metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15⋊3C32, C60.8C8, C80.3S3, C48.3D5, C240.4C2, C30.3C16, C16.2D15, C120.12C4, C40.7Dic3, C24.3Dic5, C8.3Dic15, C5⋊2(C3⋊C32), C3⋊(C5⋊2C32), C6.(C5⋊2C16), C20.5(C3⋊C8), C10.2(C3⋊C16), C2.(C15⋊3C16), C4.2(C15⋊3C8), C12.2(C5⋊2C8), SmallGroup(480,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C15⋊3C32 |
Generators and relations for C15⋊3C32
G = < a,b | a15=b32=1, bab-1=a-1 >
15C32
Smallest permutation representation of C15⋊3C32
►Regular action on 480 points
Generators in S480
(1 478 141 391 51 445 94 218 327 292 108 161 354 271 251)(2 252 272 355 162 109 293 328 219 95 446 52 392 142 479)(3 480 143 393 53 447 96 220 329 294 110 163 356 273 253)(4 254 274 357 164 111 295 330 221 65 448 54 394 144 449)(5 450 145 395 55 417 66 222 331 296 112 165 358 275 255)(6 256 276 359 166 113 297 332 223 67 418 56 396 146 451)(7 452 147 397 57 419 68 224 333 298 114 167 360 277 225)(8 226 278 361 168 115 299 334 193 69 420 58 398 148 453)(9 454 149 399 59 421 70 194 335 300 116 169 362 279 227)(10 228 280 363 170 117 301 336 195 71 422 60 400 150 455)(11 456 151 401 61 423 72 196 337 302 118 171 364 281 229)(12 230 282 365 172 119 303 338 197 73 424 62 402 152 457)(13 458 153 403 63 425 74 198 339 304 120 173 366 283 231)(14 232 284 367 174 121 305 340 199 75 426 64 404 154 459)(15 460 155 405 33 427 76 200 341 306 122 175 368 285 233)(16 234 286 369 176 123 307 342 201 77 428 34 406 156 461)(17 462 157 407 35 429 78 202 343 308 124 177 370 287 235)(18 236 288 371 178 125 309 344 203 79 430 36 408 158 463)(19 464 159 409 37 431 80 204 345 310 126 179 372 257 237)(20 238 258 373 180 127 311 346 205 81 432 38 410 160 465)(21 466 129 411 39 433 82 206 347 312 128 181 374 259 239)(22 240 260 375 182 97 313 348 207 83 434 40 412 130 467)(23 468 131 413 41 435 84 208 349 314 98 183 376 261 241)(24 242 262 377 184 99 315 350 209 85 436 42 414 132 469)(25 470 133 415 43 437 86 210 351 316 100 185 378 263 243)(26 244 264 379 186 101 317 352 211 87 438 44 416 134 471)(27 472 135 385 45 439 88 212 321 318 102 187 380 265 245)(28 246 266 381 188 103 319 322 213 89 440 46 386 136 473)(29 474 137 387 47 441 90 214 323 320 104 189 382 267 247)(30 248 268 383 190 105 289 324 215 91 442 48 388 138 475)(31 476 139 389 49 443 92 216 325 290 106 191 384 269 249)(32 250 270 353 192 107 291 326 217 93 444 50 390 140 477)
(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)| (1,478,141,391,51,445,94,218,327,292,108,161,354,271,251)(2,252,272,355,162,109,293,328,219,95,446,52,392,142,479)(3,480,143,393,53,447,96,220,329,294,110,163,356,273,253)(4,254,274,357,164,111,295,330,221,65,448,54,394,144,449)(5,450,145,395,55,417,66,222,331,296,112,165,358,275,255)(6,256,276,359,166,113,297,332,223,67,418,56,396,146,451)(7,452,147,397,57,419,68,224,333,298,114,167,360,277,225)(8,226,278,361,168,115,299,334,193,69,420,58,398,148,453)(9,454,149,399,59,421,70,194,335,300,116,169,362,279,227)(10,228,280,363,170,117,301,336,195,71,422,60,400,150,455)(11,456,151,401,61,423,72,196,337,302,118,171,364,281,229)(12,230,282,365,172,119,303,338,197,73,424,62,402,152,457)(13,458,153,403,63,425,74,198,339,304,120,173,366,283,231)(14,232,284,367,174,121,305,340,199,75,426,64,404,154,459)(15,460,155,405,33,427,76,200,341,306,122,175,368,285,233)(16,234,286,369,176,123,307,342,201,77,428,34,406,156,461)(17,462,157,407,35,429,78,202,343,308,124,177,370,287,235)(18,236,288,371,178,125,309,344,203,79,430,36,408,158,463)(19,464,159,409,37,431,80,204,345,310,126,179,372,257,237)(20,238,258,373,180,127,311,346,205,81,432,38,410,160,465)(21,466,129,411,39,433,82,206,347,312,128,181,374,259,239)(22,240,260,375,182,97,313,348,207,83,434,40,412,130,467)(23,468,131,413,41,435,84,208,349,314,98,183,376,261,241)(24,242,262,377,184,99,315,350,209,85,436,42,414,132,469)(25,470,133,415,43,437,86,210,351,316,100,185,378,263,243)(26,244,264,379,186,101,317,352,211,87,438,44,416,134,471)(27,472,135,385,45,439,88,212,321,318,102,187,380,265,245)(28,246,266,381,188,103,319,322,213,89,440,46,386,136,473)(29,474,137,387,47,441,90,214,323,320,104,189,382,267,247)(30,248,268,383,190,105,289,324,215,91,442,48,388,138,475)(31,476,139,389,49,443,92,216,325,290,106,191,384,269,249)(32,250,270,353,192,107,291,326,217,93,444,50,390,140,477), (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( (1,478,141,391,51,445,94,218,327,292,108,161,354,271,251)(2,252,272,355,162,109,293,328,219,95,446,52,392,142,479)(3,480,143,393,53,447,96,220,329,294,110,163,356,273,253)(4,254,274,357,164,111,295,330,221,65,448,54,394,144,449)(5,450,145,395,55,417,66,222,331,296,112,165,358,275,255)(6,256,276,359,166,113,297,332,223,67,418,56,396,146,451)(7,452,147,397,57,419,68,224,333,298,114,167,360,277,225)(8,226,278,361,168,115,299,334,193,69,420,58,398,148,453)(9,454,149,399,59,421,70,194,335,300,116,169,362,279,227)(10,228,280,363,170,117,301,336,195,71,422,60,400,150,455)(11,456,151,401,61,423,72,196,337,302,118,171,364,281,229)(12,230,282,365,172,119,303,338,197,73,424,62,402,152,457)(13,458,153,403,63,425,74,198,339,304,120,173,366,283,231)(14,232,284,367,174,121,305,340,199,75,426,64,404,154,459)(15,460,155,405,33,427,76,200,341,306,122,175,368,285,233)(16,234,286,369,176,123,307,342,201,77,428,34,406,156,461)(17,462,157,407,35,429,78,202,343,308,124,177,370,287,235)(18,236,288,371,178,125,309,344,203,79,430,36,408,158,463)(19,464,159,409,37,431,80,204,345,310,126,179,372,257,237)(20,238,258,373,180,127,311,346,205,81,432,38,410,160,465)(21,466,129,411,39,433,82,206,347,312,128,181,374,259,239)(22,240,260,375,182,97,313,348,207,83,434,40,412,130,467)(23,468,131,413,41,435,84,208,349,314,98,183,376,261,241)(24,242,262,377,184,99,315,350,209,85,436,42,414,132,469)(25,470,133,415,43,437,86,210,351,316,100,185,378,263,243)(26,244,264,379,186,101,317,352,211,87,438,44,416,134,471)(27,472,135,385,45,439,88,212,321,318,102,187,380,265,245)(28,246,266,381,188,103,319,322,213,89,440,46,386,136,473)(29,474,137,387,47,441,90,214,323,320,104,189,382,267,247)(30,248,268,383,190,105,289,324,215,91,442,48,388,138,475)(31,476,139,389,49,443,92,216,325,290,106,191,384,269,249)(32,250,270,353,192,107,291,326,217,93,444,50,390,140,477), (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([[(1,478,141,391,51,445,94,218,327,292,108,161,354,271,251),(2,252,272,355,162,109,293,328,219,95,446,52,392,142,479),(3,480,143,393,53,447,96,220,329,294,110,163,356,273,253),(4,254,274,357,164,111,295,330,221,65,448,54,394,144,449),(5,450,145,395,55,417,66,222,331,296,112,165,358,275,255),(6,256,276,359,166,113,297,332,223,67,418,56,396,146,451),(7,452,147,397,57,419,68,224,333,298,114,167,360,277,225),(8,226,278,361,168,115,299,334,193,69,420,58,398,148,453),(9,454,149,399,59,421,70,194,335,300,116,169,362,279,227),(10,228,280,363,170,117,301,336,195,71,422,60,400,150,455),(11,456,151,401,61,423,72,196,337,302,118,171,364,281,229),(12,230,282,365,172,119,303,338,197,73,424,62,402,152,457),(13,458,153,403,63,425,74,198,339,304,120,173,366,283,231),(14,232,284,367,174,121,305,340,199,75,426,64,404,154,459),(15,460,155,405,33,427,76,200,341,306,122,175,368,285,233),(16,234,286,369,176,123,307,342,201,77,428,34,406,156,461),(17,462,157,407,35,429,78,202,343,308,124,177,370,287,235),(18,236,288,371,178,125,309,344,203,79,430,36,408,158,463),(19,464,159,409,37,431,80,204,345,310,126,179,372,257,237),(20,238,258,373,180,127,311,346,205,81,432,38,410,160,465),(21,466,129,411,39,433,82,206,347,312,128,181,374,259,239),(22,240,260,375,182,97,313,348,207,83,434,40,412,130,467),(23,468,131,413,41,435,84,208,349,314,98,183,376,261,241),(24,242,262,377,184,99,315,350,209,85,436,42,414,132,469),(25,470,133,415,43,437,86,210,351,316,100,185,378,263,243),(26,244,264,379,186,101,317,352,211,87,438,44,416,134,471),(27,472,135,385,45,439,88,212,321,318,102,187,380,265,245),(28,246,266,381,188,103,319,322,213,89,440,46,386,136,473),(29,474,137,387,47,441,90,214,323,320,104,189,382,267,247),(30,248,268,383,190,105,289,324,215,91,442,48,388,138,475),(31,476,139,389,49,443,92,216,325,290,106,191,384,269,249),(32,250,270,353,192,107,291,326,217,93,444,50,390,140,477)], [(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)]])
144 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 32A | ··· | 32P | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 32 | ··· | 32 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |||||||||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | S3 | D5 | Dic3 | Dic5 | C3⋊C8 | D15 | C5⋊2C8 | C3⋊C16 | Dic15 | C5⋊2C16 | C3⋊C32 | C15⋊3C8 | C5⋊2C32 | C15⋊3C16 | C15⋊3C32 |
| kernel | C15⋊3C32 | C240 | C120 | C60 | C30 | C15 | C80 | C48 | C40 | C24 | C20 | C16 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15⋊3C32 ►in GL4(𝔽3361) generated by
| 3360 | 488 | 0 | 0 |
| 2873 | 2873 | 0 | 0 |
| 0 | 0 | 1548 | 1806 |
| 0 | 0 | 1555 | 2294 |
| 41 | 1386 | 0 | 0 |
| 1228 | 3320 | 0 | 0 |
| 0 | 0 | 411 | 1790 |
| 0 | 0 | 698 | 2950 |
G:=sub<GL(4,GF(3361))| [3360,2873,0,0,488,2873,0,0,0,0,1548,1555,0,0,1806,2294],[41,1228,0,0,1386,3320,0,0,0,0,411,698,0,0,1790,2950] >;
C15⋊3C32 in GAP, Magma, Sage, TeX
C_{15}\rtimes_3C_{32} % in TeX
G:=Group("C15:3C32"); // GroupNames label
G:=SmallGroup(480,3);
// by ID
G=gap.SmallGroup(480,3);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,14,36,58,80,2693,18822]);
// Polycyclic
G:=Group<a,b|a^15=b^32=1,b*a*b^-1=a^-1>;
// generators/relations
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