direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×C38, C6⋊C76, C114⋊3C4, C38.16D6, C114.21C22, C57⋊9(C2×C4), C3⋊2(C2×C76), (C2×C6).C38, C2.2(S3×C38), C6.4(C2×C38), (C2×C38).2S3, C22.(S3×C19), (C2×C114).3C2, SmallGroup(456,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C38 |
Generators and relations for Dic3×C38
G = < a,b,c | a38=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C38
►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)
(1 285 245 143 307 381)(2 286 246 144 308 382)(3 287 247 145 309 383)(4 288 248 146 310 384)(5 289 249 147 311 385)(6 290 250 148 312 386)(7 291 251 149 313 387)(8 292 252 150 314 388)(9 293 253 151 315 389)(10 294 254 152 316 390)(11 295 255 115 317 391)(12 296 256 116 318 392)(13 297 257 117 319 393)(14 298 258 118 320 394)(15 299 259 119 321 395)(16 300 260 120 322 396)(17 301 261 121 323 397)(18 302 262 122 324 398)(19 303 263 123 325 399)(20 304 264 124 326 400)(21 267 265 125 327 401)(22 268 266 126 328 402)(23 269 229 127 329 403)(24 270 230 128 330 404)(25 271 231 129 331 405)(26 272 232 130 332 406)(27 273 233 131 333 407)(28 274 234 132 334 408)(29 275 235 133 335 409)(30 276 236 134 336 410)(31 277 237 135 337 411)(32 278 238 136 338 412)(33 279 239 137 339 413)(34 280 240 138 340 414)(35 281 241 139 341 415)(36 282 242 140 342 416)(37 283 243 141 305 417)(38 284 244 142 306 418)(39 187 94 221 358 436)(40 188 95 222 359 437)(41 189 96 223 360 438)(42 190 97 224 361 439)(43 153 98 225 362 440)(44 154 99 226 363 441)(45 155 100 227 364 442)(46 156 101 228 365 443)(47 157 102 191 366 444)(48 158 103 192 367 445)(49 159 104 193 368 446)(50 160 105 194 369 447)(51 161 106 195 370 448)(52 162 107 196 371 449)(53 163 108 197 372 450)(54 164 109 198 373 451)(55 165 110 199 374 452)(56 166 111 200 375 453)(57 167 112 201 376 454)(58 168 113 202 377 455)(59 169 114 203 378 456)(60 170 77 204 379 419)(61 171 78 205 380 420)(62 172 79 206 343 421)(63 173 80 207 344 422)(64 174 81 208 345 423)(65 175 82 209 346 424)(66 176 83 210 347 425)(67 177 84 211 348 426)(68 178 85 212 349 427)(69 179 86 213 350 428)(70 180 87 214 351 429)(71 181 88 215 352 430)(72 182 89 216 353 431)(73 183 90 217 354 432)(74 184 91 218 355 433)(75 185 92 219 356 434)(76 186 93 220 357 435)
(1 106 143 448)(2 107 144 449)(3 108 145 450)(4 109 146 451)(5 110 147 452)(6 111 148 453)(7 112 149 454)(8 113 150 455)(9 114 151 456)(10 77 152 419)(11 78 115 420)(12 79 116 421)(13 80 117 422)(14 81 118 423)(15 82 119 424)(16 83 120 425)(17 84 121 426)(18 85 122 427)(19 86 123 428)(20 87 124 429)(21 88 125 430)(22 89 126 431)(23 90 127 432)(24 91 128 433)(25 92 129 434)(26 93 130 435)(27 94 131 436)(28 95 132 437)(29 96 133 438)(30 97 134 439)(31 98 135 440)(32 99 136 441)(33 100 137 442)(34 101 138 443)(35 102 139 444)(36 103 140 445)(37 104 141 446)(38 105 142 447)(39 407 221 233)(40 408 222 234)(41 409 223 235)(42 410 224 236)(43 411 225 237)(44 412 226 238)(45 413 227 239)(46 414 228 240)(47 415 191 241)(48 416 192 242)(49 417 193 243)(50 418 194 244)(51 381 195 245)(52 382 196 246)(53 383 197 247)(54 384 198 248)(55 385 199 249)(56 386 200 250)(57 387 201 251)(58 388 202 252)(59 389 203 253)(60 390 204 254)(61 391 205 255)(62 392 206 256)(63 393 207 257)(64 394 208 258)(65 395 209 259)(66 396 210 260)(67 397 211 261)(68 398 212 262)(69 399 213 263)(70 400 214 264)(71 401 215 265)(72 402 216 266)(73 403 217 229)(74 404 218 230)(75 405 219 231)(76 406 220 232)(153 337 362 277)(154 338 363 278)(155 339 364 279)(156 340 365 280)(157 341 366 281)(158 342 367 282)(159 305 368 283)(160 306 369 284)(161 307 370 285)(162 308 371 286)(163 309 372 287)(164 310 373 288)(165 311 374 289)(166 312 375 290)(167 313 376 291)(168 314 377 292)(169 315 378 293)(170 316 379 294)(171 317 380 295)(172 318 343 296)(173 319 344 297)(174 320 345 298)(175 321 346 299)(176 322 347 300)(177 323 348 301)(178 324 349 302)(179 325 350 303)(180 326 351 304)(181 327 352 267)(182 328 353 268)(183 329 354 269)(184 330 355 270)(185 331 356 271)(186 332 357 272)(187 333 358 273)(188 334 359 274)(189 335 360 275)(190 336 361 276)
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), (1,285,245,143,307,381)(2,286,246,144,308,382)(3,287,247,145,309,383)(4,288,248,146,310,384)(5,289,249,147,311,385)(6,290,250,148,312,386)(7,291,251,149,313,387)(8,292,252,150,314,388)(9,293,253,151,315,389)(10,294,254,152,316,390)(11,295,255,115,317,391)(12,296,256,116,318,392)(13,297,257,117,319,393)(14,298,258,118,320,394)(15,299,259,119,321,395)(16,300,260,120,322,396)(17,301,261,121,323,397)(18,302,262,122,324,398)(19,303,263,123,325,399)(20,304,264,124,326,400)(21,267,265,125,327,401)(22,268,266,126,328,402)(23,269,229,127,329,403)(24,270,230,128,330,404)(25,271,231,129,331,405)(26,272,232,130,332,406)(27,273,233,131,333,407)(28,274,234,132,334,408)(29,275,235,133,335,409)(30,276,236,134,336,410)(31,277,237,135,337,411)(32,278,238,136,338,412)(33,279,239,137,339,413)(34,280,240,138,340,414)(35,281,241,139,341,415)(36,282,242,140,342,416)(37,283,243,141,305,417)(38,284,244,142,306,418)(39,187,94,221,358,436)(40,188,95,222,359,437)(41,189,96,223,360,438)(42,190,97,224,361,439)(43,153,98,225,362,440)(44,154,99,226,363,441)(45,155,100,227,364,442)(46,156,101,228,365,443)(47,157,102,191,366,444)(48,158,103,192,367,445)(49,159,104,193,368,446)(50,160,105,194,369,447)(51,161,106,195,370,448)(52,162,107,196,371,449)(53,163,108,197,372,450)(54,164,109,198,373,451)(55,165,110,199,374,452)(56,166,111,200,375,453)(57,167,112,201,376,454)(58,168,113,202,377,455)(59,169,114,203,378,456)(60,170,77,204,379,419)(61,171,78,205,380,420)(62,172,79,206,343,421)(63,173,80,207,344,422)(64,174,81,208,345,423)(65,175,82,209,346,424)(66,176,83,210,347,425)(67,177,84,211,348,426)(68,178,85,212,349,427)(69,179,86,213,350,428)(70,180,87,214,351,429)(71,181,88,215,352,430)(72,182,89,216,353,431)(73,183,90,217,354,432)(74,184,91,218,355,433)(75,185,92,219,356,434)(76,186,93,220,357,435), (1,106,143,448)(2,107,144,449)(3,108,145,450)(4,109,146,451)(5,110,147,452)(6,111,148,453)(7,112,149,454)(8,113,150,455)(9,114,151,456)(10,77,152,419)(11,78,115,420)(12,79,116,421)(13,80,117,422)(14,81,118,423)(15,82,119,424)(16,83,120,425)(17,84,121,426)(18,85,122,427)(19,86,123,428)(20,87,124,429)(21,88,125,430)(22,89,126,431)(23,90,127,432)(24,91,128,433)(25,92,129,434)(26,93,130,435)(27,94,131,436)(28,95,132,437)(29,96,133,438)(30,97,134,439)(31,98,135,440)(32,99,136,441)(33,100,137,442)(34,101,138,443)(35,102,139,444)(36,103,140,445)(37,104,141,446)(38,105,142,447)(39,407,221,233)(40,408,222,234)(41,409,223,235)(42,410,224,236)(43,411,225,237)(44,412,226,238)(45,413,227,239)(46,414,228,240)(47,415,191,241)(48,416,192,242)(49,417,193,243)(50,418,194,244)(51,381,195,245)(52,382,196,246)(53,383,197,247)(54,384,198,248)(55,385,199,249)(56,386,200,250)(57,387,201,251)(58,388,202,252)(59,389,203,253)(60,390,204,254)(61,391,205,255)(62,392,206,256)(63,393,207,257)(64,394,208,258)(65,395,209,259)(66,396,210,260)(67,397,211,261)(68,398,212,262)(69,399,213,263)(70,400,214,264)(71,401,215,265)(72,402,216,266)(73,403,217,229)(74,404,218,230)(75,405,219,231)(76,406,220,232)(153,337,362,277)(154,338,363,278)(155,339,364,279)(156,340,365,280)(157,341,366,281)(158,342,367,282)(159,305,368,283)(160,306,369,284)(161,307,370,285)(162,308,371,286)(163,309,372,287)(164,310,373,288)(165,311,374,289)(166,312,375,290)(167,313,376,291)(168,314,377,292)(169,315,378,293)(170,316,379,294)(171,317,380,295)(172,318,343,296)(173,319,344,297)(174,320,345,298)(175,321,346,299)(176,322,347,300)(177,323,348,301)(178,324,349,302)(179,325,350,303)(180,326,351,304)(181,327,352,267)(182,328,353,268)(183,329,354,269)(184,330,355,270)(185,331,356,271)(186,332,357,272)(187,333,358,273)(188,334,359,274)(189,335,360,275)(190,336,361,276)>;
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), (1,285,245,143,307,381)(2,286,246,144,308,382)(3,287,247,145,309,383)(4,288,248,146,310,384)(5,289,249,147,311,385)(6,290,250,148,312,386)(7,291,251,149,313,387)(8,292,252,150,314,388)(9,293,253,151,315,389)(10,294,254,152,316,390)(11,295,255,115,317,391)(12,296,256,116,318,392)(13,297,257,117,319,393)(14,298,258,118,320,394)(15,299,259,119,321,395)(16,300,260,120,322,396)(17,301,261,121,323,397)(18,302,262,122,324,398)(19,303,263,123,325,399)(20,304,264,124,326,400)(21,267,265,125,327,401)(22,268,266,126,328,402)(23,269,229,127,329,403)(24,270,230,128,330,404)(25,271,231,129,331,405)(26,272,232,130,332,406)(27,273,233,131,333,407)(28,274,234,132,334,408)(29,275,235,133,335,409)(30,276,236,134,336,410)(31,277,237,135,337,411)(32,278,238,136,338,412)(33,279,239,137,339,413)(34,280,240,138,340,414)(35,281,241,139,341,415)(36,282,242,140,342,416)(37,283,243,141,305,417)(38,284,244,142,306,418)(39,187,94,221,358,436)(40,188,95,222,359,437)(41,189,96,223,360,438)(42,190,97,224,361,439)(43,153,98,225,362,440)(44,154,99,226,363,441)(45,155,100,227,364,442)(46,156,101,228,365,443)(47,157,102,191,366,444)(48,158,103,192,367,445)(49,159,104,193,368,446)(50,160,105,194,369,447)(51,161,106,195,370,448)(52,162,107,196,371,449)(53,163,108,197,372,450)(54,164,109,198,373,451)(55,165,110,199,374,452)(56,166,111,200,375,453)(57,167,112,201,376,454)(58,168,113,202,377,455)(59,169,114,203,378,456)(60,170,77,204,379,419)(61,171,78,205,380,420)(62,172,79,206,343,421)(63,173,80,207,344,422)(64,174,81,208,345,423)(65,175,82,209,346,424)(66,176,83,210,347,425)(67,177,84,211,348,426)(68,178,85,212,349,427)(69,179,86,213,350,428)(70,180,87,214,351,429)(71,181,88,215,352,430)(72,182,89,216,353,431)(73,183,90,217,354,432)(74,184,91,218,355,433)(75,185,92,219,356,434)(76,186,93,220,357,435), (1,106,143,448)(2,107,144,449)(3,108,145,450)(4,109,146,451)(5,110,147,452)(6,111,148,453)(7,112,149,454)(8,113,150,455)(9,114,151,456)(10,77,152,419)(11,78,115,420)(12,79,116,421)(13,80,117,422)(14,81,118,423)(15,82,119,424)(16,83,120,425)(17,84,121,426)(18,85,122,427)(19,86,123,428)(20,87,124,429)(21,88,125,430)(22,89,126,431)(23,90,127,432)(24,91,128,433)(25,92,129,434)(26,93,130,435)(27,94,131,436)(28,95,132,437)(29,96,133,438)(30,97,134,439)(31,98,135,440)(32,99,136,441)(33,100,137,442)(34,101,138,443)(35,102,139,444)(36,103,140,445)(37,104,141,446)(38,105,142,447)(39,407,221,233)(40,408,222,234)(41,409,223,235)(42,410,224,236)(43,411,225,237)(44,412,226,238)(45,413,227,239)(46,414,228,240)(47,415,191,241)(48,416,192,242)(49,417,193,243)(50,418,194,244)(51,381,195,245)(52,382,196,246)(53,383,197,247)(54,384,198,248)(55,385,199,249)(56,386,200,250)(57,387,201,251)(58,388,202,252)(59,389,203,253)(60,390,204,254)(61,391,205,255)(62,392,206,256)(63,393,207,257)(64,394,208,258)(65,395,209,259)(66,396,210,260)(67,397,211,261)(68,398,212,262)(69,399,213,263)(70,400,214,264)(71,401,215,265)(72,402,216,266)(73,403,217,229)(74,404,218,230)(75,405,219,231)(76,406,220,232)(153,337,362,277)(154,338,363,278)(155,339,364,279)(156,340,365,280)(157,341,366,281)(158,342,367,282)(159,305,368,283)(160,306,369,284)(161,307,370,285)(162,308,371,286)(163,309,372,287)(164,310,373,288)(165,311,374,289)(166,312,375,290)(167,313,376,291)(168,314,377,292)(169,315,378,293)(170,316,379,294)(171,317,380,295)(172,318,343,296)(173,319,344,297)(174,320,345,298)(175,321,346,299)(176,322,347,300)(177,323,348,301)(178,324,349,302)(179,325,350,303)(180,326,351,304)(181,327,352,267)(182,328,353,268)(183,329,354,269)(184,330,355,270)(185,331,356,271)(186,332,357,272)(187,333,358,273)(188,334,359,274)(189,335,360,275)(190,336,361,276) );
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)], [(1,285,245,143,307,381),(2,286,246,144,308,382),(3,287,247,145,309,383),(4,288,248,146,310,384),(5,289,249,147,311,385),(6,290,250,148,312,386),(7,291,251,149,313,387),(8,292,252,150,314,388),(9,293,253,151,315,389),(10,294,254,152,316,390),(11,295,255,115,317,391),(12,296,256,116,318,392),(13,297,257,117,319,393),(14,298,258,118,320,394),(15,299,259,119,321,395),(16,300,260,120,322,396),(17,301,261,121,323,397),(18,302,262,122,324,398),(19,303,263,123,325,399),(20,304,264,124,326,400),(21,267,265,125,327,401),(22,268,266,126,328,402),(23,269,229,127,329,403),(24,270,230,128,330,404),(25,271,231,129,331,405),(26,272,232,130,332,406),(27,273,233,131,333,407),(28,274,234,132,334,408),(29,275,235,133,335,409),(30,276,236,134,336,410),(31,277,237,135,337,411),(32,278,238,136,338,412),(33,279,239,137,339,413),(34,280,240,138,340,414),(35,281,241,139,341,415),(36,282,242,140,342,416),(37,283,243,141,305,417),(38,284,244,142,306,418),(39,187,94,221,358,436),(40,188,95,222,359,437),(41,189,96,223,360,438),(42,190,97,224,361,439),(43,153,98,225,362,440),(44,154,99,226,363,441),(45,155,100,227,364,442),(46,156,101,228,365,443),(47,157,102,191,366,444),(48,158,103,192,367,445),(49,159,104,193,368,446),(50,160,105,194,369,447),(51,161,106,195,370,448),(52,162,107,196,371,449),(53,163,108,197,372,450),(54,164,109,198,373,451),(55,165,110,199,374,452),(56,166,111,200,375,453),(57,167,112,201,376,454),(58,168,113,202,377,455),(59,169,114,203,378,456),(60,170,77,204,379,419),(61,171,78,205,380,420),(62,172,79,206,343,421),(63,173,80,207,344,422),(64,174,81,208,345,423),(65,175,82,209,346,424),(66,176,83,210,347,425),(67,177,84,211,348,426),(68,178,85,212,349,427),(69,179,86,213,350,428),(70,180,87,214,351,429),(71,181,88,215,352,430),(72,182,89,216,353,431),(73,183,90,217,354,432),(74,184,91,218,355,433),(75,185,92,219,356,434),(76,186,93,220,357,435)], [(1,106,143,448),(2,107,144,449),(3,108,145,450),(4,109,146,451),(5,110,147,452),(6,111,148,453),(7,112,149,454),(8,113,150,455),(9,114,151,456),(10,77,152,419),(11,78,115,420),(12,79,116,421),(13,80,117,422),(14,81,118,423),(15,82,119,424),(16,83,120,425),(17,84,121,426),(18,85,122,427),(19,86,123,428),(20,87,124,429),(21,88,125,430),(22,89,126,431),(23,90,127,432),(24,91,128,433),(25,92,129,434),(26,93,130,435),(27,94,131,436),(28,95,132,437),(29,96,133,438),(30,97,134,439),(31,98,135,440),(32,99,136,441),(33,100,137,442),(34,101,138,443),(35,102,139,444),(36,103,140,445),(37,104,141,446),(38,105,142,447),(39,407,221,233),(40,408,222,234),(41,409,223,235),(42,410,224,236),(43,411,225,237),(44,412,226,238),(45,413,227,239),(46,414,228,240),(47,415,191,241),(48,416,192,242),(49,417,193,243),(50,418,194,244),(51,381,195,245),(52,382,196,246),(53,383,197,247),(54,384,198,248),(55,385,199,249),(56,386,200,250),(57,387,201,251),(58,388,202,252),(59,389,203,253),(60,390,204,254),(61,391,205,255),(62,392,206,256),(63,393,207,257),(64,394,208,258),(65,395,209,259),(66,396,210,260),(67,397,211,261),(68,398,212,262),(69,399,213,263),(70,400,214,264),(71,401,215,265),(72,402,216,266),(73,403,217,229),(74,404,218,230),(75,405,219,231),(76,406,220,232),(153,337,362,277),(154,338,363,278),(155,339,364,279),(156,340,365,280),(157,341,366,281),(158,342,367,282),(159,305,368,283),(160,306,369,284),(161,307,370,285),(162,308,371,286),(163,309,372,287),(164,310,373,288),(165,311,374,289),(166,312,375,290),(167,313,376,291),(168,314,377,292),(169,315,378,293),(170,316,379,294),(171,317,380,295),(172,318,343,296),(173,319,344,297),(174,320,345,298),(175,321,346,299),(176,322,347,300),(177,323,348,301),(178,324,349,302),(179,325,350,303),(180,326,351,304),(181,327,352,267),(182,328,353,268),(183,329,354,269),(184,330,355,270),(185,331,356,271),(186,332,357,272),(187,333,358,273),(188,334,359,274),(189,335,360,275),(190,336,361,276)]])
228 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 19A | ··· | 19R | 38A | ··· | 38BB | 57A | ··· | 57R | 76A | ··· | 76BT | 114A | ··· | 114BB |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
228 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C19 | C38 | C38 | C76 | S3 | Dic3 | D6 | S3×C19 | Dic3×C19 | S3×C38 |
| kernel | Dic3×C38 | Dic3×C19 | C2×C114 | C114 | C2×Dic3 | Dic3 | C2×C6 | C6 | C2×C38 | C38 | C38 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 18 | 36 | 18 | 72 | 1 | 2 | 1 | 18 | 36 | 18 |
Matrix representation of Dic3×C38 ►in GL3(𝔽229) generated by
| 1 | 0 | 0 |
| 0 | 176 | 0 |
| 0 | 0 | 176 |
| 228 | 0 | 0 |
| 0 | 228 | 1 |
| 0 | 228 | 0 |
| 107 | 0 | 0 |
| 0 | 4 | 24 |
| 0 | 28 | 225 |
G:=sub<GL(3,GF(229))| [1,0,0,0,176,0,0,0,176],[228,0,0,0,228,228,0,1,0],[107,0,0,0,4,28,0,24,225] >;
Dic3×C38 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{38} % in TeX
G:=Group("Dic3xC38"); // GroupNames label
G:=SmallGroup(456,32);
// by ID
G=gap.SmallGroup(456,32);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-3,380,7604]);
// Polycyclic
G:=Group<a,b,c|a^38=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export