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## G = C6×C5⋊Q16order 480 = 25·3·5

### Direct product of C6 and C5⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C6×C5⋊Q16
 Chief series C1 — C5 — C10 — C20 — C60 — C3×Dic10 — C6×Dic10 — C6×C5⋊Q16
 Lower central C5 — C10 — C20 — C6×C5⋊Q16
 Upper central C1 — C2×C6 — C2×C12 — C6×Q8

Generators and relations for C6×C5⋊Q16
G = < a,b,c,d | a6=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 304 in 120 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], Q8 [×4], C10, C10 [×2], C12 [×2], C12 [×4], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C3×Q8 [×4], C30, C30 [×2], C2×Q16, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×C24, C3×Q16 [×4], C6×Q8, C6×Q8, C3×Dic5 [×2], C60 [×2], C60 [×2], C2×C30, C2×C52C8, C5⋊Q16 [×4], C2×Dic10, Q8×C10, C6×Q16, C3×C52C8 [×2], C3×Dic10 [×2], C3×Dic10, C6×Dic5, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C2×C5⋊Q16, C6×C52C8, C3×C5⋊Q16 [×4], C6×Dic10, Q8×C30, C6×C5⋊Q16
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], Q16 [×2], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C2×Q16, C5⋊D4 [×2], C22×D5, C3×Q16 [×2], C6×D4, C6×D5 [×3], C5⋊Q16 [×2], C2×C5⋊D4, C6×Q16, C3×C5⋊D4 [×2], D5×C2×C6, C2×C5⋊Q16, C3×C5⋊Q16 [×2], C6×C5⋊D4, C6×C5⋊Q16

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

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A ··· 6F 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 15A 15B 15C 15D 20A ··· 20L 24A ··· 24H 30A ··· 30L 60A ··· 60X order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 6 ··· 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 2 4 4 20 20 2 2 1 ··· 1 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 20 20 20 20 2 2 2 2 4 ··· 4 10 ··· 10 2 ··· 2 4 ··· 4

102 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 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 D5 Q16 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C3×Q16 C6×D5 C6×D5 C3×C5⋊D4 C3×C5⋊D4 C5⋊Q16 C3×C5⋊Q16 kernel C6×C5⋊Q16 C6×C5⋊2C8 C3×C5⋊Q16 C6×Dic10 Q8×C30 C2×C5⋊Q16 C2×C5⋊2C8 C5⋊Q16 C2×Dic10 Q8×C10 C60 C2×C30 C6×Q8 C30 C2×C12 C3×Q8 C20 C2×C10 C2×Q8 C12 C2×C6 C10 C2×C4 Q8 C4 C22 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 1 2 4 2 4 2 2 4 4 4 8 4 8 8 8 4 8

Matrix representation of C6×C5⋊Q16 in GL4(𝔽241) generated by

 16 0 0 0 0 16 0 0 0 0 225 0 0 0 0 225
,
 0 240 0 0 1 51 0 0 0 0 1 0 0 0 0 1
,
 73 16 0 0 149 168 0 0 0 0 22 129 0 0 170 0
,
 1 0 0 0 0 1 0 0 0 0 64 159 0 0 0 177
G:=sub<GL(4,GF(241))| [16,0,0,0,0,16,0,0,0,0,225,0,0,0,0,225],[0,1,0,0,240,51,0,0,0,0,1,0,0,0,0,1],[73,149,0,0,16,168,0,0,0,0,22,170,0,0,129,0],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,159,177] >;

C6×C5⋊Q16 in GAP, Magma, Sage, TeX

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

G:=Group("C6xC5:Q16");
// GroupNames label

G:=SmallGroup(480,736);
// by ID

G=gap.SmallGroup(480,736);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,590,268,2524,648,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^5=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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