direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C15×Q32, C16.C30, C80.2C6, Q16.C30, C240.5C2, C48.2C10, C30.62D8, C60.195D4, C120.105C22, C8.4(C2×C30), C4.3(D4×C15), C6.17(C5×D8), C2.5(C15×D8), C40.26(C2×C6), C10.17(C3×D8), C20.38(C3×D4), C12.38(C5×D4), (C5×Q16).2C6, C24.21(C2×C10), (C3×Q16).2C10, (C15×Q16).4C2, SmallGroup(480,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×Q32
G = < a,b,c | a15=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C15×Q32
►Regular action on 480 points
Generators in S480
(1 117 51 140 349 22 477 190 447 149 389 246 220 168 462)(2 118 52 141 350 23 478 191 448 150 390 247 221 169 463)(3 119 53 142 351 24 479 192 433 151 391 248 222 170 464)(4 120 54 143 352 25 480 177 434 152 392 249 223 171 449)(5 121 55 144 337 26 465 178 435 153 393 250 224 172 450)(6 122 56 129 338 27 466 179 436 154 394 251 209 173 451)(7 123 57 130 339 28 467 180 437 155 395 252 210 174 452)(8 124 58 131 340 29 468 181 438 156 396 253 211 175 453)(9 125 59 132 341 30 469 182 439 157 397 254 212 176 454)(10 126 60 133 342 31 470 183 440 158 398 255 213 161 455)(11 127 61 134 343 32 471 184 441 159 399 256 214 162 456)(12 128 62 135 344 17 472 185 442 160 400 241 215 163 457)(13 113 63 136 345 18 473 186 443 145 385 242 216 164 458)(14 114 64 137 346 19 474 187 444 146 386 243 217 165 459)(15 115 49 138 347 20 475 188 445 147 387 244 218 166 460)(16 116 50 139 348 21 476 189 446 148 388 245 219 167 461)(33 384 227 289 418 321 402 196 111 365 265 285 310 93 69)(34 369 228 290 419 322 403 197 112 366 266 286 311 94 70)(35 370 229 291 420 323 404 198 97 367 267 287 312 95 71)(36 371 230 292 421 324 405 199 98 368 268 288 313 96 72)(37 372 231 293 422 325 406 200 99 353 269 273 314 81 73)(38 373 232 294 423 326 407 201 100 354 270 274 315 82 74)(39 374 233 295 424 327 408 202 101 355 271 275 316 83 75)(40 375 234 296 425 328 409 203 102 356 272 276 317 84 76)(41 376 235 297 426 329 410 204 103 357 257 277 318 85 77)(42 377 236 298 427 330 411 205 104 358 258 278 319 86 78)(43 378 237 299 428 331 412 206 105 359 259 279 320 87 79)(44 379 238 300 429 332 413 207 106 360 260 280 305 88 80)(45 380 239 301 430 333 414 208 107 361 261 281 306 89 65)(46 381 240 302 431 334 415 193 108 362 262 282 307 90 66)(47 382 225 303 432 335 416 194 109 363 263 283 308 91 67)(48 383 226 304 417 336 401 195 110 364 264 284 309 92 68)
(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)
(1 269 9 261)(2 268 10 260)(3 267 11 259)(4 266 12 258)(5 265 13 257)(6 264 14 272)(7 263 15 271)(8 262 16 270)(17 42 25 34)(18 41 26 33)(19 40 27 48)(20 39 28 47)(21 38 29 46)(22 37 30 45)(23 36 31 44)(24 35 32 43)(49 316 57 308)(50 315 58 307)(51 314 59 306)(52 313 60 305)(53 312 61 320)(54 311 62 319)(55 310 63 318)(56 309 64 317)(65 349 73 341)(66 348 74 340)(67 347 75 339)(68 346 76 338)(69 345 77 337)(70 344 78 352)(71 343 79 351)(72 342 80 350)(81 132 89 140)(82 131 90 139)(83 130 91 138)(84 129 92 137)(85 144 93 136)(86 143 94 135)(87 142 95 134)(88 141 96 133)(97 162 105 170)(98 161 106 169)(99 176 107 168)(100 175 108 167)(101 174 109 166)(102 173 110 165)(103 172 111 164)(104 171 112 163)(113 277 121 285)(114 276 122 284)(115 275 123 283)(116 274 124 282)(117 273 125 281)(118 288 126 280)(119 287 127 279)(120 286 128 278)(145 426 153 418)(146 425 154 417)(147 424 155 432)(148 423 156 431)(149 422 157 430)(150 421 158 429)(151 420 159 428)(152 419 160 427)(177 228 185 236)(178 227 186 235)(179 226 187 234)(180 225 188 233)(181 240 189 232)(182 239 190 231)(183 238 191 230)(184 237 192 229)(193 219 201 211)(194 218 202 210)(195 217 203 209)(196 216 204 224)(197 215 205 223)(198 214 206 222)(199 213 207 221)(200 212 208 220)(241 411 249 403)(242 410 250 402)(243 409 251 401)(244 408 252 416)(245 407 253 415)(246 406 254 414)(247 405 255 413)(248 404 256 412)(289 443 297 435)(290 442 298 434)(291 441 299 433)(292 440 300 448)(293 439 301 447)(294 438 302 446)(295 437 303 445)(296 436 304 444)(321 385 329 393)(322 400 330 392)(323 399 331 391)(324 398 332 390)(325 397 333 389)(326 396 334 388)(327 395 335 387)(328 394 336 386)(353 454 361 462)(354 453 362 461)(355 452 363 460)(356 451 364 459)(357 450 365 458)(358 449 366 457)(359 464 367 456)(360 463 368 455)(369 472 377 480)(370 471 378 479)(371 470 379 478)(372 469 380 477)(373 468 381 476)(374 467 382 475)(375 466 383 474)(376 465 384 473)
G:=sub<Sym(480)| (1,117,51,140,349,22,477,190,447,149,389,246,220,168,462)(2,118,52,141,350,23,478,191,448,150,390,247,221,169,463)(3,119,53,142,351,24,479,192,433,151,391,248,222,170,464)(4,120,54,143,352,25,480,177,434,152,392,249,223,171,449)(5,121,55,144,337,26,465,178,435,153,393,250,224,172,450)(6,122,56,129,338,27,466,179,436,154,394,251,209,173,451)(7,123,57,130,339,28,467,180,437,155,395,252,210,174,452)(8,124,58,131,340,29,468,181,438,156,396,253,211,175,453)(9,125,59,132,341,30,469,182,439,157,397,254,212,176,454)(10,126,60,133,342,31,470,183,440,158,398,255,213,161,455)(11,127,61,134,343,32,471,184,441,159,399,256,214,162,456)(12,128,62,135,344,17,472,185,442,160,400,241,215,163,457)(13,113,63,136,345,18,473,186,443,145,385,242,216,164,458)(14,114,64,137,346,19,474,187,444,146,386,243,217,165,459)(15,115,49,138,347,20,475,188,445,147,387,244,218,166,460)(16,116,50,139,348,21,476,189,446,148,388,245,219,167,461)(33,384,227,289,418,321,402,196,111,365,265,285,310,93,69)(34,369,228,290,419,322,403,197,112,366,266,286,311,94,70)(35,370,229,291,420,323,404,198,97,367,267,287,312,95,71)(36,371,230,292,421,324,405,199,98,368,268,288,313,96,72)(37,372,231,293,422,325,406,200,99,353,269,273,314,81,73)(38,373,232,294,423,326,407,201,100,354,270,274,315,82,74)(39,374,233,295,424,327,408,202,101,355,271,275,316,83,75)(40,375,234,296,425,328,409,203,102,356,272,276,317,84,76)(41,376,235,297,426,329,410,204,103,357,257,277,318,85,77)(42,377,236,298,427,330,411,205,104,358,258,278,319,86,78)(43,378,237,299,428,331,412,206,105,359,259,279,320,87,79)(44,379,238,300,429,332,413,207,106,360,260,280,305,88,80)(45,380,239,301,430,333,414,208,107,361,261,281,306,89,65)(46,381,240,302,431,334,415,193,108,362,262,282,307,90,66)(47,382,225,303,432,335,416,194,109,363,263,283,308,91,67)(48,383,226,304,417,336,401,195,110,364,264,284,309,92,68), (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), (1,269,9,261)(2,268,10,260)(3,267,11,259)(4,266,12,258)(5,265,13,257)(6,264,14,272)(7,263,15,271)(8,262,16,270)(17,42,25,34)(18,41,26,33)(19,40,27,48)(20,39,28,47)(21,38,29,46)(22,37,30,45)(23,36,31,44)(24,35,32,43)(49,316,57,308)(50,315,58,307)(51,314,59,306)(52,313,60,305)(53,312,61,320)(54,311,62,319)(55,310,63,318)(56,309,64,317)(65,349,73,341)(66,348,74,340)(67,347,75,339)(68,346,76,338)(69,345,77,337)(70,344,78,352)(71,343,79,351)(72,342,80,350)(81,132,89,140)(82,131,90,139)(83,130,91,138)(84,129,92,137)(85,144,93,136)(86,143,94,135)(87,142,95,134)(88,141,96,133)(97,162,105,170)(98,161,106,169)(99,176,107,168)(100,175,108,167)(101,174,109,166)(102,173,110,165)(103,172,111,164)(104,171,112,163)(113,277,121,285)(114,276,122,284)(115,275,123,283)(116,274,124,282)(117,273,125,281)(118,288,126,280)(119,287,127,279)(120,286,128,278)(145,426,153,418)(146,425,154,417)(147,424,155,432)(148,423,156,431)(149,422,157,430)(150,421,158,429)(151,420,159,428)(152,419,160,427)(177,228,185,236)(178,227,186,235)(179,226,187,234)(180,225,188,233)(181,240,189,232)(182,239,190,231)(183,238,191,230)(184,237,192,229)(193,219,201,211)(194,218,202,210)(195,217,203,209)(196,216,204,224)(197,215,205,223)(198,214,206,222)(199,213,207,221)(200,212,208,220)(241,411,249,403)(242,410,250,402)(243,409,251,401)(244,408,252,416)(245,407,253,415)(246,406,254,414)(247,405,255,413)(248,404,256,412)(289,443,297,435)(290,442,298,434)(291,441,299,433)(292,440,300,448)(293,439,301,447)(294,438,302,446)(295,437,303,445)(296,436,304,444)(321,385,329,393)(322,400,330,392)(323,399,331,391)(324,398,332,390)(325,397,333,389)(326,396,334,388)(327,395,335,387)(328,394,336,386)(353,454,361,462)(354,453,362,461)(355,452,363,460)(356,451,364,459)(357,450,365,458)(358,449,366,457)(359,464,367,456)(360,463,368,455)(369,472,377,480)(370,471,378,479)(371,470,379,478)(372,469,380,477)(373,468,381,476)(374,467,382,475)(375,466,383,474)(376,465,384,473)>;
G:=Group( (1,117,51,140,349,22,477,190,447,149,389,246,220,168,462)(2,118,52,141,350,23,478,191,448,150,390,247,221,169,463)(3,119,53,142,351,24,479,192,433,151,391,248,222,170,464)(4,120,54,143,352,25,480,177,434,152,392,249,223,171,449)(5,121,55,144,337,26,465,178,435,153,393,250,224,172,450)(6,122,56,129,338,27,466,179,436,154,394,251,209,173,451)(7,123,57,130,339,28,467,180,437,155,395,252,210,174,452)(8,124,58,131,340,29,468,181,438,156,396,253,211,175,453)(9,125,59,132,341,30,469,182,439,157,397,254,212,176,454)(10,126,60,133,342,31,470,183,440,158,398,255,213,161,455)(11,127,61,134,343,32,471,184,441,159,399,256,214,162,456)(12,128,62,135,344,17,472,185,442,160,400,241,215,163,457)(13,113,63,136,345,18,473,186,443,145,385,242,216,164,458)(14,114,64,137,346,19,474,187,444,146,386,243,217,165,459)(15,115,49,138,347,20,475,188,445,147,387,244,218,166,460)(16,116,50,139,348,21,476,189,446,148,388,245,219,167,461)(33,384,227,289,418,321,402,196,111,365,265,285,310,93,69)(34,369,228,290,419,322,403,197,112,366,266,286,311,94,70)(35,370,229,291,420,323,404,198,97,367,267,287,312,95,71)(36,371,230,292,421,324,405,199,98,368,268,288,313,96,72)(37,372,231,293,422,325,406,200,99,353,269,273,314,81,73)(38,373,232,294,423,326,407,201,100,354,270,274,315,82,74)(39,374,233,295,424,327,408,202,101,355,271,275,316,83,75)(40,375,234,296,425,328,409,203,102,356,272,276,317,84,76)(41,376,235,297,426,329,410,204,103,357,257,277,318,85,77)(42,377,236,298,427,330,411,205,104,358,258,278,319,86,78)(43,378,237,299,428,331,412,206,105,359,259,279,320,87,79)(44,379,238,300,429,332,413,207,106,360,260,280,305,88,80)(45,380,239,301,430,333,414,208,107,361,261,281,306,89,65)(46,381,240,302,431,334,415,193,108,362,262,282,307,90,66)(47,382,225,303,432,335,416,194,109,363,263,283,308,91,67)(48,383,226,304,417,336,401,195,110,364,264,284,309,92,68), (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), (1,269,9,261)(2,268,10,260)(3,267,11,259)(4,266,12,258)(5,265,13,257)(6,264,14,272)(7,263,15,271)(8,262,16,270)(17,42,25,34)(18,41,26,33)(19,40,27,48)(20,39,28,47)(21,38,29,46)(22,37,30,45)(23,36,31,44)(24,35,32,43)(49,316,57,308)(50,315,58,307)(51,314,59,306)(52,313,60,305)(53,312,61,320)(54,311,62,319)(55,310,63,318)(56,309,64,317)(65,349,73,341)(66,348,74,340)(67,347,75,339)(68,346,76,338)(69,345,77,337)(70,344,78,352)(71,343,79,351)(72,342,80,350)(81,132,89,140)(82,131,90,139)(83,130,91,138)(84,129,92,137)(85,144,93,136)(86,143,94,135)(87,142,95,134)(88,141,96,133)(97,162,105,170)(98,161,106,169)(99,176,107,168)(100,175,108,167)(101,174,109,166)(102,173,110,165)(103,172,111,164)(104,171,112,163)(113,277,121,285)(114,276,122,284)(115,275,123,283)(116,274,124,282)(117,273,125,281)(118,288,126,280)(119,287,127,279)(120,286,128,278)(145,426,153,418)(146,425,154,417)(147,424,155,432)(148,423,156,431)(149,422,157,430)(150,421,158,429)(151,420,159,428)(152,419,160,427)(177,228,185,236)(178,227,186,235)(179,226,187,234)(180,225,188,233)(181,240,189,232)(182,239,190,231)(183,238,191,230)(184,237,192,229)(193,219,201,211)(194,218,202,210)(195,217,203,209)(196,216,204,224)(197,215,205,223)(198,214,206,222)(199,213,207,221)(200,212,208,220)(241,411,249,403)(242,410,250,402)(243,409,251,401)(244,408,252,416)(245,407,253,415)(246,406,254,414)(247,405,255,413)(248,404,256,412)(289,443,297,435)(290,442,298,434)(291,441,299,433)(292,440,300,448)(293,439,301,447)(294,438,302,446)(295,437,303,445)(296,436,304,444)(321,385,329,393)(322,400,330,392)(323,399,331,391)(324,398,332,390)(325,397,333,389)(326,396,334,388)(327,395,335,387)(328,394,336,386)(353,454,361,462)(354,453,362,461)(355,452,363,460)(356,451,364,459)(357,450,365,458)(358,449,366,457)(359,464,367,456)(360,463,368,455)(369,472,377,480)(370,471,378,479)(371,470,379,478)(372,469,380,477)(373,468,381,476)(374,467,382,475)(375,466,383,474)(376,465,384,473) );
G=PermutationGroup([[(1,117,51,140,349,22,477,190,447,149,389,246,220,168,462),(2,118,52,141,350,23,478,191,448,150,390,247,221,169,463),(3,119,53,142,351,24,479,192,433,151,391,248,222,170,464),(4,120,54,143,352,25,480,177,434,152,392,249,223,171,449),(5,121,55,144,337,26,465,178,435,153,393,250,224,172,450),(6,122,56,129,338,27,466,179,436,154,394,251,209,173,451),(7,123,57,130,339,28,467,180,437,155,395,252,210,174,452),(8,124,58,131,340,29,468,181,438,156,396,253,211,175,453),(9,125,59,132,341,30,469,182,439,157,397,254,212,176,454),(10,126,60,133,342,31,470,183,440,158,398,255,213,161,455),(11,127,61,134,343,32,471,184,441,159,399,256,214,162,456),(12,128,62,135,344,17,472,185,442,160,400,241,215,163,457),(13,113,63,136,345,18,473,186,443,145,385,242,216,164,458),(14,114,64,137,346,19,474,187,444,146,386,243,217,165,459),(15,115,49,138,347,20,475,188,445,147,387,244,218,166,460),(16,116,50,139,348,21,476,189,446,148,388,245,219,167,461),(33,384,227,289,418,321,402,196,111,365,265,285,310,93,69),(34,369,228,290,419,322,403,197,112,366,266,286,311,94,70),(35,370,229,291,420,323,404,198,97,367,267,287,312,95,71),(36,371,230,292,421,324,405,199,98,368,268,288,313,96,72),(37,372,231,293,422,325,406,200,99,353,269,273,314,81,73),(38,373,232,294,423,326,407,201,100,354,270,274,315,82,74),(39,374,233,295,424,327,408,202,101,355,271,275,316,83,75),(40,375,234,296,425,328,409,203,102,356,272,276,317,84,76),(41,376,235,297,426,329,410,204,103,357,257,277,318,85,77),(42,377,236,298,427,330,411,205,104,358,258,278,319,86,78),(43,378,237,299,428,331,412,206,105,359,259,279,320,87,79),(44,379,238,300,429,332,413,207,106,360,260,280,305,88,80),(45,380,239,301,430,333,414,208,107,361,261,281,306,89,65),(46,381,240,302,431,334,415,193,108,362,262,282,307,90,66),(47,382,225,303,432,335,416,194,109,363,263,283,308,91,67),(48,383,226,304,417,336,401,195,110,364,264,284,309,92,68)], [(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)], [(1,269,9,261),(2,268,10,260),(3,267,11,259),(4,266,12,258),(5,265,13,257),(6,264,14,272),(7,263,15,271),(8,262,16,270),(17,42,25,34),(18,41,26,33),(19,40,27,48),(20,39,28,47),(21,38,29,46),(22,37,30,45),(23,36,31,44),(24,35,32,43),(49,316,57,308),(50,315,58,307),(51,314,59,306),(52,313,60,305),(53,312,61,320),(54,311,62,319),(55,310,63,318),(56,309,64,317),(65,349,73,341),(66,348,74,340),(67,347,75,339),(68,346,76,338),(69,345,77,337),(70,344,78,352),(71,343,79,351),(72,342,80,350),(81,132,89,140),(82,131,90,139),(83,130,91,138),(84,129,92,137),(85,144,93,136),(86,143,94,135),(87,142,95,134),(88,141,96,133),(97,162,105,170),(98,161,106,169),(99,176,107,168),(100,175,108,167),(101,174,109,166),(102,173,110,165),(103,172,111,164),(104,171,112,163),(113,277,121,285),(114,276,122,284),(115,275,123,283),(116,274,124,282),(117,273,125,281),(118,288,126,280),(119,287,127,279),(120,286,128,278),(145,426,153,418),(146,425,154,417),(147,424,155,432),(148,423,156,431),(149,422,157,430),(150,421,158,429),(151,420,159,428),(152,419,160,427),(177,228,185,236),(178,227,186,235),(179,226,187,234),(180,225,188,233),(181,240,189,232),(182,239,190,231),(183,238,191,230),(184,237,192,229),(193,219,201,211),(194,218,202,210),(195,217,203,209),(196,216,204,224),(197,215,205,223),(198,214,206,222),(199,213,207,221),(200,212,208,220),(241,411,249,403),(242,410,250,402),(243,409,251,401),(244,408,252,416),(245,407,253,415),(246,406,254,414),(247,405,255,413),(248,404,256,412),(289,443,297,435),(290,442,298,434),(291,441,299,433),(292,440,300,448),(293,439,301,447),(294,438,302,446),(295,437,303,445),(296,436,304,444),(321,385,329,393),(322,400,330,392),(323,399,331,391),(324,398,332,390),(325,397,333,389),(326,396,334,388),(327,395,335,387),(328,394,336,386),(353,454,361,462),(354,453,362,461),(355,452,363,460),(356,451,364,459),(357,450,365,458),(358,449,366,457),(359,464,367,456),(360,463,368,455),(369,472,377,480),(370,471,378,479),(371,470,379,478),(372,469,380,477),(373,468,381,476),(374,467,382,475),(375,466,383,474),(376,465,384,473)]])
165 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | ··· | 15H | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 60I | ··· | 60X | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | D4 | D8 | C3×D4 | Q32 | C5×D4 | C3×D8 | C5×D8 | C3×Q32 | D4×C15 | C5×Q32 | C15×D8 | C15×Q32 |
| kernel | C15×Q32 | C240 | C15×Q16 | C5×Q32 | C3×Q32 | C80 | C5×Q16 | C48 | C3×Q16 | Q32 | C16 | Q16 | C60 | C30 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×Q32 ►in GL2(𝔽31) generated by
| 19 | 0 |
| 0 | 19 |
| 20 | 10 |
| 4 | 16 |
| 0 | 7 |
| 22 | 0 |
G:=sub<GL(2,GF(31))| [19,0,0,19],[20,4,10,16],[0,22,7,0] >;
C15×Q32 in GAP, Magma, Sage, TeX
C_{15}\times Q_{32} % in TeX
G:=Group("C15xQ32"); // GroupNames label
G:=SmallGroup(480,216);
// by ID
G=gap.SmallGroup(480,216);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,1680,869,1688,6304,3161,242,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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