metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊7Q32, C8.7D30, C60.6D4, Q16.D15, C30.42D8, C40.15D6, C24.15D10, Dic60.3C2, C120.12C22, C3⋊3(C5⋊Q32), C5⋊3(C3⋊Q32), (C5×Q16).1S3, (C3×Q16).1D5, C15⋊3C16.1C2, C6.20(D4⋊D5), C2.7(D4⋊D15), C4.4(C15⋊7D4), (C15×Q16).1C2, C10.20(D4⋊S3), C20.18(C3⋊D4), C12.20(C5⋊D4), SmallGroup(480,189)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊7Q32
G = < a,b,c | a15=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C15⋊7Q32
►Regular action on 480 points
Generators in S480
(1 27 393 333 263 461 384 291 343 416 35 188 278 125 443)(2 444 126 279 189 36 401 344 292 369 462 264 334 394 28)(3 29 395 335 265 463 370 293 345 402 37 190 280 127 445)(4 446 128 281 191 38 403 346 294 371 464 266 336 396 30)(5 31 397 321 267 449 372 295 347 404 39 192 282 113 447)(6 448 114 283 177 40 405 348 296 373 450 268 322 398 32)(7 17 399 323 269 451 374 297 349 406 41 178 284 115 433)(8 434 116 285 179 42 407 350 298 375 452 270 324 400 18)(9 19 385 325 271 453 376 299 351 408 43 180 286 117 435)(10 436 118 287 181 44 409 352 300 377 454 272 326 386 20)(11 21 387 327 257 455 378 301 337 410 45 182 288 119 437)(12 438 120 273 183 46 411 338 302 379 456 258 328 388 22)(13 23 389 329 259 457 380 303 339 412 47 184 274 121 439)(14 440 122 275 185 48 413 340 304 381 458 260 330 390 24)(15 25 391 331 261 459 382 289 341 414 33 186 276 123 441)(16 442 124 277 187 34 415 342 290 383 460 262 332 392 26)(49 316 204 110 217 419 237 173 157 362 139 473 95 66 244)(50 245 67 96 474 140 363 158 174 238 420 218 111 205 317)(51 318 206 112 219 421 239 175 159 364 141 475 81 68 246)(52 247 69 82 476 142 365 160 176 240 422 220 97 207 319)(53 320 208 98 221 423 225 161 145 366 143 477 83 70 248)(54 249 71 84 478 144 367 146 162 226 424 222 99 193 305)(55 306 194 100 223 425 227 163 147 368 129 479 85 72 250)(56 251 73 86 480 130 353 148 164 228 426 224 101 195 307)(57 308 196 102 209 427 229 165 149 354 131 465 87 74 252)(58 253 75 88 466 132 355 150 166 230 428 210 103 197 309)(59 310 198 104 211 429 231 167 151 356 133 467 89 76 254)(60 255 77 90 468 134 357 152 168 232 430 212 105 199 311)(61 312 200 106 213 431 233 169 153 358 135 469 91 78 256)(62 241 79 92 470 136 359 154 170 234 432 214 107 201 313)(63 314 202 108 215 417 235 171 155 360 137 471 93 80 242)(64 243 65 94 472 138 361 156 172 236 418 216 109 203 315)
(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)
(1 221 9 213)(2 220 10 212)(3 219 11 211)(4 218 12 210)(5 217 13 209)(6 216 14 224)(7 215 15 223)(8 214 16 222)(17 417 25 425)(18 432 26 424)(19 431 27 423)(20 430 28 422)(21 429 29 421)(22 428 30 420)(23 427 31 419)(24 426 32 418)(33 250 41 242)(34 249 42 241)(35 248 43 256)(36 247 44 255)(37 246 45 254)(38 245 46 253)(39 244 47 252)(40 243 48 251)(49 184 57 192)(50 183 58 191)(51 182 59 190)(52 181 60 189)(53 180 61 188)(54 179 62 187)(55 178 63 186)(56 177 64 185)(65 413 73 405)(66 412 74 404)(67 411 75 403)(68 410 76 402)(69 409 77 401)(70 408 78 416)(71 407 79 415)(72 406 80 414)(81 337 89 345)(82 352 90 344)(83 351 91 343)(84 350 92 342)(85 349 93 341)(86 348 94 340)(87 347 95 339)(88 346 96 338)(97 436 105 444)(98 435 106 443)(99 434 107 442)(100 433 108 441)(101 448 109 440)(102 447 110 439)(103 446 111 438)(104 445 112 437)(113 204 121 196)(114 203 122 195)(115 202 123 194)(116 201 124 193)(117 200 125 208)(118 199 126 207)(119 198 127 206)(120 197 128 205)(129 374 137 382)(130 373 138 381)(131 372 139 380)(132 371 140 379)(133 370 141 378)(134 369 142 377)(135 384 143 376)(136 383 144 375)(145 271 153 263)(146 270 154 262)(147 269 155 261)(148 268 156 260)(149 267 157 259)(150 266 158 258)(151 265 159 257)(152 264 160 272)(161 325 169 333)(162 324 170 332)(163 323 171 331)(164 322 172 330)(165 321 173 329)(166 336 174 328)(167 335 175 327)(168 334 176 326)(225 385 233 393)(226 400 234 392)(227 399 235 391)(228 398 236 390)(229 397 237 389)(230 396 238 388)(231 395 239 387)(232 394 240 386)(273 309 281 317)(274 308 282 316)(275 307 283 315)(276 306 284 314)(277 305 285 313)(278 320 286 312)(279 319 287 311)(280 318 288 310)(289 479 297 471)(290 478 298 470)(291 477 299 469)(292 476 300 468)(293 475 301 467)(294 474 302 466)(295 473 303 465)(296 472 304 480)(353 450 361 458)(354 449 362 457)(355 464 363 456)(356 463 364 455)(357 462 365 454)(358 461 366 453)(359 460 367 452)(360 459 368 451)
G:=sub<Sym(480)| (1,27,393,333,263,461,384,291,343,416,35,188,278,125,443)(2,444,126,279,189,36,401,344,292,369,462,264,334,394,28)(3,29,395,335,265,463,370,293,345,402,37,190,280,127,445)(4,446,128,281,191,38,403,346,294,371,464,266,336,396,30)(5,31,397,321,267,449,372,295,347,404,39,192,282,113,447)(6,448,114,283,177,40,405,348,296,373,450,268,322,398,32)(7,17,399,323,269,451,374,297,349,406,41,178,284,115,433)(8,434,116,285,179,42,407,350,298,375,452,270,324,400,18)(9,19,385,325,271,453,376,299,351,408,43,180,286,117,435)(10,436,118,287,181,44,409,352,300,377,454,272,326,386,20)(11,21,387,327,257,455,378,301,337,410,45,182,288,119,437)(12,438,120,273,183,46,411,338,302,379,456,258,328,388,22)(13,23,389,329,259,457,380,303,339,412,47,184,274,121,439)(14,440,122,275,185,48,413,340,304,381,458,260,330,390,24)(15,25,391,331,261,459,382,289,341,414,33,186,276,123,441)(16,442,124,277,187,34,415,342,290,383,460,262,332,392,26)(49,316,204,110,217,419,237,173,157,362,139,473,95,66,244)(50,245,67,96,474,140,363,158,174,238,420,218,111,205,317)(51,318,206,112,219,421,239,175,159,364,141,475,81,68,246)(52,247,69,82,476,142,365,160,176,240,422,220,97,207,319)(53,320,208,98,221,423,225,161,145,366,143,477,83,70,248)(54,249,71,84,478,144,367,146,162,226,424,222,99,193,305)(55,306,194,100,223,425,227,163,147,368,129,479,85,72,250)(56,251,73,86,480,130,353,148,164,228,426,224,101,195,307)(57,308,196,102,209,427,229,165,149,354,131,465,87,74,252)(58,253,75,88,466,132,355,150,166,230,428,210,103,197,309)(59,310,198,104,211,429,231,167,151,356,133,467,89,76,254)(60,255,77,90,468,134,357,152,168,232,430,212,105,199,311)(61,312,200,106,213,431,233,169,153,358,135,469,91,78,256)(62,241,79,92,470,136,359,154,170,234,432,214,107,201,313)(63,314,202,108,215,417,235,171,155,360,137,471,93,80,242)(64,243,65,94,472,138,361,156,172,236,418,216,109,203,315), (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), (1,221,9,213)(2,220,10,212)(3,219,11,211)(4,218,12,210)(5,217,13,209)(6,216,14,224)(7,215,15,223)(8,214,16,222)(17,417,25,425)(18,432,26,424)(19,431,27,423)(20,430,28,422)(21,429,29,421)(22,428,30,420)(23,427,31,419)(24,426,32,418)(33,250,41,242)(34,249,42,241)(35,248,43,256)(36,247,44,255)(37,246,45,254)(38,245,46,253)(39,244,47,252)(40,243,48,251)(49,184,57,192)(50,183,58,191)(51,182,59,190)(52,181,60,189)(53,180,61,188)(54,179,62,187)(55,178,63,186)(56,177,64,185)(65,413,73,405)(66,412,74,404)(67,411,75,403)(68,410,76,402)(69,409,77,401)(70,408,78,416)(71,407,79,415)(72,406,80,414)(81,337,89,345)(82,352,90,344)(83,351,91,343)(84,350,92,342)(85,349,93,341)(86,348,94,340)(87,347,95,339)(88,346,96,338)(97,436,105,444)(98,435,106,443)(99,434,107,442)(100,433,108,441)(101,448,109,440)(102,447,110,439)(103,446,111,438)(104,445,112,437)(113,204,121,196)(114,203,122,195)(115,202,123,194)(116,201,124,193)(117,200,125,208)(118,199,126,207)(119,198,127,206)(120,197,128,205)(129,374,137,382)(130,373,138,381)(131,372,139,380)(132,371,140,379)(133,370,141,378)(134,369,142,377)(135,384,143,376)(136,383,144,375)(145,271,153,263)(146,270,154,262)(147,269,155,261)(148,268,156,260)(149,267,157,259)(150,266,158,258)(151,265,159,257)(152,264,160,272)(161,325,169,333)(162,324,170,332)(163,323,171,331)(164,322,172,330)(165,321,173,329)(166,336,174,328)(167,335,175,327)(168,334,176,326)(225,385,233,393)(226,400,234,392)(227,399,235,391)(228,398,236,390)(229,397,237,389)(230,396,238,388)(231,395,239,387)(232,394,240,386)(273,309,281,317)(274,308,282,316)(275,307,283,315)(276,306,284,314)(277,305,285,313)(278,320,286,312)(279,319,287,311)(280,318,288,310)(289,479,297,471)(290,478,298,470)(291,477,299,469)(292,476,300,468)(293,475,301,467)(294,474,302,466)(295,473,303,465)(296,472,304,480)(353,450,361,458)(354,449,362,457)(355,464,363,456)(356,463,364,455)(357,462,365,454)(358,461,366,453)(359,460,367,452)(360,459,368,451)>;
G:=Group( (1,27,393,333,263,461,384,291,343,416,35,188,278,125,443)(2,444,126,279,189,36,401,344,292,369,462,264,334,394,28)(3,29,395,335,265,463,370,293,345,402,37,190,280,127,445)(4,446,128,281,191,38,403,346,294,371,464,266,336,396,30)(5,31,397,321,267,449,372,295,347,404,39,192,282,113,447)(6,448,114,283,177,40,405,348,296,373,450,268,322,398,32)(7,17,399,323,269,451,374,297,349,406,41,178,284,115,433)(8,434,116,285,179,42,407,350,298,375,452,270,324,400,18)(9,19,385,325,271,453,376,299,351,408,43,180,286,117,435)(10,436,118,287,181,44,409,352,300,377,454,272,326,386,20)(11,21,387,327,257,455,378,301,337,410,45,182,288,119,437)(12,438,120,273,183,46,411,338,302,379,456,258,328,388,22)(13,23,389,329,259,457,380,303,339,412,47,184,274,121,439)(14,440,122,275,185,48,413,340,304,381,458,260,330,390,24)(15,25,391,331,261,459,382,289,341,414,33,186,276,123,441)(16,442,124,277,187,34,415,342,290,383,460,262,332,392,26)(49,316,204,110,217,419,237,173,157,362,139,473,95,66,244)(50,245,67,96,474,140,363,158,174,238,420,218,111,205,317)(51,318,206,112,219,421,239,175,159,364,141,475,81,68,246)(52,247,69,82,476,142,365,160,176,240,422,220,97,207,319)(53,320,208,98,221,423,225,161,145,366,143,477,83,70,248)(54,249,71,84,478,144,367,146,162,226,424,222,99,193,305)(55,306,194,100,223,425,227,163,147,368,129,479,85,72,250)(56,251,73,86,480,130,353,148,164,228,426,224,101,195,307)(57,308,196,102,209,427,229,165,149,354,131,465,87,74,252)(58,253,75,88,466,132,355,150,166,230,428,210,103,197,309)(59,310,198,104,211,429,231,167,151,356,133,467,89,76,254)(60,255,77,90,468,134,357,152,168,232,430,212,105,199,311)(61,312,200,106,213,431,233,169,153,358,135,469,91,78,256)(62,241,79,92,470,136,359,154,170,234,432,214,107,201,313)(63,314,202,108,215,417,235,171,155,360,137,471,93,80,242)(64,243,65,94,472,138,361,156,172,236,418,216,109,203,315), (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), (1,221,9,213)(2,220,10,212)(3,219,11,211)(4,218,12,210)(5,217,13,209)(6,216,14,224)(7,215,15,223)(8,214,16,222)(17,417,25,425)(18,432,26,424)(19,431,27,423)(20,430,28,422)(21,429,29,421)(22,428,30,420)(23,427,31,419)(24,426,32,418)(33,250,41,242)(34,249,42,241)(35,248,43,256)(36,247,44,255)(37,246,45,254)(38,245,46,253)(39,244,47,252)(40,243,48,251)(49,184,57,192)(50,183,58,191)(51,182,59,190)(52,181,60,189)(53,180,61,188)(54,179,62,187)(55,178,63,186)(56,177,64,185)(65,413,73,405)(66,412,74,404)(67,411,75,403)(68,410,76,402)(69,409,77,401)(70,408,78,416)(71,407,79,415)(72,406,80,414)(81,337,89,345)(82,352,90,344)(83,351,91,343)(84,350,92,342)(85,349,93,341)(86,348,94,340)(87,347,95,339)(88,346,96,338)(97,436,105,444)(98,435,106,443)(99,434,107,442)(100,433,108,441)(101,448,109,440)(102,447,110,439)(103,446,111,438)(104,445,112,437)(113,204,121,196)(114,203,122,195)(115,202,123,194)(116,201,124,193)(117,200,125,208)(118,199,126,207)(119,198,127,206)(120,197,128,205)(129,374,137,382)(130,373,138,381)(131,372,139,380)(132,371,140,379)(133,370,141,378)(134,369,142,377)(135,384,143,376)(136,383,144,375)(145,271,153,263)(146,270,154,262)(147,269,155,261)(148,268,156,260)(149,267,157,259)(150,266,158,258)(151,265,159,257)(152,264,160,272)(161,325,169,333)(162,324,170,332)(163,323,171,331)(164,322,172,330)(165,321,173,329)(166,336,174,328)(167,335,175,327)(168,334,176,326)(225,385,233,393)(226,400,234,392)(227,399,235,391)(228,398,236,390)(229,397,237,389)(230,396,238,388)(231,395,239,387)(232,394,240,386)(273,309,281,317)(274,308,282,316)(275,307,283,315)(276,306,284,314)(277,305,285,313)(278,320,286,312)(279,319,287,311)(280,318,288,310)(289,479,297,471)(290,478,298,470)(291,477,299,469)(292,476,300,468)(293,475,301,467)(294,474,302,466)(295,473,303,465)(296,472,304,480)(353,450,361,458)(354,449,362,457)(355,464,363,456)(356,463,364,455)(357,462,365,454)(358,461,366,453)(359,460,367,452)(360,459,368,451) );
G=PermutationGroup([[(1,27,393,333,263,461,384,291,343,416,35,188,278,125,443),(2,444,126,279,189,36,401,344,292,369,462,264,334,394,28),(3,29,395,335,265,463,370,293,345,402,37,190,280,127,445),(4,446,128,281,191,38,403,346,294,371,464,266,336,396,30),(5,31,397,321,267,449,372,295,347,404,39,192,282,113,447),(6,448,114,283,177,40,405,348,296,373,450,268,322,398,32),(7,17,399,323,269,451,374,297,349,406,41,178,284,115,433),(8,434,116,285,179,42,407,350,298,375,452,270,324,400,18),(9,19,385,325,271,453,376,299,351,408,43,180,286,117,435),(10,436,118,287,181,44,409,352,300,377,454,272,326,386,20),(11,21,387,327,257,455,378,301,337,410,45,182,288,119,437),(12,438,120,273,183,46,411,338,302,379,456,258,328,388,22),(13,23,389,329,259,457,380,303,339,412,47,184,274,121,439),(14,440,122,275,185,48,413,340,304,381,458,260,330,390,24),(15,25,391,331,261,459,382,289,341,414,33,186,276,123,441),(16,442,124,277,187,34,415,342,290,383,460,262,332,392,26),(49,316,204,110,217,419,237,173,157,362,139,473,95,66,244),(50,245,67,96,474,140,363,158,174,238,420,218,111,205,317),(51,318,206,112,219,421,239,175,159,364,141,475,81,68,246),(52,247,69,82,476,142,365,160,176,240,422,220,97,207,319),(53,320,208,98,221,423,225,161,145,366,143,477,83,70,248),(54,249,71,84,478,144,367,146,162,226,424,222,99,193,305),(55,306,194,100,223,425,227,163,147,368,129,479,85,72,250),(56,251,73,86,480,130,353,148,164,228,426,224,101,195,307),(57,308,196,102,209,427,229,165,149,354,131,465,87,74,252),(58,253,75,88,466,132,355,150,166,230,428,210,103,197,309),(59,310,198,104,211,429,231,167,151,356,133,467,89,76,254),(60,255,77,90,468,134,357,152,168,232,430,212,105,199,311),(61,312,200,106,213,431,233,169,153,358,135,469,91,78,256),(62,241,79,92,470,136,359,154,170,234,432,214,107,201,313),(63,314,202,108,215,417,235,171,155,360,137,471,93,80,242),(64,243,65,94,472,138,361,156,172,236,418,216,109,203,315)], [(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)], [(1,221,9,213),(2,220,10,212),(3,219,11,211),(4,218,12,210),(5,217,13,209),(6,216,14,224),(7,215,15,223),(8,214,16,222),(17,417,25,425),(18,432,26,424),(19,431,27,423),(20,430,28,422),(21,429,29,421),(22,428,30,420),(23,427,31,419),(24,426,32,418),(33,250,41,242),(34,249,42,241),(35,248,43,256),(36,247,44,255),(37,246,45,254),(38,245,46,253),(39,244,47,252),(40,243,48,251),(49,184,57,192),(50,183,58,191),(51,182,59,190),(52,181,60,189),(53,180,61,188),(54,179,62,187),(55,178,63,186),(56,177,64,185),(65,413,73,405),(66,412,74,404),(67,411,75,403),(68,410,76,402),(69,409,77,401),(70,408,78,416),(71,407,79,415),(72,406,80,414),(81,337,89,345),(82,352,90,344),(83,351,91,343),(84,350,92,342),(85,349,93,341),(86,348,94,340),(87,347,95,339),(88,346,96,338),(97,436,105,444),(98,435,106,443),(99,434,107,442),(100,433,108,441),(101,448,109,440),(102,447,110,439),(103,446,111,438),(104,445,112,437),(113,204,121,196),(114,203,122,195),(115,202,123,194),(116,201,124,193),(117,200,125,208),(118,199,126,207),(119,198,127,206),(120,197,128,205),(129,374,137,382),(130,373,138,381),(131,372,139,380),(132,371,140,379),(133,370,141,378),(134,369,142,377),(135,384,143,376),(136,383,144,375),(145,271,153,263),(146,270,154,262),(147,269,155,261),(148,268,156,260),(149,267,157,259),(150,266,158,258),(151,265,159,257),(152,264,160,272),(161,325,169,333),(162,324,170,332),(163,323,171,331),(164,322,172,330),(165,321,173,329),(166,336,174,328),(167,335,175,327),(168,334,176,326),(225,385,233,393),(226,400,234,392),(227,399,235,391),(228,398,236,390),(229,397,237,389),(230,396,238,388),(231,395,239,387),(232,394,240,386),(273,309,281,317),(274,308,282,316),(275,307,283,315),(276,306,284,314),(277,305,285,313),(278,320,286,312),(279,319,287,311),(280,318,288,310),(289,479,297,471),(290,478,298,470),(291,477,299,469),(292,476,300,468),(293,475,301,467),(294,474,302,466),(295,473,303,465),(296,472,304,480),(353,450,361,458),(354,449,362,457),(355,464,363,456),(356,463,364,455),(357,462,365,454),(358,461,366,453),(359,460,367,452),(360,459,368,451)]])
60 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 30C | 30D | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | ··· | 60L | 120A | ··· | 120H |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 8 | 120 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | D15 | Q32 | C5⋊D4 | D30 | C15⋊7D4 | D4⋊S3 | D4⋊D5 | C3⋊Q32 | C5⋊Q32 | D4⋊D15 | C15⋊7Q32 |
| kernel | C15⋊7Q32 | C15⋊3C16 | Dic60 | C15×Q16 | C5×Q16 | C60 | C3×Q16 | C40 | C30 | C24 | C20 | Q16 | C15 | C12 | C8 | C4 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C15⋊7Q32 ►in GL4(𝔽241) generated by
| 148 | 80 | 0 | 0 |
| 161 | 131 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 87 | 205 | 0 | 0 |
| 63 | 154 | 0 | 0 |
| 0 | 0 | 183 | 123 |
| 0 | 0 | 188 | 129 |
| 165 | 192 | 0 | 0 |
| 49 | 76 | 0 | 0 |
| 0 | 0 | 127 | 240 |
| 0 | 0 | 224 | 114 |
G:=sub<GL(4,GF(241))| [148,161,0,0,80,131,0,0,0,0,1,0,0,0,0,1],[87,63,0,0,205,154,0,0,0,0,183,188,0,0,123,129],[165,49,0,0,192,76,0,0,0,0,127,224,0,0,240,114] >;
C15⋊7Q32 in GAP, Magma, Sage, TeX
C_{15}\rtimes_7Q_{32} % in TeX
G:=Group("C15:7Q32"); // GroupNames label
G:=SmallGroup(480,189);
// by ID
G=gap.SmallGroup(480,189);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,120,254,135,142,675,346,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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