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G = C3×Dic38order 456 = 23·3·19

Direct product of C3 and Dic38

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

Aliases: C3×Dic38, C573Q8, C76.5C6, C228.3C2, C6.13D38, C12.3D19, Dic19.2C6, C114.13C22, C4.(C3×D19), C193(C3×Q8), C38.9(C2×C6), C2.3(C6×D19), (C3×Dic19).2C2, SmallGroup(456,24)

Series: Derived Chief Lower central Upper central

C1C38 — C3×Dic38
C1C19C38C114C3×Dic19 — C3×Dic38
C19C38 — C3×Dic38
C1C6C12

Generators and relations for C3×Dic38
 G = < a,b,c | a3=b76=1, c2=b38, ab=ba, ac=ca, cbc-1=b-1 >

19C4
19C4
19Q8
19C12
19C12
19C3×Q8

Smallest permutation representation of C3×Dic38
Regular action on 456 points
Generators in S456
(1 373 87)(2 374 88)(3 375 89)(4 376 90)(5 377 91)(6 378 92)(7 379 93)(8 380 94)(9 305 95)(10 306 96)(11 307 97)(12 308 98)(13 309 99)(14 310 100)(15 311 101)(16 312 102)(17 313 103)(18 314 104)(19 315 105)(20 316 106)(21 317 107)(22 318 108)(23 319 109)(24 320 110)(25 321 111)(26 322 112)(27 323 113)(28 324 114)(29 325 115)(30 326 116)(31 327 117)(32 328 118)(33 329 119)(34 330 120)(35 331 121)(36 332 122)(37 333 123)(38 334 124)(39 335 125)(40 336 126)(41 337 127)(42 338 128)(43 339 129)(44 340 130)(45 341 131)(46 342 132)(47 343 133)(48 344 134)(49 345 135)(50 346 136)(51 347 137)(52 348 138)(53 349 139)(54 350 140)(55 351 141)(56 352 142)(57 353 143)(58 354 144)(59 355 145)(60 356 146)(61 357 147)(62 358 148)(63 359 149)(64 360 150)(65 361 151)(66 362 152)(67 363 77)(68 364 78)(69 365 79)(70 366 80)(71 367 81)(72 368 82)(73 369 83)(74 370 84)(75 371 85)(76 372 86)(153 230 396)(154 231 397)(155 232 398)(156 233 399)(157 234 400)(158 235 401)(159 236 402)(160 237 403)(161 238 404)(162 239 405)(163 240 406)(164 241 407)(165 242 408)(166 243 409)(167 244 410)(168 245 411)(169 246 412)(170 247 413)(171 248 414)(172 249 415)(173 250 416)(174 251 417)(175 252 418)(176 253 419)(177 254 420)(178 255 421)(179 256 422)(180 257 423)(181 258 424)(182 259 425)(183 260 426)(184 261 427)(185 262 428)(186 263 429)(187 264 430)(188 265 431)(189 266 432)(190 267 433)(191 268 434)(192 269 435)(193 270 436)(194 271 437)(195 272 438)(196 273 439)(197 274 440)(198 275 441)(199 276 442)(200 277 443)(201 278 444)(202 279 445)(203 280 446)(204 281 447)(205 282 448)(206 283 449)(207 284 450)(208 285 451)(209 286 452)(210 287 453)(211 288 454)(212 289 455)(213 290 456)(214 291 381)(215 292 382)(216 293 383)(217 294 384)(218 295 385)(219 296 386)(220 297 387)(221 298 388)(222 299 389)(223 300 390)(224 301 391)(225 302 392)(226 303 393)(227 304 394)(228 229 395)
(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)
(1 427 39 389)(2 426 40 388)(3 425 41 387)(4 424 42 386)(5 423 43 385)(6 422 44 384)(7 421 45 383)(8 420 46 382)(9 419 47 381)(10 418 48 456)(11 417 49 455)(12 416 50 454)(13 415 51 453)(14 414 52 452)(15 413 53 451)(16 412 54 450)(17 411 55 449)(18 410 56 448)(19 409 57 447)(20 408 58 446)(21 407 59 445)(22 406 60 444)(23 405 61 443)(24 404 62 442)(25 403 63 441)(26 402 64 440)(27 401 65 439)(28 400 66 438)(29 399 67 437)(30 398 68 436)(31 397 69 435)(32 396 70 434)(33 395 71 433)(34 394 72 432)(35 393 73 431)(36 392 74 430)(37 391 75 429)(38 390 76 428)(77 271 115 233)(78 270 116 232)(79 269 117 231)(80 268 118 230)(81 267 119 229)(82 266 120 304)(83 265 121 303)(84 264 122 302)(85 263 123 301)(86 262 124 300)(87 261 125 299)(88 260 126 298)(89 259 127 297)(90 258 128 296)(91 257 129 295)(92 256 130 294)(93 255 131 293)(94 254 132 292)(95 253 133 291)(96 252 134 290)(97 251 135 289)(98 250 136 288)(99 249 137 287)(100 248 138 286)(101 247 139 285)(102 246 140 284)(103 245 141 283)(104 244 142 282)(105 243 143 281)(106 242 144 280)(107 241 145 279)(108 240 146 278)(109 239 147 277)(110 238 148 276)(111 237 149 275)(112 236 150 274)(113 235 151 273)(114 234 152 272)(153 366 191 328)(154 365 192 327)(155 364 193 326)(156 363 194 325)(157 362 195 324)(158 361 196 323)(159 360 197 322)(160 359 198 321)(161 358 199 320)(162 357 200 319)(163 356 201 318)(164 355 202 317)(165 354 203 316)(166 353 204 315)(167 352 205 314)(168 351 206 313)(169 350 207 312)(170 349 208 311)(171 348 209 310)(172 347 210 309)(173 346 211 308)(174 345 212 307)(175 344 213 306)(176 343 214 305)(177 342 215 380)(178 341 216 379)(179 340 217 378)(180 339 218 377)(181 338 219 376)(182 337 220 375)(183 336 221 374)(184 335 222 373)(185 334 223 372)(186 333 224 371)(187 332 225 370)(188 331 226 369)(189 330 227 368)(190 329 228 367)

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

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

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

123 conjugacy classes

class 1  2 3A3B4A4B4C6A6B12A12B12C12D12E12F19A···19I38A···38I57A···57R76A···76R114A···114R228A···228AJ
order12334446612121212121219···1938···3857···5776···76114···114228···228
size1111238381122383838382···22···22···22···22···22···2

123 irreducible representations

dim11111122222222
type+++-++-
imageC1C2C2C3C6C6Q8C3×Q8D19D38C3×D19Dic38C6×D19C3×Dic38
kernelC3×Dic38C3×Dic19C228Dic38Dic19C76C57C19C12C6C4C3C2C1
# reps121242129918181836

Matrix representation of C3×Dic38 in GL3(𝔽229) generated by

13400
010
001
,
22800
0184119
0110162
,
22800
020955
05120
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

Subgroup lattice of C3×Dic38 in TeX

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