metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊1Q32, C60.75D4, C30.10D8, C40.10D6, C24.10D10, Dic12.2D5, Dic20.2S3, C120.35C22, C8.28(S3×D5), C3⋊2(C5⋊Q32), C5⋊2(C3⋊Q32), C15⋊3C16.2C2, C6.11(D4⋊D5), C4.4(C15⋊D4), C2.7(C15⋊D8), C12.4(C5⋊D4), C20.4(C3⋊D4), C10.11(D4⋊S3), (C5×Dic12).2C2, (C3×Dic20).3C2, SmallGroup(480,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊Q32
G = < a,b,c | a15=b16=1, c2=b8, bab-1=a-1, cac-1=a11, cbc-1=b-1 >
Smallest permutation representation of C15⋊Q32
►Regular action on 480 points
Generators in S480
(1 179 442 299 21 171 478 321 66 338 285 205 220 102 127)(2 128 103 221 206 286 339 67 322 479 172 22 300 443 180)(3 181 444 301 23 173 480 323 68 340 287 207 222 104 113)(4 114 105 223 208 288 341 69 324 465 174 24 302 445 182)(5 183 446 303 25 175 466 325 70 342 273 193 224 106 115)(6 116 107 209 194 274 343 71 326 467 176 26 304 447 184)(7 185 448 289 27 161 468 327 72 344 275 195 210 108 117)(8 118 109 211 196 276 345 73 328 469 162 28 290 433 186)(9 187 434 291 29 163 470 329 74 346 277 197 212 110 119)(10 120 111 213 198 278 347 75 330 471 164 30 292 435 188)(11 189 436 293 31 165 472 331 76 348 279 199 214 112 121)(12 122 97 215 200 280 349 77 332 473 166 32 294 437 190)(13 191 438 295 17 167 474 333 78 350 281 201 216 98 123)(14 124 99 217 202 282 351 79 334 475 168 18 296 439 192)(15 177 440 297 19 169 476 335 80 352 283 203 218 100 125)(16 126 101 219 204 284 337 65 336 477 170 20 298 441 178)(33 464 137 378 238 364 242 313 91 60 399 425 403 258 147)(34 148 259 404 426 400 61 92 314 243 365 239 379 138 449)(35 450 139 380 240 366 244 315 93 62 385 427 405 260 149)(36 150 261 406 428 386 63 94 316 245 367 225 381 140 451)(37 452 141 382 226 368 246 317 95 64 387 429 407 262 151)(38 152 263 408 430 388 49 96 318 247 353 227 383 142 453)(39 454 143 384 228 354 248 319 81 50 389 431 409 264 153)(40 154 265 410 432 390 51 82 320 249 355 229 369 144 455)(41 456 129 370 230 356 250 305 83 52 391 417 411 266 155)(42 156 267 412 418 392 53 84 306 251 357 231 371 130 457)(43 458 131 372 232 358 252 307 85 54 393 419 413 268 157)(44 158 269 414 420 394 55 86 308 253 359 233 373 132 459)(45 460 133 374 234 360 254 309 87 56 395 421 415 270 159)(46 160 271 416 422 396 57 88 310 255 361 235 375 134 461)(47 462 135 376 236 362 256 311 89 58 397 423 401 272 145)(48 146 257 402 424 398 59 90 312 241 363 237 377 136 463)
(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 34 9 42)(2 33 10 41)(3 48 11 40)(4 47 12 39)(5 46 13 38)(6 45 14 37)(7 44 15 36)(8 43 16 35)(17 453 25 461)(18 452 26 460)(19 451 27 459)(20 450 28 458)(21 449 29 457)(22 464 30 456)(23 463 31 455)(24 462 32 454)(49 466 57 474)(50 465 58 473)(51 480 59 472)(52 479 60 471)(53 478 61 470)(54 477 62 469)(55 476 63 468)(56 475 64 467)(65 139 73 131)(66 138 74 130)(67 137 75 129)(68 136 76 144)(69 135 77 143)(70 134 78 142)(71 133 79 141)(72 132 80 140)(81 445 89 437)(82 444 90 436)(83 443 91 435)(84 442 92 434)(85 441 93 433)(86 440 94 448)(87 439 95 447)(88 438 96 446)(97 319 105 311)(98 318 106 310)(99 317 107 309)(100 316 108 308)(101 315 109 307)(102 314 110 306)(103 313 111 305)(104 312 112 320)(113 424 121 432)(114 423 122 431)(115 422 123 430)(116 421 124 429)(117 420 125 428)(118 419 126 427)(119 418 127 426)(120 417 128 425)(145 200 153 208)(146 199 154 207)(147 198 155 206)(148 197 156 205)(149 196 157 204)(150 195 158 203)(151 194 159 202)(152 193 160 201)(161 359 169 367)(162 358 170 366)(163 357 171 365)(164 356 172 364)(165 355 173 363)(166 354 174 362)(167 353 175 361)(168 368 176 360)(177 225 185 233)(178 240 186 232)(179 239 187 231)(180 238 188 230)(181 237 189 229)(182 236 190 228)(183 235 191 227)(184 234 192 226)(209 374 217 382)(210 373 218 381)(211 372 219 380)(212 371 220 379)(213 370 221 378)(214 369 222 377)(215 384 223 376)(216 383 224 375)(241 348 249 340)(242 347 250 339)(243 346 251 338)(244 345 252 337)(245 344 253 352)(246 343 254 351)(247 342 255 350)(248 341 256 349)(257 331 265 323)(258 330 266 322)(259 329 267 321)(260 328 268 336)(261 327 269 335)(262 326 270 334)(263 325 271 333)(264 324 272 332)(273 396 281 388)(274 395 282 387)(275 394 283 386)(276 393 284 385)(277 392 285 400)(278 391 286 399)(279 390 287 398)(280 389 288 397)(289 414 297 406)(290 413 298 405)(291 412 299 404)(292 411 300 403)(293 410 301 402)(294 409 302 401)(295 408 303 416)(296 407 304 415)
G:=sub<Sym(480)| (1,179,442,299,21,171,478,321,66,338,285,205,220,102,127)(2,128,103,221,206,286,339,67,322,479,172,22,300,443,180)(3,181,444,301,23,173,480,323,68,340,287,207,222,104,113)(4,114,105,223,208,288,341,69,324,465,174,24,302,445,182)(5,183,446,303,25,175,466,325,70,342,273,193,224,106,115)(6,116,107,209,194,274,343,71,326,467,176,26,304,447,184)(7,185,448,289,27,161,468,327,72,344,275,195,210,108,117)(8,118,109,211,196,276,345,73,328,469,162,28,290,433,186)(9,187,434,291,29,163,470,329,74,346,277,197,212,110,119)(10,120,111,213,198,278,347,75,330,471,164,30,292,435,188)(11,189,436,293,31,165,472,331,76,348,279,199,214,112,121)(12,122,97,215,200,280,349,77,332,473,166,32,294,437,190)(13,191,438,295,17,167,474,333,78,350,281,201,216,98,123)(14,124,99,217,202,282,351,79,334,475,168,18,296,439,192)(15,177,440,297,19,169,476,335,80,352,283,203,218,100,125)(16,126,101,219,204,284,337,65,336,477,170,20,298,441,178)(33,464,137,378,238,364,242,313,91,60,399,425,403,258,147)(34,148,259,404,426,400,61,92,314,243,365,239,379,138,449)(35,450,139,380,240,366,244,315,93,62,385,427,405,260,149)(36,150,261,406,428,386,63,94,316,245,367,225,381,140,451)(37,452,141,382,226,368,246,317,95,64,387,429,407,262,151)(38,152,263,408,430,388,49,96,318,247,353,227,383,142,453)(39,454,143,384,228,354,248,319,81,50,389,431,409,264,153)(40,154,265,410,432,390,51,82,320,249,355,229,369,144,455)(41,456,129,370,230,356,250,305,83,52,391,417,411,266,155)(42,156,267,412,418,392,53,84,306,251,357,231,371,130,457)(43,458,131,372,232,358,252,307,85,54,393,419,413,268,157)(44,158,269,414,420,394,55,86,308,253,359,233,373,132,459)(45,460,133,374,234,360,254,309,87,56,395,421,415,270,159)(46,160,271,416,422,396,57,88,310,255,361,235,375,134,461)(47,462,135,376,236,362,256,311,89,58,397,423,401,272,145)(48,146,257,402,424,398,59,90,312,241,363,237,377,136,463), (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), (1,34,9,42)(2,33,10,41)(3,48,11,40)(4,47,12,39)(5,46,13,38)(6,45,14,37)(7,44,15,36)(8,43,16,35)(17,453,25,461)(18,452,26,460)(19,451,27,459)(20,450,28,458)(21,449,29,457)(22,464,30,456)(23,463,31,455)(24,462,32,454)(49,466,57,474)(50,465,58,473)(51,480,59,472)(52,479,60,471)(53,478,61,470)(54,477,62,469)(55,476,63,468)(56,475,64,467)(65,139,73,131)(66,138,74,130)(67,137,75,129)(68,136,76,144)(69,135,77,143)(70,134,78,142)(71,133,79,141)(72,132,80,140)(81,445,89,437)(82,444,90,436)(83,443,91,435)(84,442,92,434)(85,441,93,433)(86,440,94,448)(87,439,95,447)(88,438,96,446)(97,319,105,311)(98,318,106,310)(99,317,107,309)(100,316,108,308)(101,315,109,307)(102,314,110,306)(103,313,111,305)(104,312,112,320)(113,424,121,432)(114,423,122,431)(115,422,123,430)(116,421,124,429)(117,420,125,428)(118,419,126,427)(119,418,127,426)(120,417,128,425)(145,200,153,208)(146,199,154,207)(147,198,155,206)(148,197,156,205)(149,196,157,204)(150,195,158,203)(151,194,159,202)(152,193,160,201)(161,359,169,367)(162,358,170,366)(163,357,171,365)(164,356,172,364)(165,355,173,363)(166,354,174,362)(167,353,175,361)(168,368,176,360)(177,225,185,233)(178,240,186,232)(179,239,187,231)(180,238,188,230)(181,237,189,229)(182,236,190,228)(183,235,191,227)(184,234,192,226)(209,374,217,382)(210,373,218,381)(211,372,219,380)(212,371,220,379)(213,370,221,378)(214,369,222,377)(215,384,223,376)(216,383,224,375)(241,348,249,340)(242,347,250,339)(243,346,251,338)(244,345,252,337)(245,344,253,352)(246,343,254,351)(247,342,255,350)(248,341,256,349)(257,331,265,323)(258,330,266,322)(259,329,267,321)(260,328,268,336)(261,327,269,335)(262,326,270,334)(263,325,271,333)(264,324,272,332)(273,396,281,388)(274,395,282,387)(275,394,283,386)(276,393,284,385)(277,392,285,400)(278,391,286,399)(279,390,287,398)(280,389,288,397)(289,414,297,406)(290,413,298,405)(291,412,299,404)(292,411,300,403)(293,410,301,402)(294,409,302,401)(295,408,303,416)(296,407,304,415)>;
G:=Group( (1,179,442,299,21,171,478,321,66,338,285,205,220,102,127)(2,128,103,221,206,286,339,67,322,479,172,22,300,443,180)(3,181,444,301,23,173,480,323,68,340,287,207,222,104,113)(4,114,105,223,208,288,341,69,324,465,174,24,302,445,182)(5,183,446,303,25,175,466,325,70,342,273,193,224,106,115)(6,116,107,209,194,274,343,71,326,467,176,26,304,447,184)(7,185,448,289,27,161,468,327,72,344,275,195,210,108,117)(8,118,109,211,196,276,345,73,328,469,162,28,290,433,186)(9,187,434,291,29,163,470,329,74,346,277,197,212,110,119)(10,120,111,213,198,278,347,75,330,471,164,30,292,435,188)(11,189,436,293,31,165,472,331,76,348,279,199,214,112,121)(12,122,97,215,200,280,349,77,332,473,166,32,294,437,190)(13,191,438,295,17,167,474,333,78,350,281,201,216,98,123)(14,124,99,217,202,282,351,79,334,475,168,18,296,439,192)(15,177,440,297,19,169,476,335,80,352,283,203,218,100,125)(16,126,101,219,204,284,337,65,336,477,170,20,298,441,178)(33,464,137,378,238,364,242,313,91,60,399,425,403,258,147)(34,148,259,404,426,400,61,92,314,243,365,239,379,138,449)(35,450,139,380,240,366,244,315,93,62,385,427,405,260,149)(36,150,261,406,428,386,63,94,316,245,367,225,381,140,451)(37,452,141,382,226,368,246,317,95,64,387,429,407,262,151)(38,152,263,408,430,388,49,96,318,247,353,227,383,142,453)(39,454,143,384,228,354,248,319,81,50,389,431,409,264,153)(40,154,265,410,432,390,51,82,320,249,355,229,369,144,455)(41,456,129,370,230,356,250,305,83,52,391,417,411,266,155)(42,156,267,412,418,392,53,84,306,251,357,231,371,130,457)(43,458,131,372,232,358,252,307,85,54,393,419,413,268,157)(44,158,269,414,420,394,55,86,308,253,359,233,373,132,459)(45,460,133,374,234,360,254,309,87,56,395,421,415,270,159)(46,160,271,416,422,396,57,88,310,255,361,235,375,134,461)(47,462,135,376,236,362,256,311,89,58,397,423,401,272,145)(48,146,257,402,424,398,59,90,312,241,363,237,377,136,463), (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), (1,34,9,42)(2,33,10,41)(3,48,11,40)(4,47,12,39)(5,46,13,38)(6,45,14,37)(7,44,15,36)(8,43,16,35)(17,453,25,461)(18,452,26,460)(19,451,27,459)(20,450,28,458)(21,449,29,457)(22,464,30,456)(23,463,31,455)(24,462,32,454)(49,466,57,474)(50,465,58,473)(51,480,59,472)(52,479,60,471)(53,478,61,470)(54,477,62,469)(55,476,63,468)(56,475,64,467)(65,139,73,131)(66,138,74,130)(67,137,75,129)(68,136,76,144)(69,135,77,143)(70,134,78,142)(71,133,79,141)(72,132,80,140)(81,445,89,437)(82,444,90,436)(83,443,91,435)(84,442,92,434)(85,441,93,433)(86,440,94,448)(87,439,95,447)(88,438,96,446)(97,319,105,311)(98,318,106,310)(99,317,107,309)(100,316,108,308)(101,315,109,307)(102,314,110,306)(103,313,111,305)(104,312,112,320)(113,424,121,432)(114,423,122,431)(115,422,123,430)(116,421,124,429)(117,420,125,428)(118,419,126,427)(119,418,127,426)(120,417,128,425)(145,200,153,208)(146,199,154,207)(147,198,155,206)(148,197,156,205)(149,196,157,204)(150,195,158,203)(151,194,159,202)(152,193,160,201)(161,359,169,367)(162,358,170,366)(163,357,171,365)(164,356,172,364)(165,355,173,363)(166,354,174,362)(167,353,175,361)(168,368,176,360)(177,225,185,233)(178,240,186,232)(179,239,187,231)(180,238,188,230)(181,237,189,229)(182,236,190,228)(183,235,191,227)(184,234,192,226)(209,374,217,382)(210,373,218,381)(211,372,219,380)(212,371,220,379)(213,370,221,378)(214,369,222,377)(215,384,223,376)(216,383,224,375)(241,348,249,340)(242,347,250,339)(243,346,251,338)(244,345,252,337)(245,344,253,352)(246,343,254,351)(247,342,255,350)(248,341,256,349)(257,331,265,323)(258,330,266,322)(259,329,267,321)(260,328,268,336)(261,327,269,335)(262,326,270,334)(263,325,271,333)(264,324,272,332)(273,396,281,388)(274,395,282,387)(275,394,283,386)(276,393,284,385)(277,392,285,400)(278,391,286,399)(279,390,287,398)(280,389,288,397)(289,414,297,406)(290,413,298,405)(291,412,299,404)(292,411,300,403)(293,410,301,402)(294,409,302,401)(295,408,303,416)(296,407,304,415) );
G=PermutationGroup([[(1,179,442,299,21,171,478,321,66,338,285,205,220,102,127),(2,128,103,221,206,286,339,67,322,479,172,22,300,443,180),(3,181,444,301,23,173,480,323,68,340,287,207,222,104,113),(4,114,105,223,208,288,341,69,324,465,174,24,302,445,182),(5,183,446,303,25,175,466,325,70,342,273,193,224,106,115),(6,116,107,209,194,274,343,71,326,467,176,26,304,447,184),(7,185,448,289,27,161,468,327,72,344,275,195,210,108,117),(8,118,109,211,196,276,345,73,328,469,162,28,290,433,186),(9,187,434,291,29,163,470,329,74,346,277,197,212,110,119),(10,120,111,213,198,278,347,75,330,471,164,30,292,435,188),(11,189,436,293,31,165,472,331,76,348,279,199,214,112,121),(12,122,97,215,200,280,349,77,332,473,166,32,294,437,190),(13,191,438,295,17,167,474,333,78,350,281,201,216,98,123),(14,124,99,217,202,282,351,79,334,475,168,18,296,439,192),(15,177,440,297,19,169,476,335,80,352,283,203,218,100,125),(16,126,101,219,204,284,337,65,336,477,170,20,298,441,178),(33,464,137,378,238,364,242,313,91,60,399,425,403,258,147),(34,148,259,404,426,400,61,92,314,243,365,239,379,138,449),(35,450,139,380,240,366,244,315,93,62,385,427,405,260,149),(36,150,261,406,428,386,63,94,316,245,367,225,381,140,451),(37,452,141,382,226,368,246,317,95,64,387,429,407,262,151),(38,152,263,408,430,388,49,96,318,247,353,227,383,142,453),(39,454,143,384,228,354,248,319,81,50,389,431,409,264,153),(40,154,265,410,432,390,51,82,320,249,355,229,369,144,455),(41,456,129,370,230,356,250,305,83,52,391,417,411,266,155),(42,156,267,412,418,392,53,84,306,251,357,231,371,130,457),(43,458,131,372,232,358,252,307,85,54,393,419,413,268,157),(44,158,269,414,420,394,55,86,308,253,359,233,373,132,459),(45,460,133,374,234,360,254,309,87,56,395,421,415,270,159),(46,160,271,416,422,396,57,88,310,255,361,235,375,134,461),(47,462,135,376,236,362,256,311,89,58,397,423,401,272,145),(48,146,257,402,424,398,59,90,312,241,363,237,377,136,463)], [(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)], [(1,34,9,42),(2,33,10,41),(3,48,11,40),(4,47,12,39),(5,46,13,38),(6,45,14,37),(7,44,15,36),(8,43,16,35),(17,453,25,461),(18,452,26,460),(19,451,27,459),(20,450,28,458),(21,449,29,457),(22,464,30,456),(23,463,31,455),(24,462,32,454),(49,466,57,474),(50,465,58,473),(51,480,59,472),(52,479,60,471),(53,478,61,470),(54,477,62,469),(55,476,63,468),(56,475,64,467),(65,139,73,131),(66,138,74,130),(67,137,75,129),(68,136,76,144),(69,135,77,143),(70,134,78,142),(71,133,79,141),(72,132,80,140),(81,445,89,437),(82,444,90,436),(83,443,91,435),(84,442,92,434),(85,441,93,433),(86,440,94,448),(87,439,95,447),(88,438,96,446),(97,319,105,311),(98,318,106,310),(99,317,107,309),(100,316,108,308),(101,315,109,307),(102,314,110,306),(103,313,111,305),(104,312,112,320),(113,424,121,432),(114,423,122,431),(115,422,123,430),(116,421,124,429),(117,420,125,428),(118,419,126,427),(119,418,127,426),(120,417,128,425),(145,200,153,208),(146,199,154,207),(147,198,155,206),(148,197,156,205),(149,196,157,204),(150,195,158,203),(151,194,159,202),(152,193,160,201),(161,359,169,367),(162,358,170,366),(163,357,171,365),(164,356,172,364),(165,355,173,363),(166,354,174,362),(167,353,175,361),(168,368,176,360),(177,225,185,233),(178,240,186,232),(179,239,187,231),(180,238,188,230),(181,237,189,229),(182,236,190,228),(183,235,191,227),(184,234,192,226),(209,374,217,382),(210,373,218,381),(211,372,219,380),(212,371,220,379),(213,370,221,378),(214,369,222,377),(215,384,223,376),(216,383,224,375),(241,348,249,340),(242,347,250,339),(243,346,251,338),(244,345,252,337),(245,344,253,352),(246,343,254,351),(247,342,255,350),(248,341,256,349),(257,331,265,323),(258,330,266,322),(259,329,267,321),(260,328,268,336),(261,327,269,335),(262,326,270,334),(263,325,271,333),(264,324,272,332),(273,396,281,388),(274,395,282,387),(275,394,283,386),(276,393,284,385),(277,392,285,400),(278,391,286,399),(279,390,287,398),(280,389,288,397),(289,414,297,406),(290,413,298,405),(291,412,299,404),(292,411,300,403),(293,410,301,402),(294,409,302,401),(295,408,303,416),(296,407,304,415)]])
48 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 24 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 4 | 4 | 30 | 30 | 30 | 30 | 4 | 4 | 24 | 24 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | Q32 | C5⋊D4 | D4⋊S3 | S3×D5 | D4⋊D5 | C3⋊Q32 | C15⋊D4 | C5⋊Q32 | C15⋊D8 | C15⋊Q32 |
| kernel | C15⋊Q32 | C15⋊3C16 | C3×Dic20 | C5×Dic12 | Dic20 | C60 | Dic12 | C40 | C30 | C24 | C20 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C15⋊Q32 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 189 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 129 | 170 | 0 | 0 | 0 | 0 |
| 156 | 58 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 79 | 0 | 0 |
| 0 | 0 | 85 | 153 | 0 | 0 |
| 0 | 0 | 0 | 0 | 136 | 232 |
| 0 | 0 | 0 | 0 | 127 | 105 |
| 173 | 94 | 0 | 0 | 0 | 0 |
| 220 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 165 | 103 | 0 | 0 |
| 0 | 0 | 89 | 76 | 0 | 0 |
| 0 | 0 | 0 | 0 | 136 | 232 |
| 0 | 0 | 0 | 0 | 127 | 105 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[129,156,0,0,0,0,170,58,0,0,0,0,0,0,88,85,0,0,0,0,79,153,0,0,0,0,0,0,136,127,0,0,0,0,232,105],[173,220,0,0,0,0,94,68,0,0,0,0,0,0,165,89,0,0,0,0,103,76,0,0,0,0,0,0,136,127,0,0,0,0,232,105] >;
C15⋊Q32 in GAP, Magma, Sage, TeX
C_{15}\rtimes Q_{32} % in TeX
G:=Group("C15:Q32"); // GroupNames label
G:=SmallGroup(480,22);
// by ID
G=gap.SmallGroup(480,22);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,120,254,135,142,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=1,c^2=b^8,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=b^-1>;
// generators/relations
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