direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C31, C8.C62, Q8.C62, C248.3C2, C62.16D4, C124.19C22, C4.3(C2×C62), C2.5(D4×C31), (Q8×C31).2C2, SmallGroup(496,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C31
G = < a,b,c | a31=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q16×C31
►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)
(1 356 416 300 175 195 124 375)(2 357 417 301 176 196 94 376)(3 358 418 302 177 197 95 377)(4 359 419 303 178 198 96 378)(5 360 420 304 179 199 97 379)(6 361 421 305 180 200 98 380)(7 362 422 306 181 201 99 381)(8 363 423 307 182 202 100 382)(9 364 424 308 183 203 101 383)(10 365 425 309 184 204 102 384)(11 366 426 310 185 205 103 385)(12 367 427 280 186 206 104 386)(13 368 428 281 156 207 105 387)(14 369 429 282 157 208 106 388)(15 370 430 283 158 209 107 389)(16 371 431 284 159 210 108 390)(17 372 432 285 160 211 109 391)(18 342 433 286 161 212 110 392)(19 343 434 287 162 213 111 393)(20 344 404 288 163 214 112 394)(21 345 405 289 164 215 113 395)(22 346 406 290 165 216 114 396)(23 347 407 291 166 217 115 397)(24 348 408 292 167 187 116 398)(25 349 409 293 168 188 117 399)(26 350 410 294 169 189 118 400)(27 351 411 295 170 190 119 401)(28 352 412 296 171 191 120 402)(29 353 413 297 172 192 121 403)(30 354 414 298 173 193 122 373)(31 355 415 299 174 194 123 374)(32 136 223 340 487 441 93 268)(33 137 224 341 488 442 63 269)(34 138 225 311 489 443 64 270)(35 139 226 312 490 444 65 271)(36 140 227 313 491 445 66 272)(37 141 228 314 492 446 67 273)(38 142 229 315 493 447 68 274)(39 143 230 316 494 448 69 275)(40 144 231 317 495 449 70 276)(41 145 232 318 496 450 71 277)(42 146 233 319 466 451 72 278)(43 147 234 320 467 452 73 279)(44 148 235 321 468 453 74 249)(45 149 236 322 469 454 75 250)(46 150 237 323 470 455 76 251)(47 151 238 324 471 456 77 252)(48 152 239 325 472 457 78 253)(49 153 240 326 473 458 79 254)(50 154 241 327 474 459 80 255)(51 155 242 328 475 460 81 256)(52 125 243 329 476 461 82 257)(53 126 244 330 477 462 83 258)(54 127 245 331 478 463 84 259)(55 128 246 332 479 464 85 260)(56 129 247 333 480 465 86 261)(57 130 248 334 481 435 87 262)(58 131 218 335 482 436 88 263)(59 132 219 336 483 437 89 264)(60 133 220 337 484 438 90 265)(61 134 221 338 485 439 91 266)(62 135 222 339 486 440 92 267)
(1 46 175 470)(2 47 176 471)(3 48 177 472)(4 49 178 473)(5 50 179 474)(6 51 180 475)(7 52 181 476)(8 53 182 477)(9 54 183 478)(10 55 184 479)(11 56 185 480)(12 57 186 481)(13 58 156 482)(14 59 157 483)(15 60 158 484)(16 61 159 485)(17 62 160 486)(18 32 161 487)(19 33 162 488)(20 34 163 489)(21 35 164 490)(22 36 165 491)(23 37 166 492)(24 38 167 493)(25 39 168 494)(26 40 169 495)(27 41 170 496)(28 42 171 466)(29 43 172 467)(30 44 173 468)(31 45 174 469)(63 111 224 434)(64 112 225 404)(65 113 226 405)(66 114 227 406)(67 115 228 407)(68 116 229 408)(69 117 230 409)(70 118 231 410)(71 119 232 411)(72 120 233 412)(73 121 234 413)(74 122 235 414)(75 123 236 415)(76 124 237 416)(77 94 238 417)(78 95 239 418)(79 96 240 419)(80 97 241 420)(81 98 242 421)(82 99 243 422)(83 100 244 423)(84 101 245 424)(85 102 246 425)(86 103 247 426)(87 104 248 427)(88 105 218 428)(89 106 219 429)(90 107 220 430)(91 108 221 431)(92 109 222 432)(93 110 223 433)(125 306 461 381)(126 307 462 382)(127 308 463 383)(128 309 464 384)(129 310 465 385)(130 280 435 386)(131 281 436 387)(132 282 437 388)(133 283 438 389)(134 284 439 390)(135 285 440 391)(136 286 441 392)(137 287 442 393)(138 288 443 394)(139 289 444 395)(140 290 445 396)(141 291 446 397)(142 292 447 398)(143 293 448 399)(144 294 449 400)(145 295 450 401)(146 296 451 402)(147 297 452 403)(148 298 453 373)(149 299 454 374)(150 300 455 375)(151 301 456 376)(152 302 457 377)(153 303 458 378)(154 304 459 379)(155 305 460 380)(187 315 348 274)(188 316 349 275)(189 317 350 276)(190 318 351 277)(191 319 352 278)(192 320 353 279)(193 321 354 249)(194 322 355 250)(195 323 356 251)(196 324 357 252)(197 325 358 253)(198 326 359 254)(199 327 360 255)(200 328 361 256)(201 329 362 257)(202 330 363 258)(203 331 364 259)(204 332 365 260)(205 333 366 261)(206 334 367 262)(207 335 368 263)(208 336 369 264)(209 337 370 265)(210 338 371 266)(211 339 372 267)(212 340 342 268)(213 341 343 269)(214 311 344 270)(215 312 345 271)(216 313 346 272)(217 314 347 273)
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), (1,356,416,300,175,195,124,375)(2,357,417,301,176,196,94,376)(3,358,418,302,177,197,95,377)(4,359,419,303,178,198,96,378)(5,360,420,304,179,199,97,379)(6,361,421,305,180,200,98,380)(7,362,422,306,181,201,99,381)(8,363,423,307,182,202,100,382)(9,364,424,308,183,203,101,383)(10,365,425,309,184,204,102,384)(11,366,426,310,185,205,103,385)(12,367,427,280,186,206,104,386)(13,368,428,281,156,207,105,387)(14,369,429,282,157,208,106,388)(15,370,430,283,158,209,107,389)(16,371,431,284,159,210,108,390)(17,372,432,285,160,211,109,391)(18,342,433,286,161,212,110,392)(19,343,434,287,162,213,111,393)(20,344,404,288,163,214,112,394)(21,345,405,289,164,215,113,395)(22,346,406,290,165,216,114,396)(23,347,407,291,166,217,115,397)(24,348,408,292,167,187,116,398)(25,349,409,293,168,188,117,399)(26,350,410,294,169,189,118,400)(27,351,411,295,170,190,119,401)(28,352,412,296,171,191,120,402)(29,353,413,297,172,192,121,403)(30,354,414,298,173,193,122,373)(31,355,415,299,174,194,123,374)(32,136,223,340,487,441,93,268)(33,137,224,341,488,442,63,269)(34,138,225,311,489,443,64,270)(35,139,226,312,490,444,65,271)(36,140,227,313,491,445,66,272)(37,141,228,314,492,446,67,273)(38,142,229,315,493,447,68,274)(39,143,230,316,494,448,69,275)(40,144,231,317,495,449,70,276)(41,145,232,318,496,450,71,277)(42,146,233,319,466,451,72,278)(43,147,234,320,467,452,73,279)(44,148,235,321,468,453,74,249)(45,149,236,322,469,454,75,250)(46,150,237,323,470,455,76,251)(47,151,238,324,471,456,77,252)(48,152,239,325,472,457,78,253)(49,153,240,326,473,458,79,254)(50,154,241,327,474,459,80,255)(51,155,242,328,475,460,81,256)(52,125,243,329,476,461,82,257)(53,126,244,330,477,462,83,258)(54,127,245,331,478,463,84,259)(55,128,246,332,479,464,85,260)(56,129,247,333,480,465,86,261)(57,130,248,334,481,435,87,262)(58,131,218,335,482,436,88,263)(59,132,219,336,483,437,89,264)(60,133,220,337,484,438,90,265)(61,134,221,338,485,439,91,266)(62,135,222,339,486,440,92,267), (1,46,175,470)(2,47,176,471)(3,48,177,472)(4,49,178,473)(5,50,179,474)(6,51,180,475)(7,52,181,476)(8,53,182,477)(9,54,183,478)(10,55,184,479)(11,56,185,480)(12,57,186,481)(13,58,156,482)(14,59,157,483)(15,60,158,484)(16,61,159,485)(17,62,160,486)(18,32,161,487)(19,33,162,488)(20,34,163,489)(21,35,164,490)(22,36,165,491)(23,37,166,492)(24,38,167,493)(25,39,168,494)(26,40,169,495)(27,41,170,496)(28,42,171,466)(29,43,172,467)(30,44,173,468)(31,45,174,469)(63,111,224,434)(64,112,225,404)(65,113,226,405)(66,114,227,406)(67,115,228,407)(68,116,229,408)(69,117,230,409)(70,118,231,410)(71,119,232,411)(72,120,233,412)(73,121,234,413)(74,122,235,414)(75,123,236,415)(76,124,237,416)(77,94,238,417)(78,95,239,418)(79,96,240,419)(80,97,241,420)(81,98,242,421)(82,99,243,422)(83,100,244,423)(84,101,245,424)(85,102,246,425)(86,103,247,426)(87,104,248,427)(88,105,218,428)(89,106,219,429)(90,107,220,430)(91,108,221,431)(92,109,222,432)(93,110,223,433)(125,306,461,381)(126,307,462,382)(127,308,463,383)(128,309,464,384)(129,310,465,385)(130,280,435,386)(131,281,436,387)(132,282,437,388)(133,283,438,389)(134,284,439,390)(135,285,440,391)(136,286,441,392)(137,287,442,393)(138,288,443,394)(139,289,444,395)(140,290,445,396)(141,291,446,397)(142,292,447,398)(143,293,448,399)(144,294,449,400)(145,295,450,401)(146,296,451,402)(147,297,452,403)(148,298,453,373)(149,299,454,374)(150,300,455,375)(151,301,456,376)(152,302,457,377)(153,303,458,378)(154,304,459,379)(155,305,460,380)(187,315,348,274)(188,316,349,275)(189,317,350,276)(190,318,351,277)(191,319,352,278)(192,320,353,279)(193,321,354,249)(194,322,355,250)(195,323,356,251)(196,324,357,252)(197,325,358,253)(198,326,359,254)(199,327,360,255)(200,328,361,256)(201,329,362,257)(202,330,363,258)(203,331,364,259)(204,332,365,260)(205,333,366,261)(206,334,367,262)(207,335,368,263)(208,336,369,264)(209,337,370,265)(210,338,371,266)(211,339,372,267)(212,340,342,268)(213,341,343,269)(214,311,344,270)(215,312,345,271)(216,313,346,272)(217,314,347,273)>;
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), (1,356,416,300,175,195,124,375)(2,357,417,301,176,196,94,376)(3,358,418,302,177,197,95,377)(4,359,419,303,178,198,96,378)(5,360,420,304,179,199,97,379)(6,361,421,305,180,200,98,380)(7,362,422,306,181,201,99,381)(8,363,423,307,182,202,100,382)(9,364,424,308,183,203,101,383)(10,365,425,309,184,204,102,384)(11,366,426,310,185,205,103,385)(12,367,427,280,186,206,104,386)(13,368,428,281,156,207,105,387)(14,369,429,282,157,208,106,388)(15,370,430,283,158,209,107,389)(16,371,431,284,159,210,108,390)(17,372,432,285,160,211,109,391)(18,342,433,286,161,212,110,392)(19,343,434,287,162,213,111,393)(20,344,404,288,163,214,112,394)(21,345,405,289,164,215,113,395)(22,346,406,290,165,216,114,396)(23,347,407,291,166,217,115,397)(24,348,408,292,167,187,116,398)(25,349,409,293,168,188,117,399)(26,350,410,294,169,189,118,400)(27,351,411,295,170,190,119,401)(28,352,412,296,171,191,120,402)(29,353,413,297,172,192,121,403)(30,354,414,298,173,193,122,373)(31,355,415,299,174,194,123,374)(32,136,223,340,487,441,93,268)(33,137,224,341,488,442,63,269)(34,138,225,311,489,443,64,270)(35,139,226,312,490,444,65,271)(36,140,227,313,491,445,66,272)(37,141,228,314,492,446,67,273)(38,142,229,315,493,447,68,274)(39,143,230,316,494,448,69,275)(40,144,231,317,495,449,70,276)(41,145,232,318,496,450,71,277)(42,146,233,319,466,451,72,278)(43,147,234,320,467,452,73,279)(44,148,235,321,468,453,74,249)(45,149,236,322,469,454,75,250)(46,150,237,323,470,455,76,251)(47,151,238,324,471,456,77,252)(48,152,239,325,472,457,78,253)(49,153,240,326,473,458,79,254)(50,154,241,327,474,459,80,255)(51,155,242,328,475,460,81,256)(52,125,243,329,476,461,82,257)(53,126,244,330,477,462,83,258)(54,127,245,331,478,463,84,259)(55,128,246,332,479,464,85,260)(56,129,247,333,480,465,86,261)(57,130,248,334,481,435,87,262)(58,131,218,335,482,436,88,263)(59,132,219,336,483,437,89,264)(60,133,220,337,484,438,90,265)(61,134,221,338,485,439,91,266)(62,135,222,339,486,440,92,267), (1,46,175,470)(2,47,176,471)(3,48,177,472)(4,49,178,473)(5,50,179,474)(6,51,180,475)(7,52,181,476)(8,53,182,477)(9,54,183,478)(10,55,184,479)(11,56,185,480)(12,57,186,481)(13,58,156,482)(14,59,157,483)(15,60,158,484)(16,61,159,485)(17,62,160,486)(18,32,161,487)(19,33,162,488)(20,34,163,489)(21,35,164,490)(22,36,165,491)(23,37,166,492)(24,38,167,493)(25,39,168,494)(26,40,169,495)(27,41,170,496)(28,42,171,466)(29,43,172,467)(30,44,173,468)(31,45,174,469)(63,111,224,434)(64,112,225,404)(65,113,226,405)(66,114,227,406)(67,115,228,407)(68,116,229,408)(69,117,230,409)(70,118,231,410)(71,119,232,411)(72,120,233,412)(73,121,234,413)(74,122,235,414)(75,123,236,415)(76,124,237,416)(77,94,238,417)(78,95,239,418)(79,96,240,419)(80,97,241,420)(81,98,242,421)(82,99,243,422)(83,100,244,423)(84,101,245,424)(85,102,246,425)(86,103,247,426)(87,104,248,427)(88,105,218,428)(89,106,219,429)(90,107,220,430)(91,108,221,431)(92,109,222,432)(93,110,223,433)(125,306,461,381)(126,307,462,382)(127,308,463,383)(128,309,464,384)(129,310,465,385)(130,280,435,386)(131,281,436,387)(132,282,437,388)(133,283,438,389)(134,284,439,390)(135,285,440,391)(136,286,441,392)(137,287,442,393)(138,288,443,394)(139,289,444,395)(140,290,445,396)(141,291,446,397)(142,292,447,398)(143,293,448,399)(144,294,449,400)(145,295,450,401)(146,296,451,402)(147,297,452,403)(148,298,453,373)(149,299,454,374)(150,300,455,375)(151,301,456,376)(152,302,457,377)(153,303,458,378)(154,304,459,379)(155,305,460,380)(187,315,348,274)(188,316,349,275)(189,317,350,276)(190,318,351,277)(191,319,352,278)(192,320,353,279)(193,321,354,249)(194,322,355,250)(195,323,356,251)(196,324,357,252)(197,325,358,253)(198,326,359,254)(199,327,360,255)(200,328,361,256)(201,329,362,257)(202,330,363,258)(203,331,364,259)(204,332,365,260)(205,333,366,261)(206,334,367,262)(207,335,368,263)(208,336,369,264)(209,337,370,265)(210,338,371,266)(211,339,372,267)(212,340,342,268)(213,341,343,269)(214,311,344,270)(215,312,345,271)(216,313,346,272)(217,314,347,273) );
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)], [(1,356,416,300,175,195,124,375),(2,357,417,301,176,196,94,376),(3,358,418,302,177,197,95,377),(4,359,419,303,178,198,96,378),(5,360,420,304,179,199,97,379),(6,361,421,305,180,200,98,380),(7,362,422,306,181,201,99,381),(8,363,423,307,182,202,100,382),(9,364,424,308,183,203,101,383),(10,365,425,309,184,204,102,384),(11,366,426,310,185,205,103,385),(12,367,427,280,186,206,104,386),(13,368,428,281,156,207,105,387),(14,369,429,282,157,208,106,388),(15,370,430,283,158,209,107,389),(16,371,431,284,159,210,108,390),(17,372,432,285,160,211,109,391),(18,342,433,286,161,212,110,392),(19,343,434,287,162,213,111,393),(20,344,404,288,163,214,112,394),(21,345,405,289,164,215,113,395),(22,346,406,290,165,216,114,396),(23,347,407,291,166,217,115,397),(24,348,408,292,167,187,116,398),(25,349,409,293,168,188,117,399),(26,350,410,294,169,189,118,400),(27,351,411,295,170,190,119,401),(28,352,412,296,171,191,120,402),(29,353,413,297,172,192,121,403),(30,354,414,298,173,193,122,373),(31,355,415,299,174,194,123,374),(32,136,223,340,487,441,93,268),(33,137,224,341,488,442,63,269),(34,138,225,311,489,443,64,270),(35,139,226,312,490,444,65,271),(36,140,227,313,491,445,66,272),(37,141,228,314,492,446,67,273),(38,142,229,315,493,447,68,274),(39,143,230,316,494,448,69,275),(40,144,231,317,495,449,70,276),(41,145,232,318,496,450,71,277),(42,146,233,319,466,451,72,278),(43,147,234,320,467,452,73,279),(44,148,235,321,468,453,74,249),(45,149,236,322,469,454,75,250),(46,150,237,323,470,455,76,251),(47,151,238,324,471,456,77,252),(48,152,239,325,472,457,78,253),(49,153,240,326,473,458,79,254),(50,154,241,327,474,459,80,255),(51,155,242,328,475,460,81,256),(52,125,243,329,476,461,82,257),(53,126,244,330,477,462,83,258),(54,127,245,331,478,463,84,259),(55,128,246,332,479,464,85,260),(56,129,247,333,480,465,86,261),(57,130,248,334,481,435,87,262),(58,131,218,335,482,436,88,263),(59,132,219,336,483,437,89,264),(60,133,220,337,484,438,90,265),(61,134,221,338,485,439,91,266),(62,135,222,339,486,440,92,267)], [(1,46,175,470),(2,47,176,471),(3,48,177,472),(4,49,178,473),(5,50,179,474),(6,51,180,475),(7,52,181,476),(8,53,182,477),(9,54,183,478),(10,55,184,479),(11,56,185,480),(12,57,186,481),(13,58,156,482),(14,59,157,483),(15,60,158,484),(16,61,159,485),(17,62,160,486),(18,32,161,487),(19,33,162,488),(20,34,163,489),(21,35,164,490),(22,36,165,491),(23,37,166,492),(24,38,167,493),(25,39,168,494),(26,40,169,495),(27,41,170,496),(28,42,171,466),(29,43,172,467),(30,44,173,468),(31,45,174,469),(63,111,224,434),(64,112,225,404),(65,113,226,405),(66,114,227,406),(67,115,228,407),(68,116,229,408),(69,117,230,409),(70,118,231,410),(71,119,232,411),(72,120,233,412),(73,121,234,413),(74,122,235,414),(75,123,236,415),(76,124,237,416),(77,94,238,417),(78,95,239,418),(79,96,240,419),(80,97,241,420),(81,98,242,421),(82,99,243,422),(83,100,244,423),(84,101,245,424),(85,102,246,425),(86,103,247,426),(87,104,248,427),(88,105,218,428),(89,106,219,429),(90,107,220,430),(91,108,221,431),(92,109,222,432),(93,110,223,433),(125,306,461,381),(126,307,462,382),(127,308,463,383),(128,309,464,384),(129,310,465,385),(130,280,435,386),(131,281,436,387),(132,282,437,388),(133,283,438,389),(134,284,439,390),(135,285,440,391),(136,286,441,392),(137,287,442,393),(138,288,443,394),(139,289,444,395),(140,290,445,396),(141,291,446,397),(142,292,447,398),(143,293,448,399),(144,294,449,400),(145,295,450,401),(146,296,451,402),(147,297,452,403),(148,298,453,373),(149,299,454,374),(150,300,455,375),(151,301,456,376),(152,302,457,377),(153,303,458,378),(154,304,459,379),(155,305,460,380),(187,315,348,274),(188,316,349,275),(189,317,350,276),(190,318,351,277),(191,319,352,278),(192,320,353,279),(193,321,354,249),(194,322,355,250),(195,323,356,251),(196,324,357,252),(197,325,358,253),(198,326,359,254),(199,327,360,255),(200,328,361,256),(201,329,362,257),(202,330,363,258),(203,331,364,259),(204,332,365,260),(205,333,366,261),(206,334,367,262),(207,335,368,263),(208,336,369,264),(209,337,370,265),(210,338,371,266),(211,339,372,267),(212,340,342,268),(213,341,343,269),(214,311,344,270),(215,312,345,271),(216,313,346,272),(217,314,347,273)]])
217 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 31A | ··· | 31AD | 62A | ··· | 62AD | 124A | ··· | 124AD | 124AE | ··· | 124CL | 248A | ··· | 248BH |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
217 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C31 | C62 | C62 | D4 | Q16 | D4×C31 | Q16×C31 |
| kernel | Q16×C31 | C248 | Q8×C31 | Q16 | C8 | Q8 | C62 | C31 | C2 | C1 |
| # reps | 1 | 1 | 2 | 30 | 30 | 60 | 1 | 2 | 30 | 60 |
Matrix representation of Q16×C31 ►in GL2(𝔽1489) generated by
| 1254 | 0 |
| 0 | 1254 |
| 0 | 1107 |
| 191 | 1107 |
| 685 | 513 |
| 197 | 804 |
G:=sub<GL(2,GF(1489))| [1254,0,0,1254],[0,191,1107,1107],[685,197,513,804] >;
Q16×C31 in GAP, Magma, Sage, TeX
Q_{16}\times C_{31} % in TeX
G:=Group("Q16xC31"); // GroupNames label
G:=SmallGroup(496,26);
// by ID
G=gap.SmallGroup(496,26);
# by ID
G:=PCGroup([5,-2,-2,-31,-2,-2,1240,1261,1246,7443,3728,58]);
// Polycyclic
G:=Group<a,b,c|a^31=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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