metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic31⋊C4, C62.5D4, C62.1Q8, C2.1Dic62, C22.4D62, C31⋊1(C4⋊C4), C62.4(C2×C4), C2.4(C4×D31), (C2×C4).1D31, (C2×C124).1C2, C2.1(C31⋊D4), (C2×C62).4C22, (C2×Dic31).1C2, SmallGroup(496,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic31⋊C4
G = < a,b,c | a62=c4=1, b2=a31, bab-1=a-1, ac=ca, cbc-1=a31b >
Smallest permutation representation of Dic31⋊C4
►Regular action on 496 points
Generators in S496
(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)(435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496)
(1 342 32 311)(2 341 33 372)(3 340 34 371)(4 339 35 370)(5 338 36 369)(6 337 37 368)(7 336 38 367)(8 335 39 366)(9 334 40 365)(10 333 41 364)(11 332 42 363)(12 331 43 362)(13 330 44 361)(14 329 45 360)(15 328 46 359)(16 327 47 358)(17 326 48 357)(18 325 49 356)(19 324 50 355)(20 323 51 354)(21 322 52 353)(22 321 53 352)(23 320 54 351)(24 319 55 350)(25 318 56 349)(26 317 57 348)(27 316 58 347)(28 315 59 346)(29 314 60 345)(30 313 61 344)(31 312 62 343)(63 258 94 289)(64 257 95 288)(65 256 96 287)(66 255 97 286)(67 254 98 285)(68 253 99 284)(69 252 100 283)(70 251 101 282)(71 250 102 281)(72 249 103 280)(73 310 104 279)(74 309 105 278)(75 308 106 277)(76 307 107 276)(77 306 108 275)(78 305 109 274)(79 304 110 273)(80 303 111 272)(81 302 112 271)(82 301 113 270)(83 300 114 269)(84 299 115 268)(85 298 116 267)(86 297 117 266)(87 296 118 265)(88 295 119 264)(89 294 120 263)(90 293 121 262)(91 292 122 261)(92 291 123 260)(93 290 124 259)(125 448 156 479)(126 447 157 478)(127 446 158 477)(128 445 159 476)(129 444 160 475)(130 443 161 474)(131 442 162 473)(132 441 163 472)(133 440 164 471)(134 439 165 470)(135 438 166 469)(136 437 167 468)(137 436 168 467)(138 435 169 466)(139 496 170 465)(140 495 171 464)(141 494 172 463)(142 493 173 462)(143 492 174 461)(144 491 175 460)(145 490 176 459)(146 489 177 458)(147 488 178 457)(148 487 179 456)(149 486 180 455)(150 485 181 454)(151 484 182 453)(152 483 183 452)(153 482 184 451)(154 481 185 450)(155 480 186 449)(187 421 218 390)(188 420 219 389)(189 419 220 388)(190 418 221 387)(191 417 222 386)(192 416 223 385)(193 415 224 384)(194 414 225 383)(195 413 226 382)(196 412 227 381)(197 411 228 380)(198 410 229 379)(199 409 230 378)(200 408 231 377)(201 407 232 376)(202 406 233 375)(203 405 234 374)(204 404 235 373)(205 403 236 434)(206 402 237 433)(207 401 238 432)(208 400 239 431)(209 399 240 430)(210 398 241 429)(211 397 242 428)(212 396 243 427)(213 395 244 426)(214 394 245 425)(215 393 246 424)(216 392 247 423)(217 391 248 422)
(1 235 103 169)(2 236 104 170)(3 237 105 171)(4 238 106 172)(5 239 107 173)(6 240 108 174)(7 241 109 175)(8 242 110 176)(9 243 111 177)(10 244 112 178)(11 245 113 179)(12 246 114 180)(13 247 115 181)(14 248 116 182)(15 187 117 183)(16 188 118 184)(17 189 119 185)(18 190 120 186)(19 191 121 125)(20 192 122 126)(21 193 123 127)(22 194 124 128)(23 195 63 129)(24 196 64 130)(25 197 65 131)(26 198 66 132)(27 199 67 133)(28 200 68 134)(29 201 69 135)(30 202 70 136)(31 203 71 137)(32 204 72 138)(33 205 73 139)(34 206 74 140)(35 207 75 141)(36 208 76 142)(37 209 77 143)(38 210 78 144)(39 211 79 145)(40 212 80 146)(41 213 81 147)(42 214 82 148)(43 215 83 149)(44 216 84 150)(45 217 85 151)(46 218 86 152)(47 219 87 153)(48 220 88 154)(49 221 89 155)(50 222 90 156)(51 223 91 157)(52 224 92 158)(53 225 93 159)(54 226 94 160)(55 227 95 161)(56 228 96 162)(57 229 97 163)(58 230 98 164)(59 231 99 165)(60 232 100 166)(61 233 101 167)(62 234 102 168)(249 466 311 373)(250 467 312 374)(251 468 313 375)(252 469 314 376)(253 470 315 377)(254 471 316 378)(255 472 317 379)(256 473 318 380)(257 474 319 381)(258 475 320 382)(259 476 321 383)(260 477 322 384)(261 478 323 385)(262 479 324 386)(263 480 325 387)(264 481 326 388)(265 482 327 389)(266 483 328 390)(267 484 329 391)(268 485 330 392)(269 486 331 393)(270 487 332 394)(271 488 333 395)(272 489 334 396)(273 490 335 397)(274 491 336 398)(275 492 337 399)(276 493 338 400)(277 494 339 401)(278 495 340 402)(279 496 341 403)(280 435 342 404)(281 436 343 405)(282 437 344 406)(283 438 345 407)(284 439 346 408)(285 440 347 409)(286 441 348 410)(287 442 349 411)(288 443 350 412)(289 444 351 413)(290 445 352 414)(291 446 353 415)(292 447 354 416)(293 448 355 417)(294 449 356 418)(295 450 357 419)(296 451 358 420)(297 452 359 421)(298 453 360 422)(299 454 361 423)(300 455 362 424)(301 456 363 425)(302 457 364 426)(303 458 365 427)(304 459 366 428)(305 460 367 429)(306 461 368 430)(307 462 369 431)(308 463 370 432)(309 464 371 433)(310 465 372 434)
G:=sub<Sym(496)| (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)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,342,32,311)(2,341,33,372)(3,340,34,371)(4,339,35,370)(5,338,36,369)(6,337,37,368)(7,336,38,367)(8,335,39,366)(9,334,40,365)(10,333,41,364)(11,332,42,363)(12,331,43,362)(13,330,44,361)(14,329,45,360)(15,328,46,359)(16,327,47,358)(17,326,48,357)(18,325,49,356)(19,324,50,355)(20,323,51,354)(21,322,52,353)(22,321,53,352)(23,320,54,351)(24,319,55,350)(25,318,56,349)(26,317,57,348)(27,316,58,347)(28,315,59,346)(29,314,60,345)(30,313,61,344)(31,312,62,343)(63,258,94,289)(64,257,95,288)(65,256,96,287)(66,255,97,286)(67,254,98,285)(68,253,99,284)(69,252,100,283)(70,251,101,282)(71,250,102,281)(72,249,103,280)(73,310,104,279)(74,309,105,278)(75,308,106,277)(76,307,107,276)(77,306,108,275)(78,305,109,274)(79,304,110,273)(80,303,111,272)(81,302,112,271)(82,301,113,270)(83,300,114,269)(84,299,115,268)(85,298,116,267)(86,297,117,266)(87,296,118,265)(88,295,119,264)(89,294,120,263)(90,293,121,262)(91,292,122,261)(92,291,123,260)(93,290,124,259)(125,448,156,479)(126,447,157,478)(127,446,158,477)(128,445,159,476)(129,444,160,475)(130,443,161,474)(131,442,162,473)(132,441,163,472)(133,440,164,471)(134,439,165,470)(135,438,166,469)(136,437,167,468)(137,436,168,467)(138,435,169,466)(139,496,170,465)(140,495,171,464)(141,494,172,463)(142,493,173,462)(143,492,174,461)(144,491,175,460)(145,490,176,459)(146,489,177,458)(147,488,178,457)(148,487,179,456)(149,486,180,455)(150,485,181,454)(151,484,182,453)(152,483,183,452)(153,482,184,451)(154,481,185,450)(155,480,186,449)(187,421,218,390)(188,420,219,389)(189,419,220,388)(190,418,221,387)(191,417,222,386)(192,416,223,385)(193,415,224,384)(194,414,225,383)(195,413,226,382)(196,412,227,381)(197,411,228,380)(198,410,229,379)(199,409,230,378)(200,408,231,377)(201,407,232,376)(202,406,233,375)(203,405,234,374)(204,404,235,373)(205,403,236,434)(206,402,237,433)(207,401,238,432)(208,400,239,431)(209,399,240,430)(210,398,241,429)(211,397,242,428)(212,396,243,427)(213,395,244,426)(214,394,245,425)(215,393,246,424)(216,392,247,423)(217,391,248,422), (1,235,103,169)(2,236,104,170)(3,237,105,171)(4,238,106,172)(5,239,107,173)(6,240,108,174)(7,241,109,175)(8,242,110,176)(9,243,111,177)(10,244,112,178)(11,245,113,179)(12,246,114,180)(13,247,115,181)(14,248,116,182)(15,187,117,183)(16,188,118,184)(17,189,119,185)(18,190,120,186)(19,191,121,125)(20,192,122,126)(21,193,123,127)(22,194,124,128)(23,195,63,129)(24,196,64,130)(25,197,65,131)(26,198,66,132)(27,199,67,133)(28,200,68,134)(29,201,69,135)(30,202,70,136)(31,203,71,137)(32,204,72,138)(33,205,73,139)(34,206,74,140)(35,207,75,141)(36,208,76,142)(37,209,77,143)(38,210,78,144)(39,211,79,145)(40,212,80,146)(41,213,81,147)(42,214,82,148)(43,215,83,149)(44,216,84,150)(45,217,85,151)(46,218,86,152)(47,219,87,153)(48,220,88,154)(49,221,89,155)(50,222,90,156)(51,223,91,157)(52,224,92,158)(53,225,93,159)(54,226,94,160)(55,227,95,161)(56,228,96,162)(57,229,97,163)(58,230,98,164)(59,231,99,165)(60,232,100,166)(61,233,101,167)(62,234,102,168)(249,466,311,373)(250,467,312,374)(251,468,313,375)(252,469,314,376)(253,470,315,377)(254,471,316,378)(255,472,317,379)(256,473,318,380)(257,474,319,381)(258,475,320,382)(259,476,321,383)(260,477,322,384)(261,478,323,385)(262,479,324,386)(263,480,325,387)(264,481,326,388)(265,482,327,389)(266,483,328,390)(267,484,329,391)(268,485,330,392)(269,486,331,393)(270,487,332,394)(271,488,333,395)(272,489,334,396)(273,490,335,397)(274,491,336,398)(275,492,337,399)(276,493,338,400)(277,494,339,401)(278,495,340,402)(279,496,341,403)(280,435,342,404)(281,436,343,405)(282,437,344,406)(283,438,345,407)(284,439,346,408)(285,440,347,409)(286,441,348,410)(287,442,349,411)(288,443,350,412)(289,444,351,413)(290,445,352,414)(291,446,353,415)(292,447,354,416)(293,448,355,417)(294,449,356,418)(295,450,357,419)(296,451,358,420)(297,452,359,421)(298,453,360,422)(299,454,361,423)(300,455,362,424)(301,456,363,425)(302,457,364,426)(303,458,365,427)(304,459,366,428)(305,460,367,429)(306,461,368,430)(307,462,369,431)(308,463,370,432)(309,464,371,433)(310,465,372,434)>;
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)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,342,32,311)(2,341,33,372)(3,340,34,371)(4,339,35,370)(5,338,36,369)(6,337,37,368)(7,336,38,367)(8,335,39,366)(9,334,40,365)(10,333,41,364)(11,332,42,363)(12,331,43,362)(13,330,44,361)(14,329,45,360)(15,328,46,359)(16,327,47,358)(17,326,48,357)(18,325,49,356)(19,324,50,355)(20,323,51,354)(21,322,52,353)(22,321,53,352)(23,320,54,351)(24,319,55,350)(25,318,56,349)(26,317,57,348)(27,316,58,347)(28,315,59,346)(29,314,60,345)(30,313,61,344)(31,312,62,343)(63,258,94,289)(64,257,95,288)(65,256,96,287)(66,255,97,286)(67,254,98,285)(68,253,99,284)(69,252,100,283)(70,251,101,282)(71,250,102,281)(72,249,103,280)(73,310,104,279)(74,309,105,278)(75,308,106,277)(76,307,107,276)(77,306,108,275)(78,305,109,274)(79,304,110,273)(80,303,111,272)(81,302,112,271)(82,301,113,270)(83,300,114,269)(84,299,115,268)(85,298,116,267)(86,297,117,266)(87,296,118,265)(88,295,119,264)(89,294,120,263)(90,293,121,262)(91,292,122,261)(92,291,123,260)(93,290,124,259)(125,448,156,479)(126,447,157,478)(127,446,158,477)(128,445,159,476)(129,444,160,475)(130,443,161,474)(131,442,162,473)(132,441,163,472)(133,440,164,471)(134,439,165,470)(135,438,166,469)(136,437,167,468)(137,436,168,467)(138,435,169,466)(139,496,170,465)(140,495,171,464)(141,494,172,463)(142,493,173,462)(143,492,174,461)(144,491,175,460)(145,490,176,459)(146,489,177,458)(147,488,178,457)(148,487,179,456)(149,486,180,455)(150,485,181,454)(151,484,182,453)(152,483,183,452)(153,482,184,451)(154,481,185,450)(155,480,186,449)(187,421,218,390)(188,420,219,389)(189,419,220,388)(190,418,221,387)(191,417,222,386)(192,416,223,385)(193,415,224,384)(194,414,225,383)(195,413,226,382)(196,412,227,381)(197,411,228,380)(198,410,229,379)(199,409,230,378)(200,408,231,377)(201,407,232,376)(202,406,233,375)(203,405,234,374)(204,404,235,373)(205,403,236,434)(206,402,237,433)(207,401,238,432)(208,400,239,431)(209,399,240,430)(210,398,241,429)(211,397,242,428)(212,396,243,427)(213,395,244,426)(214,394,245,425)(215,393,246,424)(216,392,247,423)(217,391,248,422), (1,235,103,169)(2,236,104,170)(3,237,105,171)(4,238,106,172)(5,239,107,173)(6,240,108,174)(7,241,109,175)(8,242,110,176)(9,243,111,177)(10,244,112,178)(11,245,113,179)(12,246,114,180)(13,247,115,181)(14,248,116,182)(15,187,117,183)(16,188,118,184)(17,189,119,185)(18,190,120,186)(19,191,121,125)(20,192,122,126)(21,193,123,127)(22,194,124,128)(23,195,63,129)(24,196,64,130)(25,197,65,131)(26,198,66,132)(27,199,67,133)(28,200,68,134)(29,201,69,135)(30,202,70,136)(31,203,71,137)(32,204,72,138)(33,205,73,139)(34,206,74,140)(35,207,75,141)(36,208,76,142)(37,209,77,143)(38,210,78,144)(39,211,79,145)(40,212,80,146)(41,213,81,147)(42,214,82,148)(43,215,83,149)(44,216,84,150)(45,217,85,151)(46,218,86,152)(47,219,87,153)(48,220,88,154)(49,221,89,155)(50,222,90,156)(51,223,91,157)(52,224,92,158)(53,225,93,159)(54,226,94,160)(55,227,95,161)(56,228,96,162)(57,229,97,163)(58,230,98,164)(59,231,99,165)(60,232,100,166)(61,233,101,167)(62,234,102,168)(249,466,311,373)(250,467,312,374)(251,468,313,375)(252,469,314,376)(253,470,315,377)(254,471,316,378)(255,472,317,379)(256,473,318,380)(257,474,319,381)(258,475,320,382)(259,476,321,383)(260,477,322,384)(261,478,323,385)(262,479,324,386)(263,480,325,387)(264,481,326,388)(265,482,327,389)(266,483,328,390)(267,484,329,391)(268,485,330,392)(269,486,331,393)(270,487,332,394)(271,488,333,395)(272,489,334,396)(273,490,335,397)(274,491,336,398)(275,492,337,399)(276,493,338,400)(277,494,339,401)(278,495,340,402)(279,496,341,403)(280,435,342,404)(281,436,343,405)(282,437,344,406)(283,438,345,407)(284,439,346,408)(285,440,347,409)(286,441,348,410)(287,442,349,411)(288,443,350,412)(289,444,351,413)(290,445,352,414)(291,446,353,415)(292,447,354,416)(293,448,355,417)(294,449,356,418)(295,450,357,419)(296,451,358,420)(297,452,359,421)(298,453,360,422)(299,454,361,423)(300,455,362,424)(301,456,363,425)(302,457,364,426)(303,458,365,427)(304,459,366,428)(305,460,367,429)(306,461,368,430)(307,462,369,431)(308,463,370,432)(309,464,371,433)(310,465,372,434) );
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),(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)], [(1,342,32,311),(2,341,33,372),(3,340,34,371),(4,339,35,370),(5,338,36,369),(6,337,37,368),(7,336,38,367),(8,335,39,366),(9,334,40,365),(10,333,41,364),(11,332,42,363),(12,331,43,362),(13,330,44,361),(14,329,45,360),(15,328,46,359),(16,327,47,358),(17,326,48,357),(18,325,49,356),(19,324,50,355),(20,323,51,354),(21,322,52,353),(22,321,53,352),(23,320,54,351),(24,319,55,350),(25,318,56,349),(26,317,57,348),(27,316,58,347),(28,315,59,346),(29,314,60,345),(30,313,61,344),(31,312,62,343),(63,258,94,289),(64,257,95,288),(65,256,96,287),(66,255,97,286),(67,254,98,285),(68,253,99,284),(69,252,100,283),(70,251,101,282),(71,250,102,281),(72,249,103,280),(73,310,104,279),(74,309,105,278),(75,308,106,277),(76,307,107,276),(77,306,108,275),(78,305,109,274),(79,304,110,273),(80,303,111,272),(81,302,112,271),(82,301,113,270),(83,300,114,269),(84,299,115,268),(85,298,116,267),(86,297,117,266),(87,296,118,265),(88,295,119,264),(89,294,120,263),(90,293,121,262),(91,292,122,261),(92,291,123,260),(93,290,124,259),(125,448,156,479),(126,447,157,478),(127,446,158,477),(128,445,159,476),(129,444,160,475),(130,443,161,474),(131,442,162,473),(132,441,163,472),(133,440,164,471),(134,439,165,470),(135,438,166,469),(136,437,167,468),(137,436,168,467),(138,435,169,466),(139,496,170,465),(140,495,171,464),(141,494,172,463),(142,493,173,462),(143,492,174,461),(144,491,175,460),(145,490,176,459),(146,489,177,458),(147,488,178,457),(148,487,179,456),(149,486,180,455),(150,485,181,454),(151,484,182,453),(152,483,183,452),(153,482,184,451),(154,481,185,450),(155,480,186,449),(187,421,218,390),(188,420,219,389),(189,419,220,388),(190,418,221,387),(191,417,222,386),(192,416,223,385),(193,415,224,384),(194,414,225,383),(195,413,226,382),(196,412,227,381),(197,411,228,380),(198,410,229,379),(199,409,230,378),(200,408,231,377),(201,407,232,376),(202,406,233,375),(203,405,234,374),(204,404,235,373),(205,403,236,434),(206,402,237,433),(207,401,238,432),(208,400,239,431),(209,399,240,430),(210,398,241,429),(211,397,242,428),(212,396,243,427),(213,395,244,426),(214,394,245,425),(215,393,246,424),(216,392,247,423),(217,391,248,422)], [(1,235,103,169),(2,236,104,170),(3,237,105,171),(4,238,106,172),(5,239,107,173),(6,240,108,174),(7,241,109,175),(8,242,110,176),(9,243,111,177),(10,244,112,178),(11,245,113,179),(12,246,114,180),(13,247,115,181),(14,248,116,182),(15,187,117,183),(16,188,118,184),(17,189,119,185),(18,190,120,186),(19,191,121,125),(20,192,122,126),(21,193,123,127),(22,194,124,128),(23,195,63,129),(24,196,64,130),(25,197,65,131),(26,198,66,132),(27,199,67,133),(28,200,68,134),(29,201,69,135),(30,202,70,136),(31,203,71,137),(32,204,72,138),(33,205,73,139),(34,206,74,140),(35,207,75,141),(36,208,76,142),(37,209,77,143),(38,210,78,144),(39,211,79,145),(40,212,80,146),(41,213,81,147),(42,214,82,148),(43,215,83,149),(44,216,84,150),(45,217,85,151),(46,218,86,152),(47,219,87,153),(48,220,88,154),(49,221,89,155),(50,222,90,156),(51,223,91,157),(52,224,92,158),(53,225,93,159),(54,226,94,160),(55,227,95,161),(56,228,96,162),(57,229,97,163),(58,230,98,164),(59,231,99,165),(60,232,100,166),(61,233,101,167),(62,234,102,168),(249,466,311,373),(250,467,312,374),(251,468,313,375),(252,469,314,376),(253,470,315,377),(254,471,316,378),(255,472,317,379),(256,473,318,380),(257,474,319,381),(258,475,320,382),(259,476,321,383),(260,477,322,384),(261,478,323,385),(262,479,324,386),(263,480,325,387),(264,481,326,388),(265,482,327,389),(266,483,328,390),(267,484,329,391),(268,485,330,392),(269,486,331,393),(270,487,332,394),(271,488,333,395),(272,489,334,396),(273,490,335,397),(274,491,336,398),(275,492,337,399),(276,493,338,400),(277,494,339,401),(278,495,340,402),(279,496,341,403),(280,435,342,404),(281,436,343,405),(282,437,344,406),(283,438,345,407),(284,439,346,408),(285,440,347,409),(286,441,348,410),(287,442,349,411),(288,443,350,412),(289,444,351,413),(290,445,352,414),(291,446,353,415),(292,447,354,416),(293,448,355,417),(294,449,356,418),(295,450,357,419),(296,451,358,420),(297,452,359,421),(298,453,360,422),(299,454,361,423),(300,455,362,424),(301,456,363,425),(302,457,364,426),(303,458,365,427),(304,459,366,428),(305,460,367,429),(306,461,368,430),(307,462,369,431),(308,463,370,432),(309,464,371,433),(310,465,372,434)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 31A | ··· | 31O | 62A | ··· | 62AS | 124A | ··· | 124BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 62 | 62 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D31 | D62 | Dic62 | C4×D31 | C31⋊D4 |
| kernel | Dic31⋊C4 | C2×Dic31 | C2×C124 | Dic31 | C62 | C62 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 15 | 15 | 30 | 30 | 30 |
Matrix representation of Dic31⋊C4 ►in GL3(𝔽373) generated by
| 1 | 0 | 0 |
| 0 | 1 | 372 |
| 0 | 75 | 299 |
| 372 | 0 | 0 |
| 0 | 25 | 225 |
| 0 | 100 | 348 |
| 269 | 0 | 0 |
| 0 | 159 | 324 |
| 0 | 318 | 214 |
G:=sub<GL(3,GF(373))| [1,0,0,0,1,75,0,372,299],[372,0,0,0,25,100,0,225,348],[269,0,0,0,159,318,0,324,214] >;
Dic31⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{31}\rtimes C_4 % in TeX
G:=Group("Dic31:C4"); // GroupNames label
G:=SmallGroup(496,11);
// by ID
G=gap.SmallGroup(496,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,40,101,26,12004]);
// Polycyclic
G:=Group<a,b,c|a^62=c^4=1,b^2=a^31,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^31*b>;
// generators/relations
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