metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic31, C124⋊1C4, C62.4D4, C62.2Q8, C2.1D124, C2.2Dic62, C22.5D62, C31⋊2(C4⋊C4), C62.8(C2×C4), (C2×C4).3D31, (C2×C124).3C2, (C2×C62).5C22, C2.4(C2×Dic31), (C2×Dic31).2C2, SmallGroup(496,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic31
G = < a,b,c | a4=b62=1, c2=b31, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C4⋊Dic31
►Regular action on 496 points
Generators in S496
(1 214 369 148)(2 215 370 149)(3 216 371 150)(4 217 372 151)(5 218 311 152)(6 219 312 153)(7 220 313 154)(8 221 314 155)(9 222 315 156)(10 223 316 157)(11 224 317 158)(12 225 318 159)(13 226 319 160)(14 227 320 161)(15 228 321 162)(16 229 322 163)(17 230 323 164)(18 231 324 165)(19 232 325 166)(20 233 326 167)(21 234 327 168)(22 235 328 169)(23 236 329 170)(24 237 330 171)(25 238 331 172)(26 239 332 173)(27 240 333 174)(28 241 334 175)(29 242 335 176)(30 243 336 177)(31 244 337 178)(32 245 338 179)(33 246 339 180)(34 247 340 181)(35 248 341 182)(36 187 342 183)(37 188 343 184)(38 189 344 185)(39 190 345 186)(40 191 346 125)(41 192 347 126)(42 193 348 127)(43 194 349 128)(44 195 350 129)(45 196 351 130)(46 197 352 131)(47 198 353 132)(48 199 354 133)(49 200 355 134)(50 201 356 135)(51 202 357 136)(52 203 358 137)(53 204 359 138)(54 205 360 139)(55 206 361 140)(56 207 362 141)(57 208 363 142)(58 209 364 143)(59 210 365 144)(60 211 366 145)(61 212 367 146)(62 213 368 147)(63 408 295 451)(64 409 296 452)(65 410 297 453)(66 411 298 454)(67 412 299 455)(68 413 300 456)(69 414 301 457)(70 415 302 458)(71 416 303 459)(72 417 304 460)(73 418 305 461)(74 419 306 462)(75 420 307 463)(76 421 308 464)(77 422 309 465)(78 423 310 466)(79 424 249 467)(80 425 250 468)(81 426 251 469)(82 427 252 470)(83 428 253 471)(84 429 254 472)(85 430 255 473)(86 431 256 474)(87 432 257 475)(88 433 258 476)(89 434 259 477)(90 373 260 478)(91 374 261 479)(92 375 262 480)(93 376 263 481)(94 377 264 482)(95 378 265 483)(96 379 266 484)(97 380 267 485)(98 381 268 486)(99 382 269 487)(100 383 270 488)(101 384 271 489)(102 385 272 490)(103 386 273 491)(104 387 274 492)(105 388 275 493)(106 389 276 494)(107 390 277 495)(108 391 278 496)(109 392 279 435)(110 393 280 436)(111 394 281 437)(112 395 282 438)(113 396 283 439)(114 397 284 440)(115 398 285 441)(116 399 286 442)(117 400 287 443)(118 401 288 444)(119 402 289 445)(120 403 290 446)(121 404 291 447)(122 405 292 448)(123 406 293 449)(124 407 294 450)
(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)
(1 310 32 279)(2 309 33 278)(3 308 34 277)(4 307 35 276)(5 306 36 275)(6 305 37 274)(7 304 38 273)(8 303 39 272)(9 302 40 271)(10 301 41 270)(11 300 42 269)(12 299 43 268)(13 298 44 267)(14 297 45 266)(15 296 46 265)(16 295 47 264)(17 294 48 263)(18 293 49 262)(19 292 50 261)(20 291 51 260)(21 290 52 259)(22 289 53 258)(23 288 54 257)(24 287 55 256)(25 286 56 255)(26 285 57 254)(27 284 58 253)(28 283 59 252)(29 282 60 251)(30 281 61 250)(31 280 62 249)(63 353 94 322)(64 352 95 321)(65 351 96 320)(66 350 97 319)(67 349 98 318)(68 348 99 317)(69 347 100 316)(70 346 101 315)(71 345 102 314)(72 344 103 313)(73 343 104 312)(74 342 105 311)(75 341 106 372)(76 340 107 371)(77 339 108 370)(78 338 109 369)(79 337 110 368)(80 336 111 367)(81 335 112 366)(82 334 113 365)(83 333 114 364)(84 332 115 363)(85 331 116 362)(86 330 117 361)(87 329 118 360)(88 328 119 359)(89 327 120 358)(90 326 121 357)(91 325 122 356)(92 324 123 355)(93 323 124 354)(125 489 156 458)(126 488 157 457)(127 487 158 456)(128 486 159 455)(129 485 160 454)(130 484 161 453)(131 483 162 452)(132 482 163 451)(133 481 164 450)(134 480 165 449)(135 479 166 448)(136 478 167 447)(137 477 168 446)(138 476 169 445)(139 475 170 444)(140 474 171 443)(141 473 172 442)(142 472 173 441)(143 471 174 440)(144 470 175 439)(145 469 176 438)(146 468 177 437)(147 467 178 436)(148 466 179 435)(149 465 180 496)(150 464 181 495)(151 463 182 494)(152 462 183 493)(153 461 184 492)(154 460 185 491)(155 459 186 490)(187 388 218 419)(188 387 219 418)(189 386 220 417)(190 385 221 416)(191 384 222 415)(192 383 223 414)(193 382 224 413)(194 381 225 412)(195 380 226 411)(196 379 227 410)(197 378 228 409)(198 377 229 408)(199 376 230 407)(200 375 231 406)(201 374 232 405)(202 373 233 404)(203 434 234 403)(204 433 235 402)(205 432 236 401)(206 431 237 400)(207 430 238 399)(208 429 239 398)(209 428 240 397)(210 427 241 396)(211 426 242 395)(212 425 243 394)(213 424 244 393)(214 423 245 392)(215 422 246 391)(216 421 247 390)(217 420 248 389)
G:=sub<Sym(496)| (1,214,369,148)(2,215,370,149)(3,216,371,150)(4,217,372,151)(5,218,311,152)(6,219,312,153)(7,220,313,154)(8,221,314,155)(9,222,315,156)(10,223,316,157)(11,224,317,158)(12,225,318,159)(13,226,319,160)(14,227,320,161)(15,228,321,162)(16,229,322,163)(17,230,323,164)(18,231,324,165)(19,232,325,166)(20,233,326,167)(21,234,327,168)(22,235,328,169)(23,236,329,170)(24,237,330,171)(25,238,331,172)(26,239,332,173)(27,240,333,174)(28,241,334,175)(29,242,335,176)(30,243,336,177)(31,244,337,178)(32,245,338,179)(33,246,339,180)(34,247,340,181)(35,248,341,182)(36,187,342,183)(37,188,343,184)(38,189,344,185)(39,190,345,186)(40,191,346,125)(41,192,347,126)(42,193,348,127)(43,194,349,128)(44,195,350,129)(45,196,351,130)(46,197,352,131)(47,198,353,132)(48,199,354,133)(49,200,355,134)(50,201,356,135)(51,202,357,136)(52,203,358,137)(53,204,359,138)(54,205,360,139)(55,206,361,140)(56,207,362,141)(57,208,363,142)(58,209,364,143)(59,210,365,144)(60,211,366,145)(61,212,367,146)(62,213,368,147)(63,408,295,451)(64,409,296,452)(65,410,297,453)(66,411,298,454)(67,412,299,455)(68,413,300,456)(69,414,301,457)(70,415,302,458)(71,416,303,459)(72,417,304,460)(73,418,305,461)(74,419,306,462)(75,420,307,463)(76,421,308,464)(77,422,309,465)(78,423,310,466)(79,424,249,467)(80,425,250,468)(81,426,251,469)(82,427,252,470)(83,428,253,471)(84,429,254,472)(85,430,255,473)(86,431,256,474)(87,432,257,475)(88,433,258,476)(89,434,259,477)(90,373,260,478)(91,374,261,479)(92,375,262,480)(93,376,263,481)(94,377,264,482)(95,378,265,483)(96,379,266,484)(97,380,267,485)(98,381,268,486)(99,382,269,487)(100,383,270,488)(101,384,271,489)(102,385,272,490)(103,386,273,491)(104,387,274,492)(105,388,275,493)(106,389,276,494)(107,390,277,495)(108,391,278,496)(109,392,279,435)(110,393,280,436)(111,394,281,437)(112,395,282,438)(113,396,283,439)(114,397,284,440)(115,398,285,441)(116,399,286,442)(117,400,287,443)(118,401,288,444)(119,402,289,445)(120,403,290,446)(121,404,291,447)(122,405,292,448)(123,406,293,449)(124,407,294,450), (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), (1,310,32,279)(2,309,33,278)(3,308,34,277)(4,307,35,276)(5,306,36,275)(6,305,37,274)(7,304,38,273)(8,303,39,272)(9,302,40,271)(10,301,41,270)(11,300,42,269)(12,299,43,268)(13,298,44,267)(14,297,45,266)(15,296,46,265)(16,295,47,264)(17,294,48,263)(18,293,49,262)(19,292,50,261)(20,291,51,260)(21,290,52,259)(22,289,53,258)(23,288,54,257)(24,287,55,256)(25,286,56,255)(26,285,57,254)(27,284,58,253)(28,283,59,252)(29,282,60,251)(30,281,61,250)(31,280,62,249)(63,353,94,322)(64,352,95,321)(65,351,96,320)(66,350,97,319)(67,349,98,318)(68,348,99,317)(69,347,100,316)(70,346,101,315)(71,345,102,314)(72,344,103,313)(73,343,104,312)(74,342,105,311)(75,341,106,372)(76,340,107,371)(77,339,108,370)(78,338,109,369)(79,337,110,368)(80,336,111,367)(81,335,112,366)(82,334,113,365)(83,333,114,364)(84,332,115,363)(85,331,116,362)(86,330,117,361)(87,329,118,360)(88,328,119,359)(89,327,120,358)(90,326,121,357)(91,325,122,356)(92,324,123,355)(93,323,124,354)(125,489,156,458)(126,488,157,457)(127,487,158,456)(128,486,159,455)(129,485,160,454)(130,484,161,453)(131,483,162,452)(132,482,163,451)(133,481,164,450)(134,480,165,449)(135,479,166,448)(136,478,167,447)(137,477,168,446)(138,476,169,445)(139,475,170,444)(140,474,171,443)(141,473,172,442)(142,472,173,441)(143,471,174,440)(144,470,175,439)(145,469,176,438)(146,468,177,437)(147,467,178,436)(148,466,179,435)(149,465,180,496)(150,464,181,495)(151,463,182,494)(152,462,183,493)(153,461,184,492)(154,460,185,491)(155,459,186,490)(187,388,218,419)(188,387,219,418)(189,386,220,417)(190,385,221,416)(191,384,222,415)(192,383,223,414)(193,382,224,413)(194,381,225,412)(195,380,226,411)(196,379,227,410)(197,378,228,409)(198,377,229,408)(199,376,230,407)(200,375,231,406)(201,374,232,405)(202,373,233,404)(203,434,234,403)(204,433,235,402)(205,432,236,401)(206,431,237,400)(207,430,238,399)(208,429,239,398)(209,428,240,397)(210,427,241,396)(211,426,242,395)(212,425,243,394)(213,424,244,393)(214,423,245,392)(215,422,246,391)(216,421,247,390)(217,420,248,389)>;
G:=Group( (1,214,369,148)(2,215,370,149)(3,216,371,150)(4,217,372,151)(5,218,311,152)(6,219,312,153)(7,220,313,154)(8,221,314,155)(9,222,315,156)(10,223,316,157)(11,224,317,158)(12,225,318,159)(13,226,319,160)(14,227,320,161)(15,228,321,162)(16,229,322,163)(17,230,323,164)(18,231,324,165)(19,232,325,166)(20,233,326,167)(21,234,327,168)(22,235,328,169)(23,236,329,170)(24,237,330,171)(25,238,331,172)(26,239,332,173)(27,240,333,174)(28,241,334,175)(29,242,335,176)(30,243,336,177)(31,244,337,178)(32,245,338,179)(33,246,339,180)(34,247,340,181)(35,248,341,182)(36,187,342,183)(37,188,343,184)(38,189,344,185)(39,190,345,186)(40,191,346,125)(41,192,347,126)(42,193,348,127)(43,194,349,128)(44,195,350,129)(45,196,351,130)(46,197,352,131)(47,198,353,132)(48,199,354,133)(49,200,355,134)(50,201,356,135)(51,202,357,136)(52,203,358,137)(53,204,359,138)(54,205,360,139)(55,206,361,140)(56,207,362,141)(57,208,363,142)(58,209,364,143)(59,210,365,144)(60,211,366,145)(61,212,367,146)(62,213,368,147)(63,408,295,451)(64,409,296,452)(65,410,297,453)(66,411,298,454)(67,412,299,455)(68,413,300,456)(69,414,301,457)(70,415,302,458)(71,416,303,459)(72,417,304,460)(73,418,305,461)(74,419,306,462)(75,420,307,463)(76,421,308,464)(77,422,309,465)(78,423,310,466)(79,424,249,467)(80,425,250,468)(81,426,251,469)(82,427,252,470)(83,428,253,471)(84,429,254,472)(85,430,255,473)(86,431,256,474)(87,432,257,475)(88,433,258,476)(89,434,259,477)(90,373,260,478)(91,374,261,479)(92,375,262,480)(93,376,263,481)(94,377,264,482)(95,378,265,483)(96,379,266,484)(97,380,267,485)(98,381,268,486)(99,382,269,487)(100,383,270,488)(101,384,271,489)(102,385,272,490)(103,386,273,491)(104,387,274,492)(105,388,275,493)(106,389,276,494)(107,390,277,495)(108,391,278,496)(109,392,279,435)(110,393,280,436)(111,394,281,437)(112,395,282,438)(113,396,283,439)(114,397,284,440)(115,398,285,441)(116,399,286,442)(117,400,287,443)(118,401,288,444)(119,402,289,445)(120,403,290,446)(121,404,291,447)(122,405,292,448)(123,406,293,449)(124,407,294,450), (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), (1,310,32,279)(2,309,33,278)(3,308,34,277)(4,307,35,276)(5,306,36,275)(6,305,37,274)(7,304,38,273)(8,303,39,272)(9,302,40,271)(10,301,41,270)(11,300,42,269)(12,299,43,268)(13,298,44,267)(14,297,45,266)(15,296,46,265)(16,295,47,264)(17,294,48,263)(18,293,49,262)(19,292,50,261)(20,291,51,260)(21,290,52,259)(22,289,53,258)(23,288,54,257)(24,287,55,256)(25,286,56,255)(26,285,57,254)(27,284,58,253)(28,283,59,252)(29,282,60,251)(30,281,61,250)(31,280,62,249)(63,353,94,322)(64,352,95,321)(65,351,96,320)(66,350,97,319)(67,349,98,318)(68,348,99,317)(69,347,100,316)(70,346,101,315)(71,345,102,314)(72,344,103,313)(73,343,104,312)(74,342,105,311)(75,341,106,372)(76,340,107,371)(77,339,108,370)(78,338,109,369)(79,337,110,368)(80,336,111,367)(81,335,112,366)(82,334,113,365)(83,333,114,364)(84,332,115,363)(85,331,116,362)(86,330,117,361)(87,329,118,360)(88,328,119,359)(89,327,120,358)(90,326,121,357)(91,325,122,356)(92,324,123,355)(93,323,124,354)(125,489,156,458)(126,488,157,457)(127,487,158,456)(128,486,159,455)(129,485,160,454)(130,484,161,453)(131,483,162,452)(132,482,163,451)(133,481,164,450)(134,480,165,449)(135,479,166,448)(136,478,167,447)(137,477,168,446)(138,476,169,445)(139,475,170,444)(140,474,171,443)(141,473,172,442)(142,472,173,441)(143,471,174,440)(144,470,175,439)(145,469,176,438)(146,468,177,437)(147,467,178,436)(148,466,179,435)(149,465,180,496)(150,464,181,495)(151,463,182,494)(152,462,183,493)(153,461,184,492)(154,460,185,491)(155,459,186,490)(187,388,218,419)(188,387,219,418)(189,386,220,417)(190,385,221,416)(191,384,222,415)(192,383,223,414)(193,382,224,413)(194,381,225,412)(195,380,226,411)(196,379,227,410)(197,378,228,409)(198,377,229,408)(199,376,230,407)(200,375,231,406)(201,374,232,405)(202,373,233,404)(203,434,234,403)(204,433,235,402)(205,432,236,401)(206,431,237,400)(207,430,238,399)(208,429,239,398)(209,428,240,397)(210,427,241,396)(211,426,242,395)(212,425,243,394)(213,424,244,393)(214,423,245,392)(215,422,246,391)(216,421,247,390)(217,420,248,389) );
G=PermutationGroup([[(1,214,369,148),(2,215,370,149),(3,216,371,150),(4,217,372,151),(5,218,311,152),(6,219,312,153),(7,220,313,154),(8,221,314,155),(9,222,315,156),(10,223,316,157),(11,224,317,158),(12,225,318,159),(13,226,319,160),(14,227,320,161),(15,228,321,162),(16,229,322,163),(17,230,323,164),(18,231,324,165),(19,232,325,166),(20,233,326,167),(21,234,327,168),(22,235,328,169),(23,236,329,170),(24,237,330,171),(25,238,331,172),(26,239,332,173),(27,240,333,174),(28,241,334,175),(29,242,335,176),(30,243,336,177),(31,244,337,178),(32,245,338,179),(33,246,339,180),(34,247,340,181),(35,248,341,182),(36,187,342,183),(37,188,343,184),(38,189,344,185),(39,190,345,186),(40,191,346,125),(41,192,347,126),(42,193,348,127),(43,194,349,128),(44,195,350,129),(45,196,351,130),(46,197,352,131),(47,198,353,132),(48,199,354,133),(49,200,355,134),(50,201,356,135),(51,202,357,136),(52,203,358,137),(53,204,359,138),(54,205,360,139),(55,206,361,140),(56,207,362,141),(57,208,363,142),(58,209,364,143),(59,210,365,144),(60,211,366,145),(61,212,367,146),(62,213,368,147),(63,408,295,451),(64,409,296,452),(65,410,297,453),(66,411,298,454),(67,412,299,455),(68,413,300,456),(69,414,301,457),(70,415,302,458),(71,416,303,459),(72,417,304,460),(73,418,305,461),(74,419,306,462),(75,420,307,463),(76,421,308,464),(77,422,309,465),(78,423,310,466),(79,424,249,467),(80,425,250,468),(81,426,251,469),(82,427,252,470),(83,428,253,471),(84,429,254,472),(85,430,255,473),(86,431,256,474),(87,432,257,475),(88,433,258,476),(89,434,259,477),(90,373,260,478),(91,374,261,479),(92,375,262,480),(93,376,263,481),(94,377,264,482),(95,378,265,483),(96,379,266,484),(97,380,267,485),(98,381,268,486),(99,382,269,487),(100,383,270,488),(101,384,271,489),(102,385,272,490),(103,386,273,491),(104,387,274,492),(105,388,275,493),(106,389,276,494),(107,390,277,495),(108,391,278,496),(109,392,279,435),(110,393,280,436),(111,394,281,437),(112,395,282,438),(113,396,283,439),(114,397,284,440),(115,398,285,441),(116,399,286,442),(117,400,287,443),(118,401,288,444),(119,402,289,445),(120,403,290,446),(121,404,291,447),(122,405,292,448),(123,406,293,449),(124,407,294,450)], [(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)], [(1,310,32,279),(2,309,33,278),(3,308,34,277),(4,307,35,276),(5,306,36,275),(6,305,37,274),(7,304,38,273),(8,303,39,272),(9,302,40,271),(10,301,41,270),(11,300,42,269),(12,299,43,268),(13,298,44,267),(14,297,45,266),(15,296,46,265),(16,295,47,264),(17,294,48,263),(18,293,49,262),(19,292,50,261),(20,291,51,260),(21,290,52,259),(22,289,53,258),(23,288,54,257),(24,287,55,256),(25,286,56,255),(26,285,57,254),(27,284,58,253),(28,283,59,252),(29,282,60,251),(30,281,61,250),(31,280,62,249),(63,353,94,322),(64,352,95,321),(65,351,96,320),(66,350,97,319),(67,349,98,318),(68,348,99,317),(69,347,100,316),(70,346,101,315),(71,345,102,314),(72,344,103,313),(73,343,104,312),(74,342,105,311),(75,341,106,372),(76,340,107,371),(77,339,108,370),(78,338,109,369),(79,337,110,368),(80,336,111,367),(81,335,112,366),(82,334,113,365),(83,333,114,364),(84,332,115,363),(85,331,116,362),(86,330,117,361),(87,329,118,360),(88,328,119,359),(89,327,120,358),(90,326,121,357),(91,325,122,356),(92,324,123,355),(93,323,124,354),(125,489,156,458),(126,488,157,457),(127,487,158,456),(128,486,159,455),(129,485,160,454),(130,484,161,453),(131,483,162,452),(132,482,163,451),(133,481,164,450),(134,480,165,449),(135,479,166,448),(136,478,167,447),(137,477,168,446),(138,476,169,445),(139,475,170,444),(140,474,171,443),(141,473,172,442),(142,472,173,441),(143,471,174,440),(144,470,175,439),(145,469,176,438),(146,468,177,437),(147,467,178,436),(148,466,179,435),(149,465,180,496),(150,464,181,495),(151,463,182,494),(152,462,183,493),(153,461,184,492),(154,460,185,491),(155,459,186,490),(187,388,218,419),(188,387,219,418),(189,386,220,417),(190,385,221,416),(191,384,222,415),(192,383,223,414),(193,382,224,413),(194,381,225,412),(195,380,226,411),(196,379,227,410),(197,378,228,409),(198,377,229,408),(199,376,230,407),(200,375,231,406),(201,374,232,405),(202,373,233,404),(203,434,234,403),(204,433,235,402),(205,432,236,401),(206,431,237,400),(207,430,238,399),(208,429,239,398),(209,428,240,397),(210,427,241,396),(211,426,242,395),(212,425,243,394),(213,424,244,393),(214,423,245,392),(215,422,246,391),(216,421,247,390),(217,420,248,389)]])
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 | Dic31 | D62 | Dic62 | D124 |
| kernel | C4⋊Dic31 | C2×Dic31 | C2×C124 | C124 | C62 | C62 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 15 | 30 | 15 | 30 | 30 |
Matrix representation of C4⋊Dic31 ►in GL3(𝔽373) generated by
| 372 | 0 | 0 |
| 0 | 187 | 94 |
| 0 | 247 | 186 |
| 372 | 0 | 0 |
| 0 | 228 | 1 |
| 0 | 205 | 351 |
| 104 | 0 | 0 |
| 0 | 135 | 315 |
| 0 | 147 | 238 |
G:=sub<GL(3,GF(373))| [372,0,0,0,187,247,0,94,186],[372,0,0,0,228,205,0,1,351],[104,0,0,0,135,147,0,315,238] >;
C4⋊Dic31 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_{31} % in TeX
G:=Group("C4:Dic31"); // GroupNames label
G:=SmallGroup(496,12);
// by ID
G=gap.SmallGroup(496,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,20,101,46,12004]);
// Polycyclic
G:=Group<a,b,c|a^4=b^62=1,c^2=b^31,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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