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G = Q16×C31order 496 = 24·31

Direct product of C31 and Q16

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

C1C4 — Q16×C31
C1C2C4C124Q8×C31 — Q16×C31
C1C2C4 — Q16×C31
C1C62C124 — Q16×C31

Generators and relations for Q16×C31
 G = < a,b,c | a31=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C124
2C124

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

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

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

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

217 conjugacy classes

class 1  2 4A4B4C8A8B31A···31AD62A···62AD124A···124AD124AE···124CL248A···248BH
order124448831···3162···62124···124124···124248···248
size11244221···11···12···24···42···2

217 irreducible representations

dim1111112222
type++++-
imageC1C2C2C31C62C62D4Q16D4×C31Q16×C31
kernelQ16×C31C248Q8×C31Q16C8Q8C62C31C2C1
# reps112303060123060

Matrix representation of Q16×C31 in GL2(𝔽1489) generated by

12540
01254
,
01107
1911107
,
685513
197804
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

Export

Subgroup lattice of Q16×C31 in TeX

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