Copied to
clipboard

G = C153C32order 480 = 25·3·5

1st semidirect product of C15 and C32 acting via C32/C16=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C153C32, C60.8C8, C80.3S3, C48.3D5, C240.4C2, C30.3C16, C16.2D15, C120.12C4, C40.7Dic3, C24.3Dic5, C8.3Dic15, C52(C3⋊C32), C3⋊(C52C32), C6.(C52C16), C20.5(C3⋊C8), C10.2(C3⋊C16), C2.(C153C16), C4.2(C153C8), C12.2(C52C8), SmallGroup(480,3)

Series: Derived Chief Lower central Upper central

C1C15 — C153C32
C1C5C15C30C60C120C240 — C153C32
C15 — C153C32
C1C16

Generators and relations for C153C32
 G = < a,b | a15=b32=1, bab-1=a-1 >

15C32
5C3⋊C32
3C52C32

Smallest permutation representation of C153C32
Regular action on 480 points
Generators in S480
(1 157 254 310 349 169 476 271 105 376 411 434 62 70 211)(2 212 71 63 435 412 377 106 272 477 170 350 311 255 158)(3 159 256 312 351 171 478 273 107 378 413 436 64 72 213)(4 214 73 33 437 414 379 108 274 479 172 352 313 225 160)(5 129 226 314 321 173 480 275 109 380 415 438 34 74 215)(6 216 75 35 439 416 381 110 276 449 174 322 315 227 130)(7 131 228 316 323 175 450 277 111 382 385 440 36 76 217)(8 218 77 37 441 386 383 112 278 451 176 324 317 229 132)(9 133 230 318 325 177 452 279 113 384 387 442 38 78 219)(10 220 79 39 443 388 353 114 280 453 178 326 319 231 134)(11 135 232 320 327 179 454 281 115 354 389 444 40 80 221)(12 222 81 41 445 390 355 116 282 455 180 328 289 233 136)(13 137 234 290 329 181 456 283 117 356 391 446 42 82 223)(14 224 83 43 447 392 357 118 284 457 182 330 291 235 138)(15 139 236 292 331 183 458 285 119 358 393 448 44 84 193)(16 194 85 45 417 394 359 120 286 459 184 332 293 237 140)(17 141 238 294 333 185 460 287 121 360 395 418 46 86 195)(18 196 87 47 419 396 361 122 288 461 186 334 295 239 142)(19 143 240 296 335 187 462 257 123 362 397 420 48 88 197)(20 198 89 49 421 398 363 124 258 463 188 336 297 241 144)(21 145 242 298 337 189 464 259 125 364 399 422 50 90 199)(22 200 91 51 423 400 365 126 260 465 190 338 299 243 146)(23 147 244 300 339 191 466 261 127 366 401 424 52 92 201)(24 202 93 53 425 402 367 128 262 467 192 340 301 245 148)(25 149 246 302 341 161 468 263 97 368 403 426 54 94 203)(26 204 95 55 427 404 369 98 264 469 162 342 303 247 150)(27 151 248 304 343 163 470 265 99 370 405 428 56 96 205)(28 206 65 57 429 406 371 100 266 471 164 344 305 249 152)(29 153 250 306 345 165 472 267 101 372 407 430 58 66 207)(30 208 67 59 431 408 373 102 268 473 166 346 307 251 154)(31 155 252 308 347 167 474 269 103 374 409 432 60 68 209)(32 210 69 61 433 410 375 104 270 475 168 348 309 253 156)
(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)| (1,157,254,310,349,169,476,271,105,376,411,434,62,70,211)(2,212,71,63,435,412,377,106,272,477,170,350,311,255,158)(3,159,256,312,351,171,478,273,107,378,413,436,64,72,213)(4,214,73,33,437,414,379,108,274,479,172,352,313,225,160)(5,129,226,314,321,173,480,275,109,380,415,438,34,74,215)(6,216,75,35,439,416,381,110,276,449,174,322,315,227,130)(7,131,228,316,323,175,450,277,111,382,385,440,36,76,217)(8,218,77,37,441,386,383,112,278,451,176,324,317,229,132)(9,133,230,318,325,177,452,279,113,384,387,442,38,78,219)(10,220,79,39,443,388,353,114,280,453,178,326,319,231,134)(11,135,232,320,327,179,454,281,115,354,389,444,40,80,221)(12,222,81,41,445,390,355,116,282,455,180,328,289,233,136)(13,137,234,290,329,181,456,283,117,356,391,446,42,82,223)(14,224,83,43,447,392,357,118,284,457,182,330,291,235,138)(15,139,236,292,331,183,458,285,119,358,393,448,44,84,193)(16,194,85,45,417,394,359,120,286,459,184,332,293,237,140)(17,141,238,294,333,185,460,287,121,360,395,418,46,86,195)(18,196,87,47,419,396,361,122,288,461,186,334,295,239,142)(19,143,240,296,335,187,462,257,123,362,397,420,48,88,197)(20,198,89,49,421,398,363,124,258,463,188,336,297,241,144)(21,145,242,298,337,189,464,259,125,364,399,422,50,90,199)(22,200,91,51,423,400,365,126,260,465,190,338,299,243,146)(23,147,244,300,339,191,466,261,127,366,401,424,52,92,201)(24,202,93,53,425,402,367,128,262,467,192,340,301,245,148)(25,149,246,302,341,161,468,263,97,368,403,426,54,94,203)(26,204,95,55,427,404,369,98,264,469,162,342,303,247,150)(27,151,248,304,343,163,470,265,99,370,405,428,56,96,205)(28,206,65,57,429,406,371,100,266,471,164,344,305,249,152)(29,153,250,306,345,165,472,267,101,372,407,430,58,66,207)(30,208,67,59,431,408,373,102,268,473,166,346,307,251,154)(31,155,252,308,347,167,474,269,103,374,409,432,60,68,209)(32,210,69,61,433,410,375,104,270,475,168,348,309,253,156), (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( (1,157,254,310,349,169,476,271,105,376,411,434,62,70,211)(2,212,71,63,435,412,377,106,272,477,170,350,311,255,158)(3,159,256,312,351,171,478,273,107,378,413,436,64,72,213)(4,214,73,33,437,414,379,108,274,479,172,352,313,225,160)(5,129,226,314,321,173,480,275,109,380,415,438,34,74,215)(6,216,75,35,439,416,381,110,276,449,174,322,315,227,130)(7,131,228,316,323,175,450,277,111,382,385,440,36,76,217)(8,218,77,37,441,386,383,112,278,451,176,324,317,229,132)(9,133,230,318,325,177,452,279,113,384,387,442,38,78,219)(10,220,79,39,443,388,353,114,280,453,178,326,319,231,134)(11,135,232,320,327,179,454,281,115,354,389,444,40,80,221)(12,222,81,41,445,390,355,116,282,455,180,328,289,233,136)(13,137,234,290,329,181,456,283,117,356,391,446,42,82,223)(14,224,83,43,447,392,357,118,284,457,182,330,291,235,138)(15,139,236,292,331,183,458,285,119,358,393,448,44,84,193)(16,194,85,45,417,394,359,120,286,459,184,332,293,237,140)(17,141,238,294,333,185,460,287,121,360,395,418,46,86,195)(18,196,87,47,419,396,361,122,288,461,186,334,295,239,142)(19,143,240,296,335,187,462,257,123,362,397,420,48,88,197)(20,198,89,49,421,398,363,124,258,463,188,336,297,241,144)(21,145,242,298,337,189,464,259,125,364,399,422,50,90,199)(22,200,91,51,423,400,365,126,260,465,190,338,299,243,146)(23,147,244,300,339,191,466,261,127,366,401,424,52,92,201)(24,202,93,53,425,402,367,128,262,467,192,340,301,245,148)(25,149,246,302,341,161,468,263,97,368,403,426,54,94,203)(26,204,95,55,427,404,369,98,264,469,162,342,303,247,150)(27,151,248,304,343,163,470,265,99,370,405,428,56,96,205)(28,206,65,57,429,406,371,100,266,471,164,344,305,249,152)(29,153,250,306,345,165,472,267,101,372,407,430,58,66,207)(30,208,67,59,431,408,373,102,268,473,166,346,307,251,154)(31,155,252,308,347,167,474,269,103,374,409,432,60,68,209)(32,210,69,61,433,410,375,104,270,475,168,348,309,253,156), (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([(1,157,254,310,349,169,476,271,105,376,411,434,62,70,211),(2,212,71,63,435,412,377,106,272,477,170,350,311,255,158),(3,159,256,312,351,171,478,273,107,378,413,436,64,72,213),(4,214,73,33,437,414,379,108,274,479,172,352,313,225,160),(5,129,226,314,321,173,480,275,109,380,415,438,34,74,215),(6,216,75,35,439,416,381,110,276,449,174,322,315,227,130),(7,131,228,316,323,175,450,277,111,382,385,440,36,76,217),(8,218,77,37,441,386,383,112,278,451,176,324,317,229,132),(9,133,230,318,325,177,452,279,113,384,387,442,38,78,219),(10,220,79,39,443,388,353,114,280,453,178,326,319,231,134),(11,135,232,320,327,179,454,281,115,354,389,444,40,80,221),(12,222,81,41,445,390,355,116,282,455,180,328,289,233,136),(13,137,234,290,329,181,456,283,117,356,391,446,42,82,223),(14,224,83,43,447,392,357,118,284,457,182,330,291,235,138),(15,139,236,292,331,183,458,285,119,358,393,448,44,84,193),(16,194,85,45,417,394,359,120,286,459,184,332,293,237,140),(17,141,238,294,333,185,460,287,121,360,395,418,46,86,195),(18,196,87,47,419,396,361,122,288,461,186,334,295,239,142),(19,143,240,296,335,187,462,257,123,362,397,420,48,88,197),(20,198,89,49,421,398,363,124,258,463,188,336,297,241,144),(21,145,242,298,337,189,464,259,125,364,399,422,50,90,199),(22,200,91,51,423,400,365,126,260,465,190,338,299,243,146),(23,147,244,300,339,191,466,261,127,366,401,424,52,92,201),(24,202,93,53,425,402,367,128,262,467,192,340,301,245,148),(25,149,246,302,341,161,468,263,97,368,403,426,54,94,203),(26,204,95,55,427,404,369,98,264,469,162,342,303,247,150),(27,151,248,304,343,163,470,265,99,370,405,428,56,96,205),(28,206,65,57,429,406,371,100,266,471,164,344,305,249,152),(29,153,250,306,345,165,472,267,101,372,407,430,58,66,207),(30,208,67,59,431,408,373,102,268,473,166,346,307,251,154),(31,155,252,308,347,167,474,269,103,374,409,432,60,68,209),(32,210,69,61,433,410,375,104,270,475,168,348,309,253,156)], [(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 4A4B5A5B 6 8A8B8C8D10A10B12A12B15A15B15C15D16A···16H20A20B20C20D24A24B24C24D30A30B30C30D32A···32P40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order123445568888101012121515151516···1620202020242424243030303032···3240···4048···4860···6080···80120···120240···240
size112112221111222222221···122222222222215···152···22···22···22···22···22···2

144 irreducible representations

dim111111222222222222222
type++++--+-
imageC1C2C4C8C16C32S3D5Dic3Dic5C3⋊C8D15C52C8C3⋊C16Dic15C52C16C3⋊C32C153C8C52C32C153C16C153C32
kernelC153C32C240C120C60C30C15C80C48C40C24C20C16C12C10C8C6C5C4C3C2C1
# reps1124816121224444888161632

Matrix representation of C153C32 in GL4(𝔽3361) generated by

336048800
2873287300
0015481806
0015552294
,
41138600
1228332000
004111790
006982950
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] >;

C153C32 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

Export

Subgroup lattice of C153C32 in TeX

׿
×
𝔽