metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15⋊1C32, C60.2C8, C24.4F5, C120.4C4, C30.1C16, C40.3Dic3, C3⋊(C5⋊C32), C5⋊(C3⋊C32), C6.(C5⋊C16), C10.(C3⋊C16), C8.4(C3⋊F5), C12.2(C5⋊C8), C20.2(C3⋊C8), C2.(C15⋊C16), C5⋊2C16.2S3, C4.2(C15⋊C8), (C3×C5⋊2C16).3C2, SmallGroup(480,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C15⋊C32 |
Generators and relations for C15⋊C32
G = < a,b | a15=b32=1, bab-1=a2 >
Smallest permutation representation of C15⋊C32
►Regular action on 480 points
Generators in S480
(1 238 51 437 219 158 102 359 167 416 476 65 346 287 308)(2 52 220 103 168 477 347 309 239 438 159 360 385 66 288)(3 221 169 348 240 160 386 257 53 104 478 310 439 361 67)(4 170 241 387 54 479 440 68 222 349 129 258 105 311 362)(5 242 55 441 223 130 106 363 171 388 480 69 350 259 312)(6 56 224 107 172 449 351 313 243 442 131 364 389 70 260)(7 193 173 352 244 132 390 261 57 108 450 314 443 365 71)(8 174 245 391 58 451 444 72 194 321 133 262 109 315 366)(9 246 59 445 195 134 110 367 175 392 452 73 322 263 316)(10 60 196 111 176 453 323 317 247 446 135 368 393 74 264)(11 197 177 324 248 136 394 265 61 112 454 318 447 369 75)(12 178 249 395 62 455 448 76 198 325 137 266 113 319 370)(13 250 63 417 199 138 114 371 179 396 456 77 326 267 320)(14 64 200 115 180 457 327 289 251 418 139 372 397 78 268)(15 201 181 328 252 140 398 269 33 116 458 290 419 373 79)(16 182 253 399 34 459 420 80 202 329 141 270 117 291 374)(17 254 35 421 203 142 118 375 183 400 460 81 330 271 292)(18 36 204 119 184 461 331 293 255 422 143 376 401 82 272)(19 205 185 332 256 144 402 273 37 120 462 294 423 377 83)(20 186 225 403 38 463 424 84 206 333 145 274 121 295 378)(21 226 39 425 207 146 122 379 187 404 464 85 334 275 296)(22 40 208 123 188 465 335 297 227 426 147 380 405 86 276)(23 209 189 336 228 148 406 277 41 124 466 298 427 381 87)(24 190 229 407 42 467 428 88 210 337 149 278 125 299 382)(25 230 43 429 211 150 126 383 191 408 468 89 338 279 300)(26 44 212 127 192 469 339 301 231 430 151 384 409 90 280)(27 213 161 340 232 152 410 281 45 128 470 302 431 353 91)(28 162 233 411 46 471 432 92 214 341 153 282 97 303 354)(29 234 47 433 215 154 98 355 163 412 472 93 342 283 304)(30 48 216 99 164 473 343 305 235 434 155 356 413 94 284)(31 217 165 344 236 156 414 285 49 100 474 306 435 357 95)(32 166 237 415 50 475 436 96 218 345 157 286 101 307 358)
(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)
G:=sub<Sym(480)| (1,238,51,437,219,158,102,359,167,416,476,65,346,287,308)(2,52,220,103,168,477,347,309,239,438,159,360,385,66,288)(3,221,169,348,240,160,386,257,53,104,478,310,439,361,67)(4,170,241,387,54,479,440,68,222,349,129,258,105,311,362)(5,242,55,441,223,130,106,363,171,388,480,69,350,259,312)(6,56,224,107,172,449,351,313,243,442,131,364,389,70,260)(7,193,173,352,244,132,390,261,57,108,450,314,443,365,71)(8,174,245,391,58,451,444,72,194,321,133,262,109,315,366)(9,246,59,445,195,134,110,367,175,392,452,73,322,263,316)(10,60,196,111,176,453,323,317,247,446,135,368,393,74,264)(11,197,177,324,248,136,394,265,61,112,454,318,447,369,75)(12,178,249,395,62,455,448,76,198,325,137,266,113,319,370)(13,250,63,417,199,138,114,371,179,396,456,77,326,267,320)(14,64,200,115,180,457,327,289,251,418,139,372,397,78,268)(15,201,181,328,252,140,398,269,33,116,458,290,419,373,79)(16,182,253,399,34,459,420,80,202,329,141,270,117,291,374)(17,254,35,421,203,142,118,375,183,400,460,81,330,271,292)(18,36,204,119,184,461,331,293,255,422,143,376,401,82,272)(19,205,185,332,256,144,402,273,37,120,462,294,423,377,83)(20,186,225,403,38,463,424,84,206,333,145,274,121,295,378)(21,226,39,425,207,146,122,379,187,404,464,85,334,275,296)(22,40,208,123,188,465,335,297,227,426,147,380,405,86,276)(23,209,189,336,228,148,406,277,41,124,466,298,427,381,87)(24,190,229,407,42,467,428,88,210,337,149,278,125,299,382)(25,230,43,429,211,150,126,383,191,408,468,89,338,279,300)(26,44,212,127,192,469,339,301,231,430,151,384,409,90,280)(27,213,161,340,232,152,410,281,45,128,470,302,431,353,91)(28,162,233,411,46,471,432,92,214,341,153,282,97,303,354)(29,234,47,433,215,154,98,355,163,412,472,93,342,283,304)(30,48,216,99,164,473,343,305,235,434,155,356,413,94,284)(31,217,165,344,236,156,414,285,49,100,474,306,435,357,95)(32,166,237,415,50,475,436,96,218,345,157,286,101,307,358), (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)>;
G:=Group( (1,238,51,437,219,158,102,359,167,416,476,65,346,287,308)(2,52,220,103,168,477,347,309,239,438,159,360,385,66,288)(3,221,169,348,240,160,386,257,53,104,478,310,439,361,67)(4,170,241,387,54,479,440,68,222,349,129,258,105,311,362)(5,242,55,441,223,130,106,363,171,388,480,69,350,259,312)(6,56,224,107,172,449,351,313,243,442,131,364,389,70,260)(7,193,173,352,244,132,390,261,57,108,450,314,443,365,71)(8,174,245,391,58,451,444,72,194,321,133,262,109,315,366)(9,246,59,445,195,134,110,367,175,392,452,73,322,263,316)(10,60,196,111,176,453,323,317,247,446,135,368,393,74,264)(11,197,177,324,248,136,394,265,61,112,454,318,447,369,75)(12,178,249,395,62,455,448,76,198,325,137,266,113,319,370)(13,250,63,417,199,138,114,371,179,396,456,77,326,267,320)(14,64,200,115,180,457,327,289,251,418,139,372,397,78,268)(15,201,181,328,252,140,398,269,33,116,458,290,419,373,79)(16,182,253,399,34,459,420,80,202,329,141,270,117,291,374)(17,254,35,421,203,142,118,375,183,400,460,81,330,271,292)(18,36,204,119,184,461,331,293,255,422,143,376,401,82,272)(19,205,185,332,256,144,402,273,37,120,462,294,423,377,83)(20,186,225,403,38,463,424,84,206,333,145,274,121,295,378)(21,226,39,425,207,146,122,379,187,404,464,85,334,275,296)(22,40,208,123,188,465,335,297,227,426,147,380,405,86,276)(23,209,189,336,228,148,406,277,41,124,466,298,427,381,87)(24,190,229,407,42,467,428,88,210,337,149,278,125,299,382)(25,230,43,429,211,150,126,383,191,408,468,89,338,279,300)(26,44,212,127,192,469,339,301,231,430,151,384,409,90,280)(27,213,161,340,232,152,410,281,45,128,470,302,431,353,91)(28,162,233,411,46,471,432,92,214,341,153,282,97,303,354)(29,234,47,433,215,154,98,355,163,412,472,93,342,283,304)(30,48,216,99,164,473,343,305,235,434,155,356,413,94,284)(31,217,165,344,236,156,414,285,49,100,474,306,435,357,95)(32,166,237,415,50,475,436,96,218,345,157,286,101,307,358), (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) );
G=PermutationGroup([[(1,238,51,437,219,158,102,359,167,416,476,65,346,287,308),(2,52,220,103,168,477,347,309,239,438,159,360,385,66,288),(3,221,169,348,240,160,386,257,53,104,478,310,439,361,67),(4,170,241,387,54,479,440,68,222,349,129,258,105,311,362),(5,242,55,441,223,130,106,363,171,388,480,69,350,259,312),(6,56,224,107,172,449,351,313,243,442,131,364,389,70,260),(7,193,173,352,244,132,390,261,57,108,450,314,443,365,71),(8,174,245,391,58,451,444,72,194,321,133,262,109,315,366),(9,246,59,445,195,134,110,367,175,392,452,73,322,263,316),(10,60,196,111,176,453,323,317,247,446,135,368,393,74,264),(11,197,177,324,248,136,394,265,61,112,454,318,447,369,75),(12,178,249,395,62,455,448,76,198,325,137,266,113,319,370),(13,250,63,417,199,138,114,371,179,396,456,77,326,267,320),(14,64,200,115,180,457,327,289,251,418,139,372,397,78,268),(15,201,181,328,252,140,398,269,33,116,458,290,419,373,79),(16,182,253,399,34,459,420,80,202,329,141,270,117,291,374),(17,254,35,421,203,142,118,375,183,400,460,81,330,271,292),(18,36,204,119,184,461,331,293,255,422,143,376,401,82,272),(19,205,185,332,256,144,402,273,37,120,462,294,423,377,83),(20,186,225,403,38,463,424,84,206,333,145,274,121,295,378),(21,226,39,425,207,146,122,379,187,404,464,85,334,275,296),(22,40,208,123,188,465,335,297,227,426,147,380,405,86,276),(23,209,189,336,228,148,406,277,41,124,466,298,427,381,87),(24,190,229,407,42,467,428,88,210,337,149,278,125,299,382),(25,230,43,429,211,150,126,383,191,408,468,89,338,279,300),(26,44,212,127,192,469,339,301,231,430,151,384,409,90,280),(27,213,161,340,232,152,410,281,45,128,470,302,431,353,91),(28,162,233,411,46,471,432,92,214,341,153,282,97,303,354),(29,234,47,433,215,154,98,355,163,412,472,93,342,283,304),(30,48,216,99,164,473,343,305,235,434,155,356,413,94,284),(31,217,165,344,236,156,414,285,49,100,474,306,435,357,95),(32,166,237,415,50,475,436,96,218,345,157,286,101,307,358)], [(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)]])
72 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 15A | 15B | 16A | ··· | 16H | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 32A | ··· | 32P | 40A | 40B | 40C | 40D | 48A | ··· | 48H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 32 | ··· | 32 | 40 | 40 | 40 | 40 | 48 | ··· | 48 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 4 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 15 | ··· | 15 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | S3 | Dic3 | C3⋊C8 | C3⋊C16 | C3⋊C32 | F5 | C5⋊C8 | C3⋊F5 | C5⋊C16 | C15⋊C8 | C5⋊C32 | C15⋊C16 | C15⋊C32 |
| kernel | C15⋊C32 | C3×C5⋊2C16 | C120 | C60 | C30 | C15 | C5⋊2C16 | C40 | C20 | C10 | C5 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C15⋊C32 ►in GL6(𝔽3361)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 3360 | 3360 | 0 | 0 | 0 | 0 |
| 0 | 0 | 755 | 2984 | 377 | 2606 |
| 0 | 0 | 755 | 378 | 0 | 2983 |
| 0 | 0 | 0 | 378 | 755 | 2606 |
| 0 | 0 | 377 | 2984 | 755 | 0 |
| 2415 | 1885 | 0 | 0 | 0 | 0 |
| 2831 | 946 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1503 | 3337 | 1878 | 110 |
| 0 | 0 | 1613 | 3231 | 20 | 86 |
| 0 | 0 | 1589 | 1748 | 130 | 3341 |
| 0 | 0 | 1483 | 3251 | 106 | 1858 |
G:=sub<GL(6,GF(3361))| [0,3360,0,0,0,0,1,3360,0,0,0,0,0,0,755,755,0,377,0,0,2984,378,378,2984,0,0,377,0,755,755,0,0,2606,2983,2606,0],[2415,2831,0,0,0,0,1885,946,0,0,0,0,0,0,1503,1613,1589,1483,0,0,3337,3231,1748,3251,0,0,1878,20,130,106,0,0,110,86,3341,1858] >;
C15⋊C32 in GAP, Magma, Sage, TeX
C_{15}\rtimes C_{32} % in TeX
G:=Group("C15:C32"); // GroupNames label
G:=SmallGroup(480,6);
// by ID
G=gap.SmallGroup(480,6);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,14,36,58,80,2693,14118,9421]);
// Polycyclic
G:=Group<a,b|a^15=b^32=1,b*a*b^-1=a^2>;
// generators/relations
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