direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic31, C31⋊C42, C124⋊2C4, C22.3D62, C62.3(C2×C4), C2.2(C4×D31), (C2×C4).6D31, (C2×C124).7C2, (C2×C62).3C22, C2.2(C2×Dic31), (C2×Dic31).4C2, SmallGroup(496,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C4×Dic31 |
Generators and relations for C4×Dic31
G = < a,b,c | a4=b62=1, c2=b31, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic31
►Regular action on 496 points
Generators in S496
(1 175 430 333)(2 176 431 334)(3 177 432 335)(4 178 433 336)(5 179 434 337)(6 180 373 338)(7 181 374 339)(8 182 375 340)(9 183 376 341)(10 184 377 342)(11 185 378 343)(12 186 379 344)(13 125 380 345)(14 126 381 346)(15 127 382 347)(16 128 383 348)(17 129 384 349)(18 130 385 350)(19 131 386 351)(20 132 387 352)(21 133 388 353)(22 134 389 354)(23 135 390 355)(24 136 391 356)(25 137 392 357)(26 138 393 358)(27 139 394 359)(28 140 395 360)(29 141 396 361)(30 142 397 362)(31 143 398 363)(32 144 399 364)(33 145 400 365)(34 146 401 366)(35 147 402 367)(36 148 403 368)(37 149 404 369)(38 150 405 370)(39 151 406 371)(40 152 407 372)(41 153 408 311)(42 154 409 312)(43 155 410 313)(44 156 411 314)(45 157 412 315)(46 158 413 316)(47 159 414 317)(48 160 415 318)(49 161 416 319)(50 162 417 320)(51 163 418 321)(52 164 419 322)(53 165 420 323)(54 166 421 324)(55 167 422 325)(56 168 423 326)(57 169 424 327)(58 170 425 328)(59 171 426 329)(60 172 427 330)(61 173 428 331)(62 174 429 332)(63 254 465 199)(64 255 466 200)(65 256 467 201)(66 257 468 202)(67 258 469 203)(68 259 470 204)(69 260 471 205)(70 261 472 206)(71 262 473 207)(72 263 474 208)(73 264 475 209)(74 265 476 210)(75 266 477 211)(76 267 478 212)(77 268 479 213)(78 269 480 214)(79 270 481 215)(80 271 482 216)(81 272 483 217)(82 273 484 218)(83 274 485 219)(84 275 486 220)(85 276 487 221)(86 277 488 222)(87 278 489 223)(88 279 490 224)(89 280 491 225)(90 281 492 226)(91 282 493 227)(92 283 494 228)(93 284 495 229)(94 285 496 230)(95 286 435 231)(96 287 436 232)(97 288 437 233)(98 289 438 234)(99 290 439 235)(100 291 440 236)(101 292 441 237)(102 293 442 238)(103 294 443 239)(104 295 444 240)(105 296 445 241)(106 297 446 242)(107 298 447 243)(108 299 448 244)(109 300 449 245)(110 301 450 246)(111 302 451 247)(112 303 452 248)(113 304 453 187)(114 305 454 188)(115 306 455 189)(116 307 456 190)(117 308 457 191)(118 309 458 192)(119 310 459 193)(120 249 460 194)(121 250 461 195)(122 251 462 196)(123 252 463 197)(124 253 464 198)
(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)
(1 302 32 271)(2 301 33 270)(3 300 34 269)(4 299 35 268)(5 298 36 267)(6 297 37 266)(7 296 38 265)(8 295 39 264)(9 294 40 263)(10 293 41 262)(11 292 42 261)(12 291 43 260)(13 290 44 259)(14 289 45 258)(15 288 46 257)(16 287 47 256)(17 286 48 255)(18 285 49 254)(19 284 50 253)(20 283 51 252)(21 282 52 251)(22 281 53 250)(23 280 54 249)(24 279 55 310)(25 278 56 309)(26 277 57 308)(27 276 58 307)(28 275 59 306)(29 274 60 305)(30 273 61 304)(31 272 62 303)(63 350 94 319)(64 349 95 318)(65 348 96 317)(66 347 97 316)(67 346 98 315)(68 345 99 314)(69 344 100 313)(70 343 101 312)(71 342 102 311)(72 341 103 372)(73 340 104 371)(74 339 105 370)(75 338 106 369)(76 337 107 368)(77 336 108 367)(78 335 109 366)(79 334 110 365)(80 333 111 364)(81 332 112 363)(82 331 113 362)(83 330 114 361)(84 329 115 360)(85 328 116 359)(86 327 117 358)(87 326 118 357)(88 325 119 356)(89 324 120 355)(90 323 121 354)(91 322 122 353)(92 321 123 352)(93 320 124 351)(125 439 156 470)(126 438 157 469)(127 437 158 468)(128 436 159 467)(129 435 160 466)(130 496 161 465)(131 495 162 464)(132 494 163 463)(133 493 164 462)(134 492 165 461)(135 491 166 460)(136 490 167 459)(137 489 168 458)(138 488 169 457)(139 487 170 456)(140 486 171 455)(141 485 172 454)(142 484 173 453)(143 483 174 452)(144 482 175 451)(145 481 176 450)(146 480 177 449)(147 479 178 448)(148 478 179 447)(149 477 180 446)(150 476 181 445)(151 475 182 444)(152 474 183 443)(153 473 184 442)(154 472 185 441)(155 471 186 440)(187 397 218 428)(188 396 219 427)(189 395 220 426)(190 394 221 425)(191 393 222 424)(192 392 223 423)(193 391 224 422)(194 390 225 421)(195 389 226 420)(196 388 227 419)(197 387 228 418)(198 386 229 417)(199 385 230 416)(200 384 231 415)(201 383 232 414)(202 382 233 413)(203 381 234 412)(204 380 235 411)(205 379 236 410)(206 378 237 409)(207 377 238 408)(208 376 239 407)(209 375 240 406)(210 374 241 405)(211 373 242 404)(212 434 243 403)(213 433 244 402)(214 432 245 401)(215 431 246 400)(216 430 247 399)(217 429 248 398)
G:=sub<Sym(496)| (1,175,430,333)(2,176,431,334)(3,177,432,335)(4,178,433,336)(5,179,434,337)(6,180,373,338)(7,181,374,339)(8,182,375,340)(9,183,376,341)(10,184,377,342)(11,185,378,343)(12,186,379,344)(13,125,380,345)(14,126,381,346)(15,127,382,347)(16,128,383,348)(17,129,384,349)(18,130,385,350)(19,131,386,351)(20,132,387,352)(21,133,388,353)(22,134,389,354)(23,135,390,355)(24,136,391,356)(25,137,392,357)(26,138,393,358)(27,139,394,359)(28,140,395,360)(29,141,396,361)(30,142,397,362)(31,143,398,363)(32,144,399,364)(33,145,400,365)(34,146,401,366)(35,147,402,367)(36,148,403,368)(37,149,404,369)(38,150,405,370)(39,151,406,371)(40,152,407,372)(41,153,408,311)(42,154,409,312)(43,155,410,313)(44,156,411,314)(45,157,412,315)(46,158,413,316)(47,159,414,317)(48,160,415,318)(49,161,416,319)(50,162,417,320)(51,163,418,321)(52,164,419,322)(53,165,420,323)(54,166,421,324)(55,167,422,325)(56,168,423,326)(57,169,424,327)(58,170,425,328)(59,171,426,329)(60,172,427,330)(61,173,428,331)(62,174,429,332)(63,254,465,199)(64,255,466,200)(65,256,467,201)(66,257,468,202)(67,258,469,203)(68,259,470,204)(69,260,471,205)(70,261,472,206)(71,262,473,207)(72,263,474,208)(73,264,475,209)(74,265,476,210)(75,266,477,211)(76,267,478,212)(77,268,479,213)(78,269,480,214)(79,270,481,215)(80,271,482,216)(81,272,483,217)(82,273,484,218)(83,274,485,219)(84,275,486,220)(85,276,487,221)(86,277,488,222)(87,278,489,223)(88,279,490,224)(89,280,491,225)(90,281,492,226)(91,282,493,227)(92,283,494,228)(93,284,495,229)(94,285,496,230)(95,286,435,231)(96,287,436,232)(97,288,437,233)(98,289,438,234)(99,290,439,235)(100,291,440,236)(101,292,441,237)(102,293,442,238)(103,294,443,239)(104,295,444,240)(105,296,445,241)(106,297,446,242)(107,298,447,243)(108,299,448,244)(109,300,449,245)(110,301,450,246)(111,302,451,247)(112,303,452,248)(113,304,453,187)(114,305,454,188)(115,306,455,189)(116,307,456,190)(117,308,457,191)(118,309,458,192)(119,310,459,193)(120,249,460,194)(121,250,461,195)(122,251,462,196)(123,252,463,197)(124,253,464,198), (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), (1,302,32,271)(2,301,33,270)(3,300,34,269)(4,299,35,268)(5,298,36,267)(6,297,37,266)(7,296,38,265)(8,295,39,264)(9,294,40,263)(10,293,41,262)(11,292,42,261)(12,291,43,260)(13,290,44,259)(14,289,45,258)(15,288,46,257)(16,287,47,256)(17,286,48,255)(18,285,49,254)(19,284,50,253)(20,283,51,252)(21,282,52,251)(22,281,53,250)(23,280,54,249)(24,279,55,310)(25,278,56,309)(26,277,57,308)(27,276,58,307)(28,275,59,306)(29,274,60,305)(30,273,61,304)(31,272,62,303)(63,350,94,319)(64,349,95,318)(65,348,96,317)(66,347,97,316)(67,346,98,315)(68,345,99,314)(69,344,100,313)(70,343,101,312)(71,342,102,311)(72,341,103,372)(73,340,104,371)(74,339,105,370)(75,338,106,369)(76,337,107,368)(77,336,108,367)(78,335,109,366)(79,334,110,365)(80,333,111,364)(81,332,112,363)(82,331,113,362)(83,330,114,361)(84,329,115,360)(85,328,116,359)(86,327,117,358)(87,326,118,357)(88,325,119,356)(89,324,120,355)(90,323,121,354)(91,322,122,353)(92,321,123,352)(93,320,124,351)(125,439,156,470)(126,438,157,469)(127,437,158,468)(128,436,159,467)(129,435,160,466)(130,496,161,465)(131,495,162,464)(132,494,163,463)(133,493,164,462)(134,492,165,461)(135,491,166,460)(136,490,167,459)(137,489,168,458)(138,488,169,457)(139,487,170,456)(140,486,171,455)(141,485,172,454)(142,484,173,453)(143,483,174,452)(144,482,175,451)(145,481,176,450)(146,480,177,449)(147,479,178,448)(148,478,179,447)(149,477,180,446)(150,476,181,445)(151,475,182,444)(152,474,183,443)(153,473,184,442)(154,472,185,441)(155,471,186,440)(187,397,218,428)(188,396,219,427)(189,395,220,426)(190,394,221,425)(191,393,222,424)(192,392,223,423)(193,391,224,422)(194,390,225,421)(195,389,226,420)(196,388,227,419)(197,387,228,418)(198,386,229,417)(199,385,230,416)(200,384,231,415)(201,383,232,414)(202,382,233,413)(203,381,234,412)(204,380,235,411)(205,379,236,410)(206,378,237,409)(207,377,238,408)(208,376,239,407)(209,375,240,406)(210,374,241,405)(211,373,242,404)(212,434,243,403)(213,433,244,402)(214,432,245,401)(215,431,246,400)(216,430,247,399)(217,429,248,398)>;
G:=Group( (1,175,430,333)(2,176,431,334)(3,177,432,335)(4,178,433,336)(5,179,434,337)(6,180,373,338)(7,181,374,339)(8,182,375,340)(9,183,376,341)(10,184,377,342)(11,185,378,343)(12,186,379,344)(13,125,380,345)(14,126,381,346)(15,127,382,347)(16,128,383,348)(17,129,384,349)(18,130,385,350)(19,131,386,351)(20,132,387,352)(21,133,388,353)(22,134,389,354)(23,135,390,355)(24,136,391,356)(25,137,392,357)(26,138,393,358)(27,139,394,359)(28,140,395,360)(29,141,396,361)(30,142,397,362)(31,143,398,363)(32,144,399,364)(33,145,400,365)(34,146,401,366)(35,147,402,367)(36,148,403,368)(37,149,404,369)(38,150,405,370)(39,151,406,371)(40,152,407,372)(41,153,408,311)(42,154,409,312)(43,155,410,313)(44,156,411,314)(45,157,412,315)(46,158,413,316)(47,159,414,317)(48,160,415,318)(49,161,416,319)(50,162,417,320)(51,163,418,321)(52,164,419,322)(53,165,420,323)(54,166,421,324)(55,167,422,325)(56,168,423,326)(57,169,424,327)(58,170,425,328)(59,171,426,329)(60,172,427,330)(61,173,428,331)(62,174,429,332)(63,254,465,199)(64,255,466,200)(65,256,467,201)(66,257,468,202)(67,258,469,203)(68,259,470,204)(69,260,471,205)(70,261,472,206)(71,262,473,207)(72,263,474,208)(73,264,475,209)(74,265,476,210)(75,266,477,211)(76,267,478,212)(77,268,479,213)(78,269,480,214)(79,270,481,215)(80,271,482,216)(81,272,483,217)(82,273,484,218)(83,274,485,219)(84,275,486,220)(85,276,487,221)(86,277,488,222)(87,278,489,223)(88,279,490,224)(89,280,491,225)(90,281,492,226)(91,282,493,227)(92,283,494,228)(93,284,495,229)(94,285,496,230)(95,286,435,231)(96,287,436,232)(97,288,437,233)(98,289,438,234)(99,290,439,235)(100,291,440,236)(101,292,441,237)(102,293,442,238)(103,294,443,239)(104,295,444,240)(105,296,445,241)(106,297,446,242)(107,298,447,243)(108,299,448,244)(109,300,449,245)(110,301,450,246)(111,302,451,247)(112,303,452,248)(113,304,453,187)(114,305,454,188)(115,306,455,189)(116,307,456,190)(117,308,457,191)(118,309,458,192)(119,310,459,193)(120,249,460,194)(121,250,461,195)(122,251,462,196)(123,252,463,197)(124,253,464,198), (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), (1,302,32,271)(2,301,33,270)(3,300,34,269)(4,299,35,268)(5,298,36,267)(6,297,37,266)(7,296,38,265)(8,295,39,264)(9,294,40,263)(10,293,41,262)(11,292,42,261)(12,291,43,260)(13,290,44,259)(14,289,45,258)(15,288,46,257)(16,287,47,256)(17,286,48,255)(18,285,49,254)(19,284,50,253)(20,283,51,252)(21,282,52,251)(22,281,53,250)(23,280,54,249)(24,279,55,310)(25,278,56,309)(26,277,57,308)(27,276,58,307)(28,275,59,306)(29,274,60,305)(30,273,61,304)(31,272,62,303)(63,350,94,319)(64,349,95,318)(65,348,96,317)(66,347,97,316)(67,346,98,315)(68,345,99,314)(69,344,100,313)(70,343,101,312)(71,342,102,311)(72,341,103,372)(73,340,104,371)(74,339,105,370)(75,338,106,369)(76,337,107,368)(77,336,108,367)(78,335,109,366)(79,334,110,365)(80,333,111,364)(81,332,112,363)(82,331,113,362)(83,330,114,361)(84,329,115,360)(85,328,116,359)(86,327,117,358)(87,326,118,357)(88,325,119,356)(89,324,120,355)(90,323,121,354)(91,322,122,353)(92,321,123,352)(93,320,124,351)(125,439,156,470)(126,438,157,469)(127,437,158,468)(128,436,159,467)(129,435,160,466)(130,496,161,465)(131,495,162,464)(132,494,163,463)(133,493,164,462)(134,492,165,461)(135,491,166,460)(136,490,167,459)(137,489,168,458)(138,488,169,457)(139,487,170,456)(140,486,171,455)(141,485,172,454)(142,484,173,453)(143,483,174,452)(144,482,175,451)(145,481,176,450)(146,480,177,449)(147,479,178,448)(148,478,179,447)(149,477,180,446)(150,476,181,445)(151,475,182,444)(152,474,183,443)(153,473,184,442)(154,472,185,441)(155,471,186,440)(187,397,218,428)(188,396,219,427)(189,395,220,426)(190,394,221,425)(191,393,222,424)(192,392,223,423)(193,391,224,422)(194,390,225,421)(195,389,226,420)(196,388,227,419)(197,387,228,418)(198,386,229,417)(199,385,230,416)(200,384,231,415)(201,383,232,414)(202,382,233,413)(203,381,234,412)(204,380,235,411)(205,379,236,410)(206,378,237,409)(207,377,238,408)(208,376,239,407)(209,375,240,406)(210,374,241,405)(211,373,242,404)(212,434,243,403)(213,433,244,402)(214,432,245,401)(215,431,246,400)(216,430,247,399)(217,429,248,398) );
G=PermutationGroup([[(1,175,430,333),(2,176,431,334),(3,177,432,335),(4,178,433,336),(5,179,434,337),(6,180,373,338),(7,181,374,339),(8,182,375,340),(9,183,376,341),(10,184,377,342),(11,185,378,343),(12,186,379,344),(13,125,380,345),(14,126,381,346),(15,127,382,347),(16,128,383,348),(17,129,384,349),(18,130,385,350),(19,131,386,351),(20,132,387,352),(21,133,388,353),(22,134,389,354),(23,135,390,355),(24,136,391,356),(25,137,392,357),(26,138,393,358),(27,139,394,359),(28,140,395,360),(29,141,396,361),(30,142,397,362),(31,143,398,363),(32,144,399,364),(33,145,400,365),(34,146,401,366),(35,147,402,367),(36,148,403,368),(37,149,404,369),(38,150,405,370),(39,151,406,371),(40,152,407,372),(41,153,408,311),(42,154,409,312),(43,155,410,313),(44,156,411,314),(45,157,412,315),(46,158,413,316),(47,159,414,317),(48,160,415,318),(49,161,416,319),(50,162,417,320),(51,163,418,321),(52,164,419,322),(53,165,420,323),(54,166,421,324),(55,167,422,325),(56,168,423,326),(57,169,424,327),(58,170,425,328),(59,171,426,329),(60,172,427,330),(61,173,428,331),(62,174,429,332),(63,254,465,199),(64,255,466,200),(65,256,467,201),(66,257,468,202),(67,258,469,203),(68,259,470,204),(69,260,471,205),(70,261,472,206),(71,262,473,207),(72,263,474,208),(73,264,475,209),(74,265,476,210),(75,266,477,211),(76,267,478,212),(77,268,479,213),(78,269,480,214),(79,270,481,215),(80,271,482,216),(81,272,483,217),(82,273,484,218),(83,274,485,219),(84,275,486,220),(85,276,487,221),(86,277,488,222),(87,278,489,223),(88,279,490,224),(89,280,491,225),(90,281,492,226),(91,282,493,227),(92,283,494,228),(93,284,495,229),(94,285,496,230),(95,286,435,231),(96,287,436,232),(97,288,437,233),(98,289,438,234),(99,290,439,235),(100,291,440,236),(101,292,441,237),(102,293,442,238),(103,294,443,239),(104,295,444,240),(105,296,445,241),(106,297,446,242),(107,298,447,243),(108,299,448,244),(109,300,449,245),(110,301,450,246),(111,302,451,247),(112,303,452,248),(113,304,453,187),(114,305,454,188),(115,306,455,189),(116,307,456,190),(117,308,457,191),(118,309,458,192),(119,310,459,193),(120,249,460,194),(121,250,461,195),(122,251,462,196),(123,252,463,197),(124,253,464,198)], [(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)], [(1,302,32,271),(2,301,33,270),(3,300,34,269),(4,299,35,268),(5,298,36,267),(6,297,37,266),(7,296,38,265),(8,295,39,264),(9,294,40,263),(10,293,41,262),(11,292,42,261),(12,291,43,260),(13,290,44,259),(14,289,45,258),(15,288,46,257),(16,287,47,256),(17,286,48,255),(18,285,49,254),(19,284,50,253),(20,283,51,252),(21,282,52,251),(22,281,53,250),(23,280,54,249),(24,279,55,310),(25,278,56,309),(26,277,57,308),(27,276,58,307),(28,275,59,306),(29,274,60,305),(30,273,61,304),(31,272,62,303),(63,350,94,319),(64,349,95,318),(65,348,96,317),(66,347,97,316),(67,346,98,315),(68,345,99,314),(69,344,100,313),(70,343,101,312),(71,342,102,311),(72,341,103,372),(73,340,104,371),(74,339,105,370),(75,338,106,369),(76,337,107,368),(77,336,108,367),(78,335,109,366),(79,334,110,365),(80,333,111,364),(81,332,112,363),(82,331,113,362),(83,330,114,361),(84,329,115,360),(85,328,116,359),(86,327,117,358),(87,326,118,357),(88,325,119,356),(89,324,120,355),(90,323,121,354),(91,322,122,353),(92,321,123,352),(93,320,124,351),(125,439,156,470),(126,438,157,469),(127,437,158,468),(128,436,159,467),(129,435,160,466),(130,496,161,465),(131,495,162,464),(132,494,163,463),(133,493,164,462),(134,492,165,461),(135,491,166,460),(136,490,167,459),(137,489,168,458),(138,488,169,457),(139,487,170,456),(140,486,171,455),(141,485,172,454),(142,484,173,453),(143,483,174,452),(144,482,175,451),(145,481,176,450),(146,480,177,449),(147,479,178,448),(148,478,179,447),(149,477,180,446),(150,476,181,445),(151,475,182,444),(152,474,183,443),(153,473,184,442),(154,472,185,441),(155,471,186,440),(187,397,218,428),(188,396,219,427),(189,395,220,426),(190,394,221,425),(191,393,222,424),(192,392,223,423),(193,391,224,422),(194,390,225,421),(195,389,226,420),(196,388,227,419),(197,387,228,418),(198,386,229,417),(199,385,230,416),(200,384,231,415),(201,383,232,414),(202,382,233,413),(203,381,234,412),(204,380,235,411),(205,379,236,410),(206,378,237,409),(207,377,238,408),(208,376,239,407),(209,375,240,406),(210,374,241,405),(211,373,242,404),(212,434,243,403),(213,433,244,402),(214,432,245,401),(215,431,246,400),(216,430,247,399),(217,429,248,398)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 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 | 1 | 1 | 1 | 1 | 31 | ··· | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D31 | Dic31 | D62 | C4×D31 |
| kernel | C4×Dic31 | C2×Dic31 | C2×C124 | Dic31 | C124 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 15 | 30 | 15 | 60 |
Matrix representation of C4×Dic31 ►in GL3(𝔽373) generated by
| 269 | 0 | 0 |
| 0 | 372 | 0 |
| 0 | 0 | 372 |
| 1 | 0 | 0 |
| 0 | 0 | 372 |
| 0 | 1 | 19 |
| 1 | 0 | 0 |
| 0 | 281 | 35 |
| 0 | 291 | 92 |
G:=sub<GL(3,GF(373))| [269,0,0,0,372,0,0,0,372],[1,0,0,0,0,1,0,372,19],[1,0,0,0,281,291,0,35,92] >;
C4×Dic31 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{31} % in TeX
G:=Group("C4xDic31"); // GroupNames label
G:=SmallGroup(496,10);
// by ID
G=gap.SmallGroup(496,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,20,46,12004]);
// Polycyclic
G:=Group<a,b,c|a^4=b^62=1,c^2=b^31,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export