metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C31⋊C16, C62.C8, C8.2D31, C124.2C4, C248.2C2, C4.2Dic31, C2.(C31⋊C8), SmallGroup(496,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C31⋊C16 |
Generators and relations for C31⋊C16
G = < a,b | a31=b16=1, bab-1=a-1 >
Smallest permutation representation of C31⋊C16
►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)
(1 466 219 342 110 404 173 280 57 435 211 311 74 373 139 249)(2 496 220 372 111 434 174 310 58 465 212 341 75 403 140 279)(3 495 221 371 112 433 175 309 59 464 213 340 76 402 141 278)(4 494 222 370 113 432 176 308 60 463 214 339 77 401 142 277)(5 493 223 369 114 431 177 307 61 462 215 338 78 400 143 276)(6 492 224 368 115 430 178 306 62 461 216 337 79 399 144 275)(7 491 225 367 116 429 179 305 32 460 217 336 80 398 145 274)(8 490 226 366 117 428 180 304 33 459 187 335 81 397 146 273)(9 489 227 365 118 427 181 303 34 458 188 334 82 396 147 272)(10 488 228 364 119 426 182 302 35 457 189 333 83 395 148 271)(11 487 229 363 120 425 183 301 36 456 190 332 84 394 149 270)(12 486 230 362 121 424 184 300 37 455 191 331 85 393 150 269)(13 485 231 361 122 423 185 299 38 454 192 330 86 392 151 268)(14 484 232 360 123 422 186 298 39 453 193 329 87 391 152 267)(15 483 233 359 124 421 156 297 40 452 194 328 88 390 153 266)(16 482 234 358 94 420 157 296 41 451 195 327 89 389 154 265)(17 481 235 357 95 419 158 295 42 450 196 326 90 388 155 264)(18 480 236 356 96 418 159 294 43 449 197 325 91 387 125 263)(19 479 237 355 97 417 160 293 44 448 198 324 92 386 126 262)(20 478 238 354 98 416 161 292 45 447 199 323 93 385 127 261)(21 477 239 353 99 415 162 291 46 446 200 322 63 384 128 260)(22 476 240 352 100 414 163 290 47 445 201 321 64 383 129 259)(23 475 241 351 101 413 164 289 48 444 202 320 65 382 130 258)(24 474 242 350 102 412 165 288 49 443 203 319 66 381 131 257)(25 473 243 349 103 411 166 287 50 442 204 318 67 380 132 256)(26 472 244 348 104 410 167 286 51 441 205 317 68 379 133 255)(27 471 245 347 105 409 168 285 52 440 206 316 69 378 134 254)(28 470 246 346 106 408 169 284 53 439 207 315 70 377 135 253)(29 469 247 345 107 407 170 283 54 438 208 314 71 376 136 252)(30 468 248 344 108 406 171 282 55 437 209 313 72 375 137 251)(31 467 218 343 109 405 172 281 56 436 210 312 73 374 138 250)
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), (1,466,219,342,110,404,173,280,57,435,211,311,74,373,139,249)(2,496,220,372,111,434,174,310,58,465,212,341,75,403,140,279)(3,495,221,371,112,433,175,309,59,464,213,340,76,402,141,278)(4,494,222,370,113,432,176,308,60,463,214,339,77,401,142,277)(5,493,223,369,114,431,177,307,61,462,215,338,78,400,143,276)(6,492,224,368,115,430,178,306,62,461,216,337,79,399,144,275)(7,491,225,367,116,429,179,305,32,460,217,336,80,398,145,274)(8,490,226,366,117,428,180,304,33,459,187,335,81,397,146,273)(9,489,227,365,118,427,181,303,34,458,188,334,82,396,147,272)(10,488,228,364,119,426,182,302,35,457,189,333,83,395,148,271)(11,487,229,363,120,425,183,301,36,456,190,332,84,394,149,270)(12,486,230,362,121,424,184,300,37,455,191,331,85,393,150,269)(13,485,231,361,122,423,185,299,38,454,192,330,86,392,151,268)(14,484,232,360,123,422,186,298,39,453,193,329,87,391,152,267)(15,483,233,359,124,421,156,297,40,452,194,328,88,390,153,266)(16,482,234,358,94,420,157,296,41,451,195,327,89,389,154,265)(17,481,235,357,95,419,158,295,42,450,196,326,90,388,155,264)(18,480,236,356,96,418,159,294,43,449,197,325,91,387,125,263)(19,479,237,355,97,417,160,293,44,448,198,324,92,386,126,262)(20,478,238,354,98,416,161,292,45,447,199,323,93,385,127,261)(21,477,239,353,99,415,162,291,46,446,200,322,63,384,128,260)(22,476,240,352,100,414,163,290,47,445,201,321,64,383,129,259)(23,475,241,351,101,413,164,289,48,444,202,320,65,382,130,258)(24,474,242,350,102,412,165,288,49,443,203,319,66,381,131,257)(25,473,243,349,103,411,166,287,50,442,204,318,67,380,132,256)(26,472,244,348,104,410,167,286,51,441,205,317,68,379,133,255)(27,471,245,347,105,409,168,285,52,440,206,316,69,378,134,254)(28,470,246,346,106,408,169,284,53,439,207,315,70,377,135,253)(29,469,247,345,107,407,170,283,54,438,208,314,71,376,136,252)(30,468,248,344,108,406,171,282,55,437,209,313,72,375,137,251)(31,467,218,343,109,405,172,281,56,436,210,312,73,374,138,250)>;
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), (1,466,219,342,110,404,173,280,57,435,211,311,74,373,139,249)(2,496,220,372,111,434,174,310,58,465,212,341,75,403,140,279)(3,495,221,371,112,433,175,309,59,464,213,340,76,402,141,278)(4,494,222,370,113,432,176,308,60,463,214,339,77,401,142,277)(5,493,223,369,114,431,177,307,61,462,215,338,78,400,143,276)(6,492,224,368,115,430,178,306,62,461,216,337,79,399,144,275)(7,491,225,367,116,429,179,305,32,460,217,336,80,398,145,274)(8,490,226,366,117,428,180,304,33,459,187,335,81,397,146,273)(9,489,227,365,118,427,181,303,34,458,188,334,82,396,147,272)(10,488,228,364,119,426,182,302,35,457,189,333,83,395,148,271)(11,487,229,363,120,425,183,301,36,456,190,332,84,394,149,270)(12,486,230,362,121,424,184,300,37,455,191,331,85,393,150,269)(13,485,231,361,122,423,185,299,38,454,192,330,86,392,151,268)(14,484,232,360,123,422,186,298,39,453,193,329,87,391,152,267)(15,483,233,359,124,421,156,297,40,452,194,328,88,390,153,266)(16,482,234,358,94,420,157,296,41,451,195,327,89,389,154,265)(17,481,235,357,95,419,158,295,42,450,196,326,90,388,155,264)(18,480,236,356,96,418,159,294,43,449,197,325,91,387,125,263)(19,479,237,355,97,417,160,293,44,448,198,324,92,386,126,262)(20,478,238,354,98,416,161,292,45,447,199,323,93,385,127,261)(21,477,239,353,99,415,162,291,46,446,200,322,63,384,128,260)(22,476,240,352,100,414,163,290,47,445,201,321,64,383,129,259)(23,475,241,351,101,413,164,289,48,444,202,320,65,382,130,258)(24,474,242,350,102,412,165,288,49,443,203,319,66,381,131,257)(25,473,243,349,103,411,166,287,50,442,204,318,67,380,132,256)(26,472,244,348,104,410,167,286,51,441,205,317,68,379,133,255)(27,471,245,347,105,409,168,285,52,440,206,316,69,378,134,254)(28,470,246,346,106,408,169,284,53,439,207,315,70,377,135,253)(29,469,247,345,107,407,170,283,54,438,208,314,71,376,136,252)(30,468,248,344,108,406,171,282,55,437,209,313,72,375,137,251)(31,467,218,343,109,405,172,281,56,436,210,312,73,374,138,250) );
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)], [(1,466,219,342,110,404,173,280,57,435,211,311,74,373,139,249),(2,496,220,372,111,434,174,310,58,465,212,341,75,403,140,279),(3,495,221,371,112,433,175,309,59,464,213,340,76,402,141,278),(4,494,222,370,113,432,176,308,60,463,214,339,77,401,142,277),(5,493,223,369,114,431,177,307,61,462,215,338,78,400,143,276),(6,492,224,368,115,430,178,306,62,461,216,337,79,399,144,275),(7,491,225,367,116,429,179,305,32,460,217,336,80,398,145,274),(8,490,226,366,117,428,180,304,33,459,187,335,81,397,146,273),(9,489,227,365,118,427,181,303,34,458,188,334,82,396,147,272),(10,488,228,364,119,426,182,302,35,457,189,333,83,395,148,271),(11,487,229,363,120,425,183,301,36,456,190,332,84,394,149,270),(12,486,230,362,121,424,184,300,37,455,191,331,85,393,150,269),(13,485,231,361,122,423,185,299,38,454,192,330,86,392,151,268),(14,484,232,360,123,422,186,298,39,453,193,329,87,391,152,267),(15,483,233,359,124,421,156,297,40,452,194,328,88,390,153,266),(16,482,234,358,94,420,157,296,41,451,195,327,89,389,154,265),(17,481,235,357,95,419,158,295,42,450,196,326,90,388,155,264),(18,480,236,356,96,418,159,294,43,449,197,325,91,387,125,263),(19,479,237,355,97,417,160,293,44,448,198,324,92,386,126,262),(20,478,238,354,98,416,161,292,45,447,199,323,93,385,127,261),(21,477,239,353,99,415,162,291,46,446,200,322,63,384,128,260),(22,476,240,352,100,414,163,290,47,445,201,321,64,383,129,259),(23,475,241,351,101,413,164,289,48,444,202,320,65,382,130,258),(24,474,242,350,102,412,165,288,49,443,203,319,66,381,131,257),(25,473,243,349,103,411,166,287,50,442,204,318,67,380,132,256),(26,472,244,348,104,410,167,286,51,441,205,317,68,379,133,255),(27,471,245,347,105,409,168,285,52,440,206,316,69,378,134,254),(28,470,246,346,106,408,169,284,53,439,207,315,70,377,135,253),(29,469,247,345,107,407,170,283,54,438,208,314,71,376,136,252),(30,468,248,344,108,406,171,282,55,437,209,313,72,375,137,251),(31,467,218,343,109,405,172,281,56,436,210,312,73,374,138,250)]])
136 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD | 248A | ··· | 248BH |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 31 | ··· | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D31 | Dic31 | C31⋊C8 | C31⋊C16 |
| kernel | C31⋊C16 | C248 | C124 | C62 | C31 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 15 | 15 | 30 | 60 |
Matrix representation of C31⋊C16 ►in GL2(𝔽1489) generated by
| 0 | 1 |
| 1488 | 246 |
| 1080 | 566 |
| 1204 | 409 |
G:=sub<GL(2,GF(1489))| [0,1488,1,246],[1080,1204,566,409] >;
C31⋊C16 in GAP, Magma, Sage, TeX
C_{31}\rtimes C_{16} % in TeX
G:=Group("C31:C16"); // GroupNames label
G:=SmallGroup(496,1);
// by ID
G=gap.SmallGroup(496,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,10,26,42,12004]);
// Polycyclic
G:=Group<a,b|a^31=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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