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## G = C5×C6.Q16order 480 = 25·3·5

### Direct product of C5 and C6.Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C6.Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C10×C3⋊C8 — C5×C6.Q16
 Lower central C3 — C6 — C12 — C5×C6.Q16
 Upper central C1 — C2×C10 — C2×C20 — C5×C4⋊C4

Generators and relations for C5×C6.Q16
G = < a,b,c,d | a5=b12=c4=1, d2=b9c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b9c-1 >

Subgroups: 148 in 72 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3, C12 [×2], C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, C20 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C2×Dic3, C2×C12, C2×C12, C30 [×3], C2.D8, C40 [×2], C2×C20, C2×C20 [×2], C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C60 [×2], C60, C2×C30, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C6.Q16, C5×C3⋊C8 [×2], C10×Dic3, C2×C60, C2×C60, C5×C2.D8, C10×C3⋊C8, C5×C4⋊Dic3, C15×C4⋊C4, C5×C6.Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4, Q8, C10 [×3], D6, C4⋊C4, D8, Q16, C20 [×2], C2×C10, Dic6, C4×S3, C3⋊D4, C5×S3, C2.D8, C2×C20, C5×D4, C5×Q8, Dic3⋊C4, D4⋊S3, C3⋊Q16, S3×C10, C5×C4⋊C4, C5×D8, C5×Q16, C6.Q16, C5×Dic6, S3×C20, C5×C3⋊D4, C5×C2.D8, C5×Dic3⋊C4, C5×D4⋊S3, C5×C3⋊Q16, C5×C6.Q16

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 12A ··· 12F 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 20Q ··· 20X 30A ··· 30L 40A ··· 40P 60A ··· 60X order 1 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 2 2 4 4 12 12 1 1 1 1 2 2 2 6 6 6 6 1 ··· 1 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 12 ··· 12 2 ··· 2 6 ··· 6 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + + - - + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 Q8 D4 D6 D8 Q16 Dic6 C4×S3 C3⋊D4 C5×S3 C5×Q8 C5×D4 S3×C10 C5×D8 C5×Q16 C5×Dic6 S3×C20 C5×C3⋊D4 D4⋊S3 C3⋊Q16 C5×D4⋊S3 C5×C3⋊Q16 kernel C5×C6.Q16 C10×C3⋊C8 C5×C4⋊Dic3 C15×C4⋊C4 C5×C3⋊C8 C6.Q16 C2×C3⋊C8 C4⋊Dic3 C3×C4⋊C4 C3⋊C8 C5×C4⋊C4 C60 C2×C30 C2×C20 C30 C30 C20 C20 C2×C10 C4⋊C4 C12 C2×C6 C2×C4 C6 C6 C4 C4 C22 C10 C10 C2 C2 # reps 1 1 1 1 4 4 4 4 4 16 1 1 1 1 2 2 2 2 2 4 4 4 4 8 8 8 8 8 1 1 4 4

Matrix representation of C5×C6.Q16 in GL5(𝔽241)

 1 0 0 0 0 0 98 0 0 0 0 0 98 0 0 0 0 0 205 0 0 0 0 0 205
,
 240 0 0 0 0 0 240 240 0 0 0 1 0 0 0 0 0 0 1 3 0 0 0 160 240
,
 177 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 15 44 0 0 0 214 226
,
 1 0 0 0 0 0 142 43 0 0 0 142 99 0 0 0 0 0 22 33 0 0 0 73 0

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,98,0,0,0,0,0,98,0,0,0,0,0,205,0,0,0,0,0,205],[240,0,0,0,0,0,240,1,0,0,0,240,0,0,0,0,0,0,1,160,0,0,0,3,240],[177,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,15,214,0,0,0,44,226],[1,0,0,0,0,0,142,142,0,0,0,43,99,0,0,0,0,0,22,73,0,0,0,33,0] >;

C5×C6.Q16 in GAP, Magma, Sage, TeX

C_5\times C_6.Q_{16}
% in TeX

G:=Group("C5xC6.Q16");
// GroupNames label

G:=SmallGroup(480,126);
// by ID

G=gap.SmallGroup(480,126);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,280,589,148,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^4=1,d^2=b^9*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^9*c^-1>;
// generators/relations

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