direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic38, C57⋊3Q8, C76.5C6, C228.3C2, C6.13D38, C12.3D19, Dic19.2C6, C114.13C22, C4.(C3×D19), C19⋊3(C3×Q8), C38.9(C2×C6), C2.3(C6×D19), (C3×Dic19).2C2, SmallGroup(456,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic38
G = < a,b,c | a3=b76=1, c2=b38, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic38
►Regular action on 456 points
Generators in S456
(1 143 161)(2 144 162)(3 145 163)(4 146 164)(5 147 165)(6 148 166)(7 149 167)(8 150 168)(9 151 169)(10 152 170)(11 77 171)(12 78 172)(13 79 173)(14 80 174)(15 81 175)(16 82 176)(17 83 177)(18 84 178)(19 85 179)(20 86 180)(21 87 181)(22 88 182)(23 89 183)(24 90 184)(25 91 185)(26 92 186)(27 93 187)(28 94 188)(29 95 189)(30 96 190)(31 97 191)(32 98 192)(33 99 193)(34 100 194)(35 101 195)(36 102 196)(37 103 197)(38 104 198)(39 105 199)(40 106 200)(41 107 201)(42 108 202)(43 109 203)(44 110 204)(45 111 205)(46 112 206)(47 113 207)(48 114 208)(49 115 209)(50 116 210)(51 117 211)(52 118 212)(53 119 213)(54 120 214)(55 121 215)(56 122 216)(57 123 217)(58 124 218)(59 125 219)(60 126 220)(61 127 221)(62 128 222)(63 129 223)(64 130 224)(65 131 225)(66 132 226)(67 133 227)(68 134 228)(69 135 153)(70 136 154)(71 137 155)(72 138 156)(73 139 157)(74 140 158)(75 141 159)(76 142 160)(229 431 357)(230 432 358)(231 433 359)(232 434 360)(233 435 361)(234 436 362)(235 437 363)(236 438 364)(237 439 365)(238 440 366)(239 441 367)(240 442 368)(241 443 369)(242 444 370)(243 445 371)(244 446 372)(245 447 373)(246 448 374)(247 449 375)(248 450 376)(249 451 377)(250 452 378)(251 453 379)(252 454 380)(253 455 305)(254 456 306)(255 381 307)(256 382 308)(257 383 309)(258 384 310)(259 385 311)(260 386 312)(261 387 313)(262 388 314)(263 389 315)(264 390 316)(265 391 317)(266 392 318)(267 393 319)(268 394 320)(269 395 321)(270 396 322)(271 397 323)(272 398 324)(273 399 325)(274 400 326)(275 401 327)(276 402 328)(277 403 329)(278 404 330)(279 405 331)(280 406 332)(281 407 333)(282 408 334)(283 409 335)(284 410 336)(285 411 337)(286 412 338)(287 413 339)(288 414 340)(289 415 341)(290 416 342)(291 417 343)(292 418 344)(293 419 345)(294 420 346)(295 421 347)(296 422 348)(297 423 349)(298 424 350)(299 425 351)(300 426 352)(301 427 353)(302 428 354)(303 429 355)(304 430 356)
(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)
(1 398 39 436)(2 397 40 435)(3 396 41 434)(4 395 42 433)(5 394 43 432)(6 393 44 431)(7 392 45 430)(8 391 46 429)(9 390 47 428)(10 389 48 427)(11 388 49 426)(12 387 50 425)(13 386 51 424)(14 385 52 423)(15 384 53 422)(16 383 54 421)(17 382 55 420)(18 381 56 419)(19 456 57 418)(20 455 58 417)(21 454 59 416)(22 453 60 415)(23 452 61 414)(24 451 62 413)(25 450 63 412)(26 449 64 411)(27 448 65 410)(28 447 66 409)(29 446 67 408)(30 445 68 407)(31 444 69 406)(32 443 70 405)(33 442 71 404)(34 441 72 403)(35 440 73 402)(36 439 74 401)(37 438 75 400)(38 437 76 399)(77 314 115 352)(78 313 116 351)(79 312 117 350)(80 311 118 349)(81 310 119 348)(82 309 120 347)(83 308 121 346)(84 307 122 345)(85 306 123 344)(86 305 124 343)(87 380 125 342)(88 379 126 341)(89 378 127 340)(90 377 128 339)(91 376 129 338)(92 375 130 337)(93 374 131 336)(94 373 132 335)(95 372 133 334)(96 371 134 333)(97 370 135 332)(98 369 136 331)(99 368 137 330)(100 367 138 329)(101 366 139 328)(102 365 140 327)(103 364 141 326)(104 363 142 325)(105 362 143 324)(106 361 144 323)(107 360 145 322)(108 359 146 321)(109 358 147 320)(110 357 148 319)(111 356 149 318)(112 355 150 317)(113 354 151 316)(114 353 152 315)(153 280 191 242)(154 279 192 241)(155 278 193 240)(156 277 194 239)(157 276 195 238)(158 275 196 237)(159 274 197 236)(160 273 198 235)(161 272 199 234)(162 271 200 233)(163 270 201 232)(164 269 202 231)(165 268 203 230)(166 267 204 229)(167 266 205 304)(168 265 206 303)(169 264 207 302)(170 263 208 301)(171 262 209 300)(172 261 210 299)(173 260 211 298)(174 259 212 297)(175 258 213 296)(176 257 214 295)(177 256 215 294)(178 255 216 293)(179 254 217 292)(180 253 218 291)(181 252 219 290)(182 251 220 289)(183 250 221 288)(184 249 222 287)(185 248 223 286)(186 247 224 285)(187 246 225 284)(188 245 226 283)(189 244 227 282)(190 243 228 281)
G:=sub<Sym(456)| (1,143,161)(2,144,162)(3,145,163)(4,146,164)(5,147,165)(6,148,166)(7,149,167)(8,150,168)(9,151,169)(10,152,170)(11,77,171)(12,78,172)(13,79,173)(14,80,174)(15,81,175)(16,82,176)(17,83,177)(18,84,178)(19,85,179)(20,86,180)(21,87,181)(22,88,182)(23,89,183)(24,90,184)(25,91,185)(26,92,186)(27,93,187)(28,94,188)(29,95,189)(30,96,190)(31,97,191)(32,98,192)(33,99,193)(34,100,194)(35,101,195)(36,102,196)(37,103,197)(38,104,198)(39,105,199)(40,106,200)(41,107,201)(42,108,202)(43,109,203)(44,110,204)(45,111,205)(46,112,206)(47,113,207)(48,114,208)(49,115,209)(50,116,210)(51,117,211)(52,118,212)(53,119,213)(54,120,214)(55,121,215)(56,122,216)(57,123,217)(58,124,218)(59,125,219)(60,126,220)(61,127,221)(62,128,222)(63,129,223)(64,130,224)(65,131,225)(66,132,226)(67,133,227)(68,134,228)(69,135,153)(70,136,154)(71,137,155)(72,138,156)(73,139,157)(74,140,158)(75,141,159)(76,142,160)(229,431,357)(230,432,358)(231,433,359)(232,434,360)(233,435,361)(234,436,362)(235,437,363)(236,438,364)(237,439,365)(238,440,366)(239,441,367)(240,442,368)(241,443,369)(242,444,370)(243,445,371)(244,446,372)(245,447,373)(246,448,374)(247,449,375)(248,450,376)(249,451,377)(250,452,378)(251,453,379)(252,454,380)(253,455,305)(254,456,306)(255,381,307)(256,382,308)(257,383,309)(258,384,310)(259,385,311)(260,386,312)(261,387,313)(262,388,314)(263,389,315)(264,390,316)(265,391,317)(266,392,318)(267,393,319)(268,394,320)(269,395,321)(270,396,322)(271,397,323)(272,398,324)(273,399,325)(274,400,326)(275,401,327)(276,402,328)(277,403,329)(278,404,330)(279,405,331)(280,406,332)(281,407,333)(282,408,334)(283,409,335)(284,410,336)(285,411,337)(286,412,338)(287,413,339)(288,414,340)(289,415,341)(290,416,342)(291,417,343)(292,418,344)(293,419,345)(294,420,346)(295,421,347)(296,422,348)(297,423,349)(298,424,350)(299,425,351)(300,426,352)(301,427,353)(302,428,354)(303,429,355)(304,430,356), (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), (1,398,39,436)(2,397,40,435)(3,396,41,434)(4,395,42,433)(5,394,43,432)(6,393,44,431)(7,392,45,430)(8,391,46,429)(9,390,47,428)(10,389,48,427)(11,388,49,426)(12,387,50,425)(13,386,51,424)(14,385,52,423)(15,384,53,422)(16,383,54,421)(17,382,55,420)(18,381,56,419)(19,456,57,418)(20,455,58,417)(21,454,59,416)(22,453,60,415)(23,452,61,414)(24,451,62,413)(25,450,63,412)(26,449,64,411)(27,448,65,410)(28,447,66,409)(29,446,67,408)(30,445,68,407)(31,444,69,406)(32,443,70,405)(33,442,71,404)(34,441,72,403)(35,440,73,402)(36,439,74,401)(37,438,75,400)(38,437,76,399)(77,314,115,352)(78,313,116,351)(79,312,117,350)(80,311,118,349)(81,310,119,348)(82,309,120,347)(83,308,121,346)(84,307,122,345)(85,306,123,344)(86,305,124,343)(87,380,125,342)(88,379,126,341)(89,378,127,340)(90,377,128,339)(91,376,129,338)(92,375,130,337)(93,374,131,336)(94,373,132,335)(95,372,133,334)(96,371,134,333)(97,370,135,332)(98,369,136,331)(99,368,137,330)(100,367,138,329)(101,366,139,328)(102,365,140,327)(103,364,141,326)(104,363,142,325)(105,362,143,324)(106,361,144,323)(107,360,145,322)(108,359,146,321)(109,358,147,320)(110,357,148,319)(111,356,149,318)(112,355,150,317)(113,354,151,316)(114,353,152,315)(153,280,191,242)(154,279,192,241)(155,278,193,240)(156,277,194,239)(157,276,195,238)(158,275,196,237)(159,274,197,236)(160,273,198,235)(161,272,199,234)(162,271,200,233)(163,270,201,232)(164,269,202,231)(165,268,203,230)(166,267,204,229)(167,266,205,304)(168,265,206,303)(169,264,207,302)(170,263,208,301)(171,262,209,300)(172,261,210,299)(173,260,211,298)(174,259,212,297)(175,258,213,296)(176,257,214,295)(177,256,215,294)(178,255,216,293)(179,254,217,292)(180,253,218,291)(181,252,219,290)(182,251,220,289)(183,250,221,288)(184,249,222,287)(185,248,223,286)(186,247,224,285)(187,246,225,284)(188,245,226,283)(189,244,227,282)(190,243,228,281)>;
G:=Group( (1,143,161)(2,144,162)(3,145,163)(4,146,164)(5,147,165)(6,148,166)(7,149,167)(8,150,168)(9,151,169)(10,152,170)(11,77,171)(12,78,172)(13,79,173)(14,80,174)(15,81,175)(16,82,176)(17,83,177)(18,84,178)(19,85,179)(20,86,180)(21,87,181)(22,88,182)(23,89,183)(24,90,184)(25,91,185)(26,92,186)(27,93,187)(28,94,188)(29,95,189)(30,96,190)(31,97,191)(32,98,192)(33,99,193)(34,100,194)(35,101,195)(36,102,196)(37,103,197)(38,104,198)(39,105,199)(40,106,200)(41,107,201)(42,108,202)(43,109,203)(44,110,204)(45,111,205)(46,112,206)(47,113,207)(48,114,208)(49,115,209)(50,116,210)(51,117,211)(52,118,212)(53,119,213)(54,120,214)(55,121,215)(56,122,216)(57,123,217)(58,124,218)(59,125,219)(60,126,220)(61,127,221)(62,128,222)(63,129,223)(64,130,224)(65,131,225)(66,132,226)(67,133,227)(68,134,228)(69,135,153)(70,136,154)(71,137,155)(72,138,156)(73,139,157)(74,140,158)(75,141,159)(76,142,160)(229,431,357)(230,432,358)(231,433,359)(232,434,360)(233,435,361)(234,436,362)(235,437,363)(236,438,364)(237,439,365)(238,440,366)(239,441,367)(240,442,368)(241,443,369)(242,444,370)(243,445,371)(244,446,372)(245,447,373)(246,448,374)(247,449,375)(248,450,376)(249,451,377)(250,452,378)(251,453,379)(252,454,380)(253,455,305)(254,456,306)(255,381,307)(256,382,308)(257,383,309)(258,384,310)(259,385,311)(260,386,312)(261,387,313)(262,388,314)(263,389,315)(264,390,316)(265,391,317)(266,392,318)(267,393,319)(268,394,320)(269,395,321)(270,396,322)(271,397,323)(272,398,324)(273,399,325)(274,400,326)(275,401,327)(276,402,328)(277,403,329)(278,404,330)(279,405,331)(280,406,332)(281,407,333)(282,408,334)(283,409,335)(284,410,336)(285,411,337)(286,412,338)(287,413,339)(288,414,340)(289,415,341)(290,416,342)(291,417,343)(292,418,344)(293,419,345)(294,420,346)(295,421,347)(296,422,348)(297,423,349)(298,424,350)(299,425,351)(300,426,352)(301,427,353)(302,428,354)(303,429,355)(304,430,356), (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), (1,398,39,436)(2,397,40,435)(3,396,41,434)(4,395,42,433)(5,394,43,432)(6,393,44,431)(7,392,45,430)(8,391,46,429)(9,390,47,428)(10,389,48,427)(11,388,49,426)(12,387,50,425)(13,386,51,424)(14,385,52,423)(15,384,53,422)(16,383,54,421)(17,382,55,420)(18,381,56,419)(19,456,57,418)(20,455,58,417)(21,454,59,416)(22,453,60,415)(23,452,61,414)(24,451,62,413)(25,450,63,412)(26,449,64,411)(27,448,65,410)(28,447,66,409)(29,446,67,408)(30,445,68,407)(31,444,69,406)(32,443,70,405)(33,442,71,404)(34,441,72,403)(35,440,73,402)(36,439,74,401)(37,438,75,400)(38,437,76,399)(77,314,115,352)(78,313,116,351)(79,312,117,350)(80,311,118,349)(81,310,119,348)(82,309,120,347)(83,308,121,346)(84,307,122,345)(85,306,123,344)(86,305,124,343)(87,380,125,342)(88,379,126,341)(89,378,127,340)(90,377,128,339)(91,376,129,338)(92,375,130,337)(93,374,131,336)(94,373,132,335)(95,372,133,334)(96,371,134,333)(97,370,135,332)(98,369,136,331)(99,368,137,330)(100,367,138,329)(101,366,139,328)(102,365,140,327)(103,364,141,326)(104,363,142,325)(105,362,143,324)(106,361,144,323)(107,360,145,322)(108,359,146,321)(109,358,147,320)(110,357,148,319)(111,356,149,318)(112,355,150,317)(113,354,151,316)(114,353,152,315)(153,280,191,242)(154,279,192,241)(155,278,193,240)(156,277,194,239)(157,276,195,238)(158,275,196,237)(159,274,197,236)(160,273,198,235)(161,272,199,234)(162,271,200,233)(163,270,201,232)(164,269,202,231)(165,268,203,230)(166,267,204,229)(167,266,205,304)(168,265,206,303)(169,264,207,302)(170,263,208,301)(171,262,209,300)(172,261,210,299)(173,260,211,298)(174,259,212,297)(175,258,213,296)(176,257,214,295)(177,256,215,294)(178,255,216,293)(179,254,217,292)(180,253,218,291)(181,252,219,290)(182,251,220,289)(183,250,221,288)(184,249,222,287)(185,248,223,286)(186,247,224,285)(187,246,225,284)(188,245,226,283)(189,244,227,282)(190,243,228,281) );
G=PermutationGroup([[(1,143,161),(2,144,162),(3,145,163),(4,146,164),(5,147,165),(6,148,166),(7,149,167),(8,150,168),(9,151,169),(10,152,170),(11,77,171),(12,78,172),(13,79,173),(14,80,174),(15,81,175),(16,82,176),(17,83,177),(18,84,178),(19,85,179),(20,86,180),(21,87,181),(22,88,182),(23,89,183),(24,90,184),(25,91,185),(26,92,186),(27,93,187),(28,94,188),(29,95,189),(30,96,190),(31,97,191),(32,98,192),(33,99,193),(34,100,194),(35,101,195),(36,102,196),(37,103,197),(38,104,198),(39,105,199),(40,106,200),(41,107,201),(42,108,202),(43,109,203),(44,110,204),(45,111,205),(46,112,206),(47,113,207),(48,114,208),(49,115,209),(50,116,210),(51,117,211),(52,118,212),(53,119,213),(54,120,214),(55,121,215),(56,122,216),(57,123,217),(58,124,218),(59,125,219),(60,126,220),(61,127,221),(62,128,222),(63,129,223),(64,130,224),(65,131,225),(66,132,226),(67,133,227),(68,134,228),(69,135,153),(70,136,154),(71,137,155),(72,138,156),(73,139,157),(74,140,158),(75,141,159),(76,142,160),(229,431,357),(230,432,358),(231,433,359),(232,434,360),(233,435,361),(234,436,362),(235,437,363),(236,438,364),(237,439,365),(238,440,366),(239,441,367),(240,442,368),(241,443,369),(242,444,370),(243,445,371),(244,446,372),(245,447,373),(246,448,374),(247,449,375),(248,450,376),(249,451,377),(250,452,378),(251,453,379),(252,454,380),(253,455,305),(254,456,306),(255,381,307),(256,382,308),(257,383,309),(258,384,310),(259,385,311),(260,386,312),(261,387,313),(262,388,314),(263,389,315),(264,390,316),(265,391,317),(266,392,318),(267,393,319),(268,394,320),(269,395,321),(270,396,322),(271,397,323),(272,398,324),(273,399,325),(274,400,326),(275,401,327),(276,402,328),(277,403,329),(278,404,330),(279,405,331),(280,406,332),(281,407,333),(282,408,334),(283,409,335),(284,410,336),(285,411,337),(286,412,338),(287,413,339),(288,414,340),(289,415,341),(290,416,342),(291,417,343),(292,418,344),(293,419,345),(294,420,346),(295,421,347),(296,422,348),(297,423,349),(298,424,350),(299,425,351),(300,426,352),(301,427,353),(302,428,354),(303,429,355),(304,430,356)], [(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)], [(1,398,39,436),(2,397,40,435),(3,396,41,434),(4,395,42,433),(5,394,43,432),(6,393,44,431),(7,392,45,430),(8,391,46,429),(9,390,47,428),(10,389,48,427),(11,388,49,426),(12,387,50,425),(13,386,51,424),(14,385,52,423),(15,384,53,422),(16,383,54,421),(17,382,55,420),(18,381,56,419),(19,456,57,418),(20,455,58,417),(21,454,59,416),(22,453,60,415),(23,452,61,414),(24,451,62,413),(25,450,63,412),(26,449,64,411),(27,448,65,410),(28,447,66,409),(29,446,67,408),(30,445,68,407),(31,444,69,406),(32,443,70,405),(33,442,71,404),(34,441,72,403),(35,440,73,402),(36,439,74,401),(37,438,75,400),(38,437,76,399),(77,314,115,352),(78,313,116,351),(79,312,117,350),(80,311,118,349),(81,310,119,348),(82,309,120,347),(83,308,121,346),(84,307,122,345),(85,306,123,344),(86,305,124,343),(87,380,125,342),(88,379,126,341),(89,378,127,340),(90,377,128,339),(91,376,129,338),(92,375,130,337),(93,374,131,336),(94,373,132,335),(95,372,133,334),(96,371,134,333),(97,370,135,332),(98,369,136,331),(99,368,137,330),(100,367,138,329),(101,366,139,328),(102,365,140,327),(103,364,141,326),(104,363,142,325),(105,362,143,324),(106,361,144,323),(107,360,145,322),(108,359,146,321),(109,358,147,320),(110,357,148,319),(111,356,149,318),(112,355,150,317),(113,354,151,316),(114,353,152,315),(153,280,191,242),(154,279,192,241),(155,278,193,240),(156,277,194,239),(157,276,195,238),(158,275,196,237),(159,274,197,236),(160,273,198,235),(161,272,199,234),(162,271,200,233),(163,270,201,232),(164,269,202,231),(165,268,203,230),(166,267,204,229),(167,266,205,304),(168,265,206,303),(169,264,207,302),(170,263,208,301),(171,262,209,300),(172,261,210,299),(173,260,211,298),(174,259,212,297),(175,258,213,296),(176,257,214,295),(177,256,215,294),(178,255,216,293),(179,254,217,292),(180,253,218,291),(181,252,219,290),(182,251,220,289),(183,250,221,288),(184,249,222,287),(185,248,223,286),(186,247,224,285),(187,246,225,284),(188,245,226,283),(189,244,227,282),(190,243,228,281)]])
123 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 1 | 1 | 2 | 38 | 38 | 1 | 1 | 2 | 2 | 38 | 38 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | C3×Q8 | D19 | D38 | C3×D19 | Dic38 | C6×D19 | C3×Dic38 |
| kernel | C3×Dic38 | C3×Dic19 | C228 | Dic38 | Dic19 | C76 | C57 | C19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C3×Dic38 ►in GL3(𝔽229) generated by
| 134 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 228 | 0 | 0 |
| 0 | 184 | 119 |
| 0 | 110 | 162 |
| 228 | 0 | 0 |
| 0 | 209 | 55 |
| 0 | 51 | 20 |
G:=sub<GL(3,GF(229))| [134,0,0,0,1,0,0,0,1],[228,0,0,0,184,110,0,119,162],[228,0,0,0,209,51,0,55,20] >;
C3×Dic38 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{38} % in TeX
G:=Group("C3xDic38"); // GroupNames label
G:=SmallGroup(456,24);
// by ID
G=gap.SmallGroup(456,24);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,60,141,66,10804]);
// Polycyclic
G:=Group<a,b,c|a^3=b^76=1,c^2=b^38,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export