direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C29, C8.C58, Q8.C58, C232.3C2, C58.16D4, C116.19C22, C4.3(C2×C58), C2.5(D4×C29), (Q8×C29).2C2, SmallGroup(464,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C29
G = < a,b,c | a29=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q16×C29
►Regular action on 464 points
Generators in S464
(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)
(1 171 223 265 442 251 422 124)(2 172 224 266 443 252 423 125)(3 173 225 267 444 253 424 126)(4 174 226 268 445 254 425 127)(5 146 227 269 446 255 426 128)(6 147 228 270 447 256 427 129)(7 148 229 271 448 257 428 130)(8 149 230 272 449 258 429 131)(9 150 231 273 450 259 430 132)(10 151 232 274 451 260 431 133)(11 152 204 275 452 261 432 134)(12 153 205 276 453 233 433 135)(13 154 206 277 454 234 434 136)(14 155 207 278 455 235 435 137)(15 156 208 279 456 236 407 138)(16 157 209 280 457 237 408 139)(17 158 210 281 458 238 409 140)(18 159 211 282 459 239 410 141)(19 160 212 283 460 240 411 142)(20 161 213 284 461 241 412 143)(21 162 214 285 462 242 413 144)(22 163 215 286 463 243 414 145)(23 164 216 287 464 244 415 117)(24 165 217 288 436 245 416 118)(25 166 218 289 437 246 417 119)(26 167 219 290 438 247 418 120)(27 168 220 262 439 248 419 121)(28 169 221 263 440 249 420 122)(29 170 222 264 441 250 421 123)(30 294 339 189 389 362 84 88)(31 295 340 190 390 363 85 89)(32 296 341 191 391 364 86 90)(33 297 342 192 392 365 87 91)(34 298 343 193 393 366 59 92)(35 299 344 194 394 367 60 93)(36 300 345 195 395 368 61 94)(37 301 346 196 396 369 62 95)(38 302 347 197 397 370 63 96)(39 303 348 198 398 371 64 97)(40 304 320 199 399 372 65 98)(41 305 321 200 400 373 66 99)(42 306 322 201 401 374 67 100)(43 307 323 202 402 375 68 101)(44 308 324 203 403 376 69 102)(45 309 325 175 404 377 70 103)(46 310 326 176 405 349 71 104)(47 311 327 177 406 350 72 105)(48 312 328 178 378 351 73 106)(49 313 329 179 379 352 74 107)(50 314 330 180 380 353 75 108)(51 315 331 181 381 354 76 109)(52 316 332 182 382 355 77 110)(53 317 333 183 383 356 78 111)(54 318 334 184 384 357 79 112)(55 319 335 185 385 358 80 113)(56 291 336 186 386 359 81 114)(57 292 337 187 387 360 82 115)(58 293 338 188 388 361 83 116)
(1 177 442 105)(2 178 443 106)(3 179 444 107)(4 180 445 108)(5 181 446 109)(6 182 447 110)(7 183 448 111)(8 184 449 112)(9 185 450 113)(10 186 451 114)(11 187 452 115)(12 188 453 116)(13 189 454 88)(14 190 455 89)(15 191 456 90)(16 192 457 91)(17 193 458 92)(18 194 459 93)(19 195 460 94)(20 196 461 95)(21 197 462 96)(22 198 463 97)(23 199 464 98)(24 200 436 99)(25 201 437 100)(26 202 438 101)(27 203 439 102)(28 175 440 103)(29 176 441 104)(30 136 389 277)(31 137 390 278)(32 138 391 279)(33 139 392 280)(34 140 393 281)(35 141 394 282)(36 142 395 283)(37 143 396 284)(38 144 397 285)(39 145 398 286)(40 117 399 287)(41 118 400 288)(42 119 401 289)(43 120 402 290)(44 121 403 262)(45 122 404 263)(46 123 405 264)(47 124 406 265)(48 125 378 266)(49 126 379 267)(50 127 380 268)(51 128 381 269)(52 129 382 270)(53 130 383 271)(54 131 384 272)(55 132 385 273)(56 133 386 274)(57 134 387 275)(58 135 388 276)(59 158 343 238)(60 159 344 239)(61 160 345 240)(62 161 346 241)(63 162 347 242)(64 163 348 243)(65 164 320 244)(66 165 321 245)(67 166 322 246)(68 167 323 247)(69 168 324 248)(70 169 325 249)(71 170 326 250)(72 171 327 251)(73 172 328 252)(74 173 329 253)(75 174 330 254)(76 146 331 255)(77 147 332 256)(78 148 333 257)(79 149 334 258)(80 150 335 259)(81 151 336 260)(82 152 337 261)(83 153 338 233)(84 154 339 234)(85 155 340 235)(86 156 341 236)(87 157 342 237)(204 292 432 360)(205 293 433 361)(206 294 434 362)(207 295 435 363)(208 296 407 364)(209 297 408 365)(210 298 409 366)(211 299 410 367)(212 300 411 368)(213 301 412 369)(214 302 413 370)(215 303 414 371)(216 304 415 372)(217 305 416 373)(218 306 417 374)(219 307 418 375)(220 308 419 376)(221 309 420 377)(222 310 421 349)(223 311 422 350)(224 312 423 351)(225 313 424 352)(226 314 425 353)(227 315 426 354)(228 316 427 355)(229 317 428 356)(230 318 429 357)(231 319 430 358)(232 291 431 359)
G:=sub<Sym(464)| (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), (1,171,223,265,442,251,422,124)(2,172,224,266,443,252,423,125)(3,173,225,267,444,253,424,126)(4,174,226,268,445,254,425,127)(5,146,227,269,446,255,426,128)(6,147,228,270,447,256,427,129)(7,148,229,271,448,257,428,130)(8,149,230,272,449,258,429,131)(9,150,231,273,450,259,430,132)(10,151,232,274,451,260,431,133)(11,152,204,275,452,261,432,134)(12,153,205,276,453,233,433,135)(13,154,206,277,454,234,434,136)(14,155,207,278,455,235,435,137)(15,156,208,279,456,236,407,138)(16,157,209,280,457,237,408,139)(17,158,210,281,458,238,409,140)(18,159,211,282,459,239,410,141)(19,160,212,283,460,240,411,142)(20,161,213,284,461,241,412,143)(21,162,214,285,462,242,413,144)(22,163,215,286,463,243,414,145)(23,164,216,287,464,244,415,117)(24,165,217,288,436,245,416,118)(25,166,218,289,437,246,417,119)(26,167,219,290,438,247,418,120)(27,168,220,262,439,248,419,121)(28,169,221,263,440,249,420,122)(29,170,222,264,441,250,421,123)(30,294,339,189,389,362,84,88)(31,295,340,190,390,363,85,89)(32,296,341,191,391,364,86,90)(33,297,342,192,392,365,87,91)(34,298,343,193,393,366,59,92)(35,299,344,194,394,367,60,93)(36,300,345,195,395,368,61,94)(37,301,346,196,396,369,62,95)(38,302,347,197,397,370,63,96)(39,303,348,198,398,371,64,97)(40,304,320,199,399,372,65,98)(41,305,321,200,400,373,66,99)(42,306,322,201,401,374,67,100)(43,307,323,202,402,375,68,101)(44,308,324,203,403,376,69,102)(45,309,325,175,404,377,70,103)(46,310,326,176,405,349,71,104)(47,311,327,177,406,350,72,105)(48,312,328,178,378,351,73,106)(49,313,329,179,379,352,74,107)(50,314,330,180,380,353,75,108)(51,315,331,181,381,354,76,109)(52,316,332,182,382,355,77,110)(53,317,333,183,383,356,78,111)(54,318,334,184,384,357,79,112)(55,319,335,185,385,358,80,113)(56,291,336,186,386,359,81,114)(57,292,337,187,387,360,82,115)(58,293,338,188,388,361,83,116), (1,177,442,105)(2,178,443,106)(3,179,444,107)(4,180,445,108)(5,181,446,109)(6,182,447,110)(7,183,448,111)(8,184,449,112)(9,185,450,113)(10,186,451,114)(11,187,452,115)(12,188,453,116)(13,189,454,88)(14,190,455,89)(15,191,456,90)(16,192,457,91)(17,193,458,92)(18,194,459,93)(19,195,460,94)(20,196,461,95)(21,197,462,96)(22,198,463,97)(23,199,464,98)(24,200,436,99)(25,201,437,100)(26,202,438,101)(27,203,439,102)(28,175,440,103)(29,176,441,104)(30,136,389,277)(31,137,390,278)(32,138,391,279)(33,139,392,280)(34,140,393,281)(35,141,394,282)(36,142,395,283)(37,143,396,284)(38,144,397,285)(39,145,398,286)(40,117,399,287)(41,118,400,288)(42,119,401,289)(43,120,402,290)(44,121,403,262)(45,122,404,263)(46,123,405,264)(47,124,406,265)(48,125,378,266)(49,126,379,267)(50,127,380,268)(51,128,381,269)(52,129,382,270)(53,130,383,271)(54,131,384,272)(55,132,385,273)(56,133,386,274)(57,134,387,275)(58,135,388,276)(59,158,343,238)(60,159,344,239)(61,160,345,240)(62,161,346,241)(63,162,347,242)(64,163,348,243)(65,164,320,244)(66,165,321,245)(67,166,322,246)(68,167,323,247)(69,168,324,248)(70,169,325,249)(71,170,326,250)(72,171,327,251)(73,172,328,252)(74,173,329,253)(75,174,330,254)(76,146,331,255)(77,147,332,256)(78,148,333,257)(79,149,334,258)(80,150,335,259)(81,151,336,260)(82,152,337,261)(83,153,338,233)(84,154,339,234)(85,155,340,235)(86,156,341,236)(87,157,342,237)(204,292,432,360)(205,293,433,361)(206,294,434,362)(207,295,435,363)(208,296,407,364)(209,297,408,365)(210,298,409,366)(211,299,410,367)(212,300,411,368)(213,301,412,369)(214,302,413,370)(215,303,414,371)(216,304,415,372)(217,305,416,373)(218,306,417,374)(219,307,418,375)(220,308,419,376)(221,309,420,377)(222,310,421,349)(223,311,422,350)(224,312,423,351)(225,313,424,352)(226,314,425,353)(227,315,426,354)(228,316,427,355)(229,317,428,356)(230,318,429,357)(231,319,430,358)(232,291,431,359)>;
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), (1,171,223,265,442,251,422,124)(2,172,224,266,443,252,423,125)(3,173,225,267,444,253,424,126)(4,174,226,268,445,254,425,127)(5,146,227,269,446,255,426,128)(6,147,228,270,447,256,427,129)(7,148,229,271,448,257,428,130)(8,149,230,272,449,258,429,131)(9,150,231,273,450,259,430,132)(10,151,232,274,451,260,431,133)(11,152,204,275,452,261,432,134)(12,153,205,276,453,233,433,135)(13,154,206,277,454,234,434,136)(14,155,207,278,455,235,435,137)(15,156,208,279,456,236,407,138)(16,157,209,280,457,237,408,139)(17,158,210,281,458,238,409,140)(18,159,211,282,459,239,410,141)(19,160,212,283,460,240,411,142)(20,161,213,284,461,241,412,143)(21,162,214,285,462,242,413,144)(22,163,215,286,463,243,414,145)(23,164,216,287,464,244,415,117)(24,165,217,288,436,245,416,118)(25,166,218,289,437,246,417,119)(26,167,219,290,438,247,418,120)(27,168,220,262,439,248,419,121)(28,169,221,263,440,249,420,122)(29,170,222,264,441,250,421,123)(30,294,339,189,389,362,84,88)(31,295,340,190,390,363,85,89)(32,296,341,191,391,364,86,90)(33,297,342,192,392,365,87,91)(34,298,343,193,393,366,59,92)(35,299,344,194,394,367,60,93)(36,300,345,195,395,368,61,94)(37,301,346,196,396,369,62,95)(38,302,347,197,397,370,63,96)(39,303,348,198,398,371,64,97)(40,304,320,199,399,372,65,98)(41,305,321,200,400,373,66,99)(42,306,322,201,401,374,67,100)(43,307,323,202,402,375,68,101)(44,308,324,203,403,376,69,102)(45,309,325,175,404,377,70,103)(46,310,326,176,405,349,71,104)(47,311,327,177,406,350,72,105)(48,312,328,178,378,351,73,106)(49,313,329,179,379,352,74,107)(50,314,330,180,380,353,75,108)(51,315,331,181,381,354,76,109)(52,316,332,182,382,355,77,110)(53,317,333,183,383,356,78,111)(54,318,334,184,384,357,79,112)(55,319,335,185,385,358,80,113)(56,291,336,186,386,359,81,114)(57,292,337,187,387,360,82,115)(58,293,338,188,388,361,83,116), (1,177,442,105)(2,178,443,106)(3,179,444,107)(4,180,445,108)(5,181,446,109)(6,182,447,110)(7,183,448,111)(8,184,449,112)(9,185,450,113)(10,186,451,114)(11,187,452,115)(12,188,453,116)(13,189,454,88)(14,190,455,89)(15,191,456,90)(16,192,457,91)(17,193,458,92)(18,194,459,93)(19,195,460,94)(20,196,461,95)(21,197,462,96)(22,198,463,97)(23,199,464,98)(24,200,436,99)(25,201,437,100)(26,202,438,101)(27,203,439,102)(28,175,440,103)(29,176,441,104)(30,136,389,277)(31,137,390,278)(32,138,391,279)(33,139,392,280)(34,140,393,281)(35,141,394,282)(36,142,395,283)(37,143,396,284)(38,144,397,285)(39,145,398,286)(40,117,399,287)(41,118,400,288)(42,119,401,289)(43,120,402,290)(44,121,403,262)(45,122,404,263)(46,123,405,264)(47,124,406,265)(48,125,378,266)(49,126,379,267)(50,127,380,268)(51,128,381,269)(52,129,382,270)(53,130,383,271)(54,131,384,272)(55,132,385,273)(56,133,386,274)(57,134,387,275)(58,135,388,276)(59,158,343,238)(60,159,344,239)(61,160,345,240)(62,161,346,241)(63,162,347,242)(64,163,348,243)(65,164,320,244)(66,165,321,245)(67,166,322,246)(68,167,323,247)(69,168,324,248)(70,169,325,249)(71,170,326,250)(72,171,327,251)(73,172,328,252)(74,173,329,253)(75,174,330,254)(76,146,331,255)(77,147,332,256)(78,148,333,257)(79,149,334,258)(80,150,335,259)(81,151,336,260)(82,152,337,261)(83,153,338,233)(84,154,339,234)(85,155,340,235)(86,156,341,236)(87,157,342,237)(204,292,432,360)(205,293,433,361)(206,294,434,362)(207,295,435,363)(208,296,407,364)(209,297,408,365)(210,298,409,366)(211,299,410,367)(212,300,411,368)(213,301,412,369)(214,302,413,370)(215,303,414,371)(216,304,415,372)(217,305,416,373)(218,306,417,374)(219,307,418,375)(220,308,419,376)(221,309,420,377)(222,310,421,349)(223,311,422,350)(224,312,423,351)(225,313,424,352)(226,314,425,353)(227,315,426,354)(228,316,427,355)(229,317,428,356)(230,318,429,357)(231,319,430,358)(232,291,431,359) );
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)], [(1,171,223,265,442,251,422,124),(2,172,224,266,443,252,423,125),(3,173,225,267,444,253,424,126),(4,174,226,268,445,254,425,127),(5,146,227,269,446,255,426,128),(6,147,228,270,447,256,427,129),(7,148,229,271,448,257,428,130),(8,149,230,272,449,258,429,131),(9,150,231,273,450,259,430,132),(10,151,232,274,451,260,431,133),(11,152,204,275,452,261,432,134),(12,153,205,276,453,233,433,135),(13,154,206,277,454,234,434,136),(14,155,207,278,455,235,435,137),(15,156,208,279,456,236,407,138),(16,157,209,280,457,237,408,139),(17,158,210,281,458,238,409,140),(18,159,211,282,459,239,410,141),(19,160,212,283,460,240,411,142),(20,161,213,284,461,241,412,143),(21,162,214,285,462,242,413,144),(22,163,215,286,463,243,414,145),(23,164,216,287,464,244,415,117),(24,165,217,288,436,245,416,118),(25,166,218,289,437,246,417,119),(26,167,219,290,438,247,418,120),(27,168,220,262,439,248,419,121),(28,169,221,263,440,249,420,122),(29,170,222,264,441,250,421,123),(30,294,339,189,389,362,84,88),(31,295,340,190,390,363,85,89),(32,296,341,191,391,364,86,90),(33,297,342,192,392,365,87,91),(34,298,343,193,393,366,59,92),(35,299,344,194,394,367,60,93),(36,300,345,195,395,368,61,94),(37,301,346,196,396,369,62,95),(38,302,347,197,397,370,63,96),(39,303,348,198,398,371,64,97),(40,304,320,199,399,372,65,98),(41,305,321,200,400,373,66,99),(42,306,322,201,401,374,67,100),(43,307,323,202,402,375,68,101),(44,308,324,203,403,376,69,102),(45,309,325,175,404,377,70,103),(46,310,326,176,405,349,71,104),(47,311,327,177,406,350,72,105),(48,312,328,178,378,351,73,106),(49,313,329,179,379,352,74,107),(50,314,330,180,380,353,75,108),(51,315,331,181,381,354,76,109),(52,316,332,182,382,355,77,110),(53,317,333,183,383,356,78,111),(54,318,334,184,384,357,79,112),(55,319,335,185,385,358,80,113),(56,291,336,186,386,359,81,114),(57,292,337,187,387,360,82,115),(58,293,338,188,388,361,83,116)], [(1,177,442,105),(2,178,443,106),(3,179,444,107),(4,180,445,108),(5,181,446,109),(6,182,447,110),(7,183,448,111),(8,184,449,112),(9,185,450,113),(10,186,451,114),(11,187,452,115),(12,188,453,116),(13,189,454,88),(14,190,455,89),(15,191,456,90),(16,192,457,91),(17,193,458,92),(18,194,459,93),(19,195,460,94),(20,196,461,95),(21,197,462,96),(22,198,463,97),(23,199,464,98),(24,200,436,99),(25,201,437,100),(26,202,438,101),(27,203,439,102),(28,175,440,103),(29,176,441,104),(30,136,389,277),(31,137,390,278),(32,138,391,279),(33,139,392,280),(34,140,393,281),(35,141,394,282),(36,142,395,283),(37,143,396,284),(38,144,397,285),(39,145,398,286),(40,117,399,287),(41,118,400,288),(42,119,401,289),(43,120,402,290),(44,121,403,262),(45,122,404,263),(46,123,405,264),(47,124,406,265),(48,125,378,266),(49,126,379,267),(50,127,380,268),(51,128,381,269),(52,129,382,270),(53,130,383,271),(54,131,384,272),(55,132,385,273),(56,133,386,274),(57,134,387,275),(58,135,388,276),(59,158,343,238),(60,159,344,239),(61,160,345,240),(62,161,346,241),(63,162,347,242),(64,163,348,243),(65,164,320,244),(66,165,321,245),(67,166,322,246),(68,167,323,247),(69,168,324,248),(70,169,325,249),(71,170,326,250),(72,171,327,251),(73,172,328,252),(74,173,329,253),(75,174,330,254),(76,146,331,255),(77,147,332,256),(78,148,333,257),(79,149,334,258),(80,150,335,259),(81,151,336,260),(82,152,337,261),(83,153,338,233),(84,154,339,234),(85,155,340,235),(86,156,341,236),(87,157,342,237),(204,292,432,360),(205,293,433,361),(206,294,434,362),(207,295,435,363),(208,296,407,364),(209,297,408,365),(210,298,409,366),(211,299,410,367),(212,300,411,368),(213,301,412,369),(214,302,413,370),(215,303,414,371),(216,304,415,372),(217,305,416,373),(218,306,417,374),(219,307,418,375),(220,308,419,376),(221,309,420,377),(222,310,421,349),(223,311,422,350),(224,312,423,351),(225,313,424,352),(226,314,425,353),(227,315,426,354),(228,316,427,355),(229,317,428,356),(230,318,429,357),(231,319,430,358),(232,291,431,359)]])
203 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 29A | ··· | 29AB | 58A | ··· | 58AB | 116A | ··· | 116AB | 116AC | ··· | 116CF | 232A | ··· | 232BD |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
203 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C29 | C58 | C58 | D4 | Q16 | D4×C29 | Q16×C29 |
| kernel | Q16×C29 | C232 | Q8×C29 | Q16 | C8 | Q8 | C58 | C29 | C2 | C1 |
| # reps | 1 | 1 | 2 | 28 | 28 | 56 | 1 | 2 | 28 | 56 |
Matrix representation of Q16×C29 ►in GL2(𝔽233) generated by
| 2 | 0 |
| 0 | 2 |
| 0 | 148 |
| 159 | 148 |
| 53 | 137 |
| 5 | 180 |
G:=sub<GL(2,GF(233))| [2,0,0,2],[0,159,148,148],[53,5,137,180] >;
Q16×C29 in GAP, Magma, Sage, TeX
Q_{16}\times C_{29} % in TeX
G:=Group("Q16xC29"); // GroupNames label
G:=SmallGroup(464,27);
// by ID
G=gap.SmallGroup(464,27);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,1160,1181,1166,6963,3488,58]);
// Polycyclic
G:=Group<a,b,c|a^29=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export