direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×Dic19, C38⋊3C12, C114⋊2C4, C6.16D38, C114.16C22, C57⋊8(C2×C4), C19⋊5(C2×C12), (C2×C38).3C6, (C2×C6).2D19, C2.2(C6×D19), C22.(C3×D19), (C2×C114).2C2, C38.12(C2×C6), SmallGroup(456,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C6×Dic19 |
Generators and relations for C6×Dic19
G = < a,b,c | a6=b38=1, c2=b19, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6×Dic19
►Regular action on 456 points
Generators in S456
(1 174 92 135 50 194)(2 175 93 136 51 195)(3 176 94 137 52 196)(4 177 95 138 53 197)(5 178 96 139 54 198)(6 179 97 140 55 199)(7 180 98 141 56 200)(8 181 99 142 57 201)(9 182 100 143 58 202)(10 183 101 144 59 203)(11 184 102 145 60 204)(12 185 103 146 61 205)(13 186 104 147 62 206)(14 187 105 148 63 207)(15 188 106 149 64 208)(16 189 107 150 65 209)(17 190 108 151 66 210)(18 153 109 152 67 211)(19 154 110 115 68 212)(20 155 111 116 69 213)(21 156 112 117 70 214)(22 157 113 118 71 215)(23 158 114 119 72 216)(24 159 77 120 73 217)(25 160 78 121 74 218)(26 161 79 122 75 219)(27 162 80 123 76 220)(28 163 81 124 39 221)(29 164 82 125 40 222)(30 165 83 126 41 223)(31 166 84 127 42 224)(32 167 85 128 43 225)(33 168 86 129 44 226)(34 169 87 130 45 227)(35 170 88 131 46 228)(36 171 89 132 47 191)(37 172 90 133 48 192)(38 173 91 134 49 193)(229 400 305 343 286 419)(230 401 306 344 287 420)(231 402 307 345 288 421)(232 403 308 346 289 422)(233 404 309 347 290 423)(234 405 310 348 291 424)(235 406 311 349 292 425)(236 407 312 350 293 426)(237 408 313 351 294 427)(238 409 314 352 295 428)(239 410 315 353 296 429)(240 411 316 354 297 430)(241 412 317 355 298 431)(242 413 318 356 299 432)(243 414 319 357 300 433)(244 415 320 358 301 434)(245 416 321 359 302 435)(246 417 322 360 303 436)(247 418 323 361 304 437)(248 381 324 362 267 438)(249 382 325 363 268 439)(250 383 326 364 269 440)(251 384 327 365 270 441)(252 385 328 366 271 442)(253 386 329 367 272 443)(254 387 330 368 273 444)(255 388 331 369 274 445)(256 389 332 370 275 446)(257 390 333 371 276 447)(258 391 334 372 277 448)(259 392 335 373 278 449)(260 393 336 374 279 450)(261 394 337 375 280 451)(262 395 338 376 281 452)(263 396 339 377 282 453)(264 397 340 378 283 454)(265 398 341 379 284 455)(266 399 342 380 285 456)
(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)
(1 248 20 229)(2 247 21 266)(3 246 22 265)(4 245 23 264)(5 244 24 263)(6 243 25 262)(7 242 26 261)(8 241 27 260)(9 240 28 259)(10 239 29 258)(11 238 30 257)(12 237 31 256)(13 236 32 255)(14 235 33 254)(15 234 34 253)(16 233 35 252)(17 232 36 251)(18 231 37 250)(19 230 38 249)(39 278 58 297)(40 277 59 296)(41 276 60 295)(42 275 61 294)(43 274 62 293)(44 273 63 292)(45 272 64 291)(46 271 65 290)(47 270 66 289)(48 269 67 288)(49 268 68 287)(50 267 69 286)(51 304 70 285)(52 303 71 284)(53 302 72 283)(54 301 73 282)(55 300 74 281)(56 299 75 280)(57 298 76 279)(77 339 96 320)(78 338 97 319)(79 337 98 318)(80 336 99 317)(81 335 100 316)(82 334 101 315)(83 333 102 314)(84 332 103 313)(85 331 104 312)(86 330 105 311)(87 329 106 310)(88 328 107 309)(89 327 108 308)(90 326 109 307)(91 325 110 306)(92 324 111 305)(93 323 112 342)(94 322 113 341)(95 321 114 340)(115 344 134 363)(116 343 135 362)(117 380 136 361)(118 379 137 360)(119 378 138 359)(120 377 139 358)(121 376 140 357)(122 375 141 356)(123 374 142 355)(124 373 143 354)(125 372 144 353)(126 371 145 352)(127 370 146 351)(128 369 147 350)(129 368 148 349)(130 367 149 348)(131 366 150 347)(132 365 151 346)(133 364 152 345)(153 402 172 383)(154 401 173 382)(155 400 174 381)(156 399 175 418)(157 398 176 417)(158 397 177 416)(159 396 178 415)(160 395 179 414)(161 394 180 413)(162 393 181 412)(163 392 182 411)(164 391 183 410)(165 390 184 409)(166 389 185 408)(167 388 186 407)(168 387 187 406)(169 386 188 405)(170 385 189 404)(171 384 190 403)(191 441 210 422)(192 440 211 421)(193 439 212 420)(194 438 213 419)(195 437 214 456)(196 436 215 455)(197 435 216 454)(198 434 217 453)(199 433 218 452)(200 432 219 451)(201 431 220 450)(202 430 221 449)(203 429 222 448)(204 428 223 447)(205 427 224 446)(206 426 225 445)(207 425 226 444)(208 424 227 443)(209 423 228 442)
G:=sub<Sym(456)| (1,174,92,135,50,194)(2,175,93,136,51,195)(3,176,94,137,52,196)(4,177,95,138,53,197)(5,178,96,139,54,198)(6,179,97,140,55,199)(7,180,98,141,56,200)(8,181,99,142,57,201)(9,182,100,143,58,202)(10,183,101,144,59,203)(11,184,102,145,60,204)(12,185,103,146,61,205)(13,186,104,147,62,206)(14,187,105,148,63,207)(15,188,106,149,64,208)(16,189,107,150,65,209)(17,190,108,151,66,210)(18,153,109,152,67,211)(19,154,110,115,68,212)(20,155,111,116,69,213)(21,156,112,117,70,214)(22,157,113,118,71,215)(23,158,114,119,72,216)(24,159,77,120,73,217)(25,160,78,121,74,218)(26,161,79,122,75,219)(27,162,80,123,76,220)(28,163,81,124,39,221)(29,164,82,125,40,222)(30,165,83,126,41,223)(31,166,84,127,42,224)(32,167,85,128,43,225)(33,168,86,129,44,226)(34,169,87,130,45,227)(35,170,88,131,46,228)(36,171,89,132,47,191)(37,172,90,133,48,192)(38,173,91,134,49,193)(229,400,305,343,286,419)(230,401,306,344,287,420)(231,402,307,345,288,421)(232,403,308,346,289,422)(233,404,309,347,290,423)(234,405,310,348,291,424)(235,406,311,349,292,425)(236,407,312,350,293,426)(237,408,313,351,294,427)(238,409,314,352,295,428)(239,410,315,353,296,429)(240,411,316,354,297,430)(241,412,317,355,298,431)(242,413,318,356,299,432)(243,414,319,357,300,433)(244,415,320,358,301,434)(245,416,321,359,302,435)(246,417,322,360,303,436)(247,418,323,361,304,437)(248,381,324,362,267,438)(249,382,325,363,268,439)(250,383,326,364,269,440)(251,384,327,365,270,441)(252,385,328,366,271,442)(253,386,329,367,272,443)(254,387,330,368,273,444)(255,388,331,369,274,445)(256,389,332,370,275,446)(257,390,333,371,276,447)(258,391,334,372,277,448)(259,392,335,373,278,449)(260,393,336,374,279,450)(261,394,337,375,280,451)(262,395,338,376,281,452)(263,396,339,377,282,453)(264,397,340,378,283,454)(265,398,341,379,284,455)(266,399,342,380,285,456), (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), (1,248,20,229)(2,247,21,266)(3,246,22,265)(4,245,23,264)(5,244,24,263)(6,243,25,262)(7,242,26,261)(8,241,27,260)(9,240,28,259)(10,239,29,258)(11,238,30,257)(12,237,31,256)(13,236,32,255)(14,235,33,254)(15,234,34,253)(16,233,35,252)(17,232,36,251)(18,231,37,250)(19,230,38,249)(39,278,58,297)(40,277,59,296)(41,276,60,295)(42,275,61,294)(43,274,62,293)(44,273,63,292)(45,272,64,291)(46,271,65,290)(47,270,66,289)(48,269,67,288)(49,268,68,287)(50,267,69,286)(51,304,70,285)(52,303,71,284)(53,302,72,283)(54,301,73,282)(55,300,74,281)(56,299,75,280)(57,298,76,279)(77,339,96,320)(78,338,97,319)(79,337,98,318)(80,336,99,317)(81,335,100,316)(82,334,101,315)(83,333,102,314)(84,332,103,313)(85,331,104,312)(86,330,105,311)(87,329,106,310)(88,328,107,309)(89,327,108,308)(90,326,109,307)(91,325,110,306)(92,324,111,305)(93,323,112,342)(94,322,113,341)(95,321,114,340)(115,344,134,363)(116,343,135,362)(117,380,136,361)(118,379,137,360)(119,378,138,359)(120,377,139,358)(121,376,140,357)(122,375,141,356)(123,374,142,355)(124,373,143,354)(125,372,144,353)(126,371,145,352)(127,370,146,351)(128,369,147,350)(129,368,148,349)(130,367,149,348)(131,366,150,347)(132,365,151,346)(133,364,152,345)(153,402,172,383)(154,401,173,382)(155,400,174,381)(156,399,175,418)(157,398,176,417)(158,397,177,416)(159,396,178,415)(160,395,179,414)(161,394,180,413)(162,393,181,412)(163,392,182,411)(164,391,183,410)(165,390,184,409)(166,389,185,408)(167,388,186,407)(168,387,187,406)(169,386,188,405)(170,385,189,404)(171,384,190,403)(191,441,210,422)(192,440,211,421)(193,439,212,420)(194,438,213,419)(195,437,214,456)(196,436,215,455)(197,435,216,454)(198,434,217,453)(199,433,218,452)(200,432,219,451)(201,431,220,450)(202,430,221,449)(203,429,222,448)(204,428,223,447)(205,427,224,446)(206,426,225,445)(207,425,226,444)(208,424,227,443)(209,423,228,442)>;
G:=Group( (1,174,92,135,50,194)(2,175,93,136,51,195)(3,176,94,137,52,196)(4,177,95,138,53,197)(5,178,96,139,54,198)(6,179,97,140,55,199)(7,180,98,141,56,200)(8,181,99,142,57,201)(9,182,100,143,58,202)(10,183,101,144,59,203)(11,184,102,145,60,204)(12,185,103,146,61,205)(13,186,104,147,62,206)(14,187,105,148,63,207)(15,188,106,149,64,208)(16,189,107,150,65,209)(17,190,108,151,66,210)(18,153,109,152,67,211)(19,154,110,115,68,212)(20,155,111,116,69,213)(21,156,112,117,70,214)(22,157,113,118,71,215)(23,158,114,119,72,216)(24,159,77,120,73,217)(25,160,78,121,74,218)(26,161,79,122,75,219)(27,162,80,123,76,220)(28,163,81,124,39,221)(29,164,82,125,40,222)(30,165,83,126,41,223)(31,166,84,127,42,224)(32,167,85,128,43,225)(33,168,86,129,44,226)(34,169,87,130,45,227)(35,170,88,131,46,228)(36,171,89,132,47,191)(37,172,90,133,48,192)(38,173,91,134,49,193)(229,400,305,343,286,419)(230,401,306,344,287,420)(231,402,307,345,288,421)(232,403,308,346,289,422)(233,404,309,347,290,423)(234,405,310,348,291,424)(235,406,311,349,292,425)(236,407,312,350,293,426)(237,408,313,351,294,427)(238,409,314,352,295,428)(239,410,315,353,296,429)(240,411,316,354,297,430)(241,412,317,355,298,431)(242,413,318,356,299,432)(243,414,319,357,300,433)(244,415,320,358,301,434)(245,416,321,359,302,435)(246,417,322,360,303,436)(247,418,323,361,304,437)(248,381,324,362,267,438)(249,382,325,363,268,439)(250,383,326,364,269,440)(251,384,327,365,270,441)(252,385,328,366,271,442)(253,386,329,367,272,443)(254,387,330,368,273,444)(255,388,331,369,274,445)(256,389,332,370,275,446)(257,390,333,371,276,447)(258,391,334,372,277,448)(259,392,335,373,278,449)(260,393,336,374,279,450)(261,394,337,375,280,451)(262,395,338,376,281,452)(263,396,339,377,282,453)(264,397,340,378,283,454)(265,398,341,379,284,455)(266,399,342,380,285,456), (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), (1,248,20,229)(2,247,21,266)(3,246,22,265)(4,245,23,264)(5,244,24,263)(6,243,25,262)(7,242,26,261)(8,241,27,260)(9,240,28,259)(10,239,29,258)(11,238,30,257)(12,237,31,256)(13,236,32,255)(14,235,33,254)(15,234,34,253)(16,233,35,252)(17,232,36,251)(18,231,37,250)(19,230,38,249)(39,278,58,297)(40,277,59,296)(41,276,60,295)(42,275,61,294)(43,274,62,293)(44,273,63,292)(45,272,64,291)(46,271,65,290)(47,270,66,289)(48,269,67,288)(49,268,68,287)(50,267,69,286)(51,304,70,285)(52,303,71,284)(53,302,72,283)(54,301,73,282)(55,300,74,281)(56,299,75,280)(57,298,76,279)(77,339,96,320)(78,338,97,319)(79,337,98,318)(80,336,99,317)(81,335,100,316)(82,334,101,315)(83,333,102,314)(84,332,103,313)(85,331,104,312)(86,330,105,311)(87,329,106,310)(88,328,107,309)(89,327,108,308)(90,326,109,307)(91,325,110,306)(92,324,111,305)(93,323,112,342)(94,322,113,341)(95,321,114,340)(115,344,134,363)(116,343,135,362)(117,380,136,361)(118,379,137,360)(119,378,138,359)(120,377,139,358)(121,376,140,357)(122,375,141,356)(123,374,142,355)(124,373,143,354)(125,372,144,353)(126,371,145,352)(127,370,146,351)(128,369,147,350)(129,368,148,349)(130,367,149,348)(131,366,150,347)(132,365,151,346)(133,364,152,345)(153,402,172,383)(154,401,173,382)(155,400,174,381)(156,399,175,418)(157,398,176,417)(158,397,177,416)(159,396,178,415)(160,395,179,414)(161,394,180,413)(162,393,181,412)(163,392,182,411)(164,391,183,410)(165,390,184,409)(166,389,185,408)(167,388,186,407)(168,387,187,406)(169,386,188,405)(170,385,189,404)(171,384,190,403)(191,441,210,422)(192,440,211,421)(193,439,212,420)(194,438,213,419)(195,437,214,456)(196,436,215,455)(197,435,216,454)(198,434,217,453)(199,433,218,452)(200,432,219,451)(201,431,220,450)(202,430,221,449)(203,429,222,448)(204,428,223,447)(205,427,224,446)(206,426,225,445)(207,425,226,444)(208,424,227,443)(209,423,228,442) );
G=PermutationGroup([[(1,174,92,135,50,194),(2,175,93,136,51,195),(3,176,94,137,52,196),(4,177,95,138,53,197),(5,178,96,139,54,198),(6,179,97,140,55,199),(7,180,98,141,56,200),(8,181,99,142,57,201),(9,182,100,143,58,202),(10,183,101,144,59,203),(11,184,102,145,60,204),(12,185,103,146,61,205),(13,186,104,147,62,206),(14,187,105,148,63,207),(15,188,106,149,64,208),(16,189,107,150,65,209),(17,190,108,151,66,210),(18,153,109,152,67,211),(19,154,110,115,68,212),(20,155,111,116,69,213),(21,156,112,117,70,214),(22,157,113,118,71,215),(23,158,114,119,72,216),(24,159,77,120,73,217),(25,160,78,121,74,218),(26,161,79,122,75,219),(27,162,80,123,76,220),(28,163,81,124,39,221),(29,164,82,125,40,222),(30,165,83,126,41,223),(31,166,84,127,42,224),(32,167,85,128,43,225),(33,168,86,129,44,226),(34,169,87,130,45,227),(35,170,88,131,46,228),(36,171,89,132,47,191),(37,172,90,133,48,192),(38,173,91,134,49,193),(229,400,305,343,286,419),(230,401,306,344,287,420),(231,402,307,345,288,421),(232,403,308,346,289,422),(233,404,309,347,290,423),(234,405,310,348,291,424),(235,406,311,349,292,425),(236,407,312,350,293,426),(237,408,313,351,294,427),(238,409,314,352,295,428),(239,410,315,353,296,429),(240,411,316,354,297,430),(241,412,317,355,298,431),(242,413,318,356,299,432),(243,414,319,357,300,433),(244,415,320,358,301,434),(245,416,321,359,302,435),(246,417,322,360,303,436),(247,418,323,361,304,437),(248,381,324,362,267,438),(249,382,325,363,268,439),(250,383,326,364,269,440),(251,384,327,365,270,441),(252,385,328,366,271,442),(253,386,329,367,272,443),(254,387,330,368,273,444),(255,388,331,369,274,445),(256,389,332,370,275,446),(257,390,333,371,276,447),(258,391,334,372,277,448),(259,392,335,373,278,449),(260,393,336,374,279,450),(261,394,337,375,280,451),(262,395,338,376,281,452),(263,396,339,377,282,453),(264,397,340,378,283,454),(265,398,341,379,284,455),(266,399,342,380,285,456)], [(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)], [(1,248,20,229),(2,247,21,266),(3,246,22,265),(4,245,23,264),(5,244,24,263),(6,243,25,262),(7,242,26,261),(8,241,27,260),(9,240,28,259),(10,239,29,258),(11,238,30,257),(12,237,31,256),(13,236,32,255),(14,235,33,254),(15,234,34,253),(16,233,35,252),(17,232,36,251),(18,231,37,250),(19,230,38,249),(39,278,58,297),(40,277,59,296),(41,276,60,295),(42,275,61,294),(43,274,62,293),(44,273,63,292),(45,272,64,291),(46,271,65,290),(47,270,66,289),(48,269,67,288),(49,268,68,287),(50,267,69,286),(51,304,70,285),(52,303,71,284),(53,302,72,283),(54,301,73,282),(55,300,74,281),(56,299,75,280),(57,298,76,279),(77,339,96,320),(78,338,97,319),(79,337,98,318),(80,336,99,317),(81,335,100,316),(82,334,101,315),(83,333,102,314),(84,332,103,313),(85,331,104,312),(86,330,105,311),(87,329,106,310),(88,328,107,309),(89,327,108,308),(90,326,109,307),(91,325,110,306),(92,324,111,305),(93,323,112,342),(94,322,113,341),(95,321,114,340),(115,344,134,363),(116,343,135,362),(117,380,136,361),(118,379,137,360),(119,378,138,359),(120,377,139,358),(121,376,140,357),(122,375,141,356),(123,374,142,355),(124,373,143,354),(125,372,144,353),(126,371,145,352),(127,370,146,351),(128,369,147,350),(129,368,148,349),(130,367,149,348),(131,366,150,347),(132,365,151,346),(133,364,152,345),(153,402,172,383),(154,401,173,382),(155,400,174,381),(156,399,175,418),(157,398,176,417),(158,397,177,416),(159,396,178,415),(160,395,179,414),(161,394,180,413),(162,393,181,412),(163,392,182,411),(164,391,183,410),(165,390,184,409),(166,389,185,408),(167,388,186,407),(168,387,187,406),(169,386,188,405),(170,385,189,404),(171,384,190,403),(191,441,210,422),(192,440,211,421),(193,439,212,420),(194,438,213,419),(195,437,214,456),(196,436,215,455),(197,435,216,454),(198,434,217,453),(199,433,218,452),(200,432,219,451),(201,431,220,450),(202,430,221,449),(203,429,222,448),(204,428,223,447),(205,427,224,446),(206,426,225,445),(207,425,226,444),(208,424,227,443),(209,423,228,442)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 19A | ··· | 19I | 38A | ··· | 38AA | 57A | ··· | 57R | 114A | ··· | 114BB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 1 | ··· | 1 | 19 | ··· | 19 | 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 | C2 | C3 | C4 | C6 | C6 | C12 | D19 | Dic19 | D38 | C3×D19 | C3×Dic19 | C6×D19 |
| kernel | C6×Dic19 | C3×Dic19 | C2×C114 | C2×Dic19 | C114 | Dic19 | C2×C38 | C38 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 9 | 18 | 9 | 18 | 36 | 18 |
Matrix representation of C6×Dic19 ►in GL3(𝔽229) generated by
| 95 | 0 | 0 |
| 0 | 135 | 0 |
| 0 | 0 | 135 |
| 228 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 228 | 18 |
| 122 | 0 | 0 |
| 0 | 168 | 41 |
| 0 | 88 | 61 |
G:=sub<GL(3,GF(229))| [95,0,0,0,135,0,0,0,135],[228,0,0,0,0,228,0,1,18],[122,0,0,0,168,88,0,41,61] >;
C6×Dic19 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{19} % in TeX
G:=Group("C6xDic19"); // GroupNames label
G:=SmallGroup(456,27);
// by ID
G=gap.SmallGroup(456,27);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,60,10804]);
// Polycyclic
G:=Group<a,b,c|a^6=b^38=1,c^2=b^19,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export