)(11,92,33,407)(12,93,34,408)(13,94,35,409)(14,95,36,410)(15,77,37,411)(16,78,38,412)(17,79,20,413)(18,80,21,414)(19,81,22,415)(39,202,279,353)(40,203,280,354)(41,204,281,355)(42,205,282,356)(43,206,283,357)(44,207,284,358)(45,208,285,359)(46,209,267,360)(47,191,268,361)(48,192,269,343)(49,193,270,344)(50,194,271,345)(51,195,272,346)(52,196,273,347)(53,197,274,348)(54,198,275,349)(55,199,276,350)(56,200,277,351)(57,201,278,352)(58,398,287,177)(59,399,288,178)(60,381,289,179)(61,382,290,180)(62,383,291,181)(63,384,292,182)(64,385,293,183)(65,386,294,184)(66,387,295,185)(67,388,296,186)(68,389,297,187)(69,390,298,188)(70,391,299,189)(71,392,300,190)(72,393,301,172)(73,394,302,173)(74,395,303,174)(75,396,304,175)(76,397,286,176)(96,163,232,261)(97,164,233,262)(98,165,234,263)(99,166,235,264)(100,167,236,265)(101,168,237,266)(102,169,238,248)(103,170,239,249)(104,171,240,250)(105,153,241,251)(106,154,242,252)(107,155,243,253)(108,156,244,254)(109,157,245,255)(110,158,246,256)(111,159,247,257)(112,160,229,258)(113,161,230,259)(114,162,231,260)(115,210,372,332)(116,211,373,333)(117,212,374,334)(118,213,375,335)(119,214,376,336)(120,215,377,337)(121,216,378,338)(122,217,379,339)(123,218,380,340)(124,219,362,341)(125,220,363,342)(126,221,364,324)(127,222,365,325)(128,223,366,326)(129,224,367,327)(130,225,368,328)(131,226,369,329)(132,227,370,330)(133,228,371,331)(134,430,316,448)(135,431,317,449)(136,432,318,450)(137,433,319,451)(138,434,320,452)(139,435,321,453)(140,436,322,454)(141,437,323,455)(142,419,305,456)(143,420,306,438)(144,421,307,439)(145,422,308,440)(146,423,309,441)(147,424,310,442)(148,425,311,443)(149,426,312,444)(150,427,313,445)(151,428,314,446)(152,429,315,447) );
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)], [(1,341,255,51,419,72,23,219,157,272,456,301),(2,342,256,52,420,73,24,220,158,273,438,302),(3,324,257,53,421,74,25,221,159,274,439,303),(4,325,258,54,422,75,26,222,160,275,440,304),(5,326,259,55,423,76,27,223,161,276,441,286),(6,327,260,56,424,58,28,224,162,277,442,287),(7,328,261,57,425,59,29,225,163,278,443,288),(8,329,262,39,426,60,30,226,164,279,444,289),(9,330,263,40,427,61,31,227,165,280,445,290),(10,331,264,41,428,62,32,228,166,281,446,291),(11,332,265,42,429,63,33,210,167,282,447,292),(12,333,266,43,430,64,34,211,168,283,448,293),(13,334,248,44,431,65,35,212,169,284,449,294),(14,335,249,45,432,66,36,213,170,285,450,295),(15,336,250,46,433,67,37,214,171,267,451,296),(16,337,251,47,434,68,38,215,153,268,452,297),(17,338,252,48,435,69,20,216,154,269,453,298),(18,339,253,49,436,70,21,217,155,270,454,299),(19,340,254,50,437,71,22,218,156,271,455,300),(77,186,137,360,240,376,411,388,319,209,104,119),(78,187,138,361,241,377,412,389,320,191,105,120),(79,188,139,343,242,378,413,390,321,192,106,121),(80,189,140,344,243,379,414,391,322,193,107,122),(81,190,141,345,244,380,415,392,323,194,108,123),(82,172,142,346,245,362,416,393,305,195,109,124),(83,173,143,347,246,363,417,394,306,196,110,125),(84,174,144,348,247,364,418,395,307,197,111,126),(85,175,145,349,229,365,400,396,308,198,112,127),(86,176,146,350,230,366,401,397,309,199,113,128),(87,177,147,351,231,367,402,398,310,200,114,129),(88,178,148,352,232,368,403,399,311,201,96,130),(89,179,149,353,233,369,404,381,312,202,97,131),(90,180,150,354,234,370,405,382,313,203,98,132),(91,181,151,355,235,371,406,383,314,204,99,133),(92,182,152,356,236,372,407,384,315,205,100,115),(93,183,134,357,237,373,408,385,316,206,101,116),(94,184,135,358,238,374,409,386,317,207,102,117),(95,185,136,359,239,375,410,387,318,208,103,118)], [(1,82,23,416),(2,83,24,417),(3,84,25,418),(4,85,26,400),(5,86,27,401),(6,87,28,402),(7,88,29,403),(8,89,30,404),(9,90,31,405),(10,91,32,406),(11,92,33,407),(12,93,34,408),(13,94,35,409),(14,95,36,410),(15,77,37,411),(16,78,38,412),(17,79,20,413),(18,80,21,414),(19,81,22,415),(39,202,279,353),(40,203,280,354),(41,204,281,355),(42,205,282,356),(43,206,283,357),(44,207,284,358),(45,208,285,359),(46,209,267,360),(47,191,268,361),(48,192,269,343),(49,193,270,344),(50,194,271,345),(51,195,272,346),(52,196,273,347),(53,197,274,348),(54,198,275,349),(55,199,276,350),(56,200,277,351),(57,201,278,352),(58,398,287,177),(59,399,288,178),(60,381,289,179),(61,382,290,180),(62,383,291,181),(63,384,292,182),(64,385,293,183),(65,386,294,184),(66,387,295,185),(67,388,296,186),(68,389,297,187),(69,390,298,188),(70,391,299,189),(71,392,300,190),(72,393,301,172),(73,394,302,173),(74,395,303,174),(75,396,304,175),(76,397,286,176),(96,163,232,261),(97,164,233,262),(98,165,234,263),(99,166,235,264),(100,167,236,265),(101,168,237,266),(102,169,238,248),(103,170,239,249),(104,171,240,250),(105,153,241,251),(106,154,242,252),(107,155,243,253),(108,156,244,254),(109,157,245,255),(110,158,246,256),(111,159,247,257),(112,160,229,258),(113,161,230,259),(114,162,231,260),(115,210,372,332),(116,211,373,333),(117,212,374,334),(118,213,375,335),(119,214,376,336),(120,215,377,337),(121,216,378,338),(122,217,379,339),(123,218,380,340),(124,219,362,341),(125,220,363,342),(126,221,364,324),(127,222,365,325),(128,223,366,326),(129,224,367,327),(130,225,368,328),(131,226,369,329),(132,227,370,330),(133,228,371,331),(134,430,316,448),(135,431,317,449),(136,432,318,450),(137,433,319,451),(138,434,320,452),(139,435,321,453),(140,436,322,454),(141,437,323,455),(142,419,305,456),(143,420,306,438),(144,421,307,439),(145,422,308,440),(146,423,309,441),(147,424,310,442),(148,425,311,443),(149,426,312,444),(150,427,313,445),(151,428,314,446),(152,429,315,447)]])
171 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 19A | ··· | 19R | 38A | ··· | 38R | 57A | ··· | 57R | 76A | ··· | 76R | 76S | ··· | 76BB | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
171 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C19 | C38 | C38 | S3 | Q8 | D6 | Dic6 | S3×C19 | Q8×C19 | S3×C38 | C19×Dic6 |
| kernel | C19×Dic6 | Dic3×C19 | C228 | Dic6 | Dic3 | C12 | C76 | C57 | C38 | C19 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 18 | 36 | 18 | 1 | 1 | 1 | 2 | 18 | 18 | 18 | 36 |
Matrix representation of C19×Dic6 ►in GL2(𝔽229) generated by
| 60 | 0 |
| 0 | 60 |
| 29 | 129 |
| 100 | 129 |
| 0 | 107 |
| 107 | 0 |
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