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## G = C10×C3⋊Q16order 480 = 25·3·5

### Direct product of C10 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C10×C3⋊Q16
 Chief series C1 — C3 — C6 — C12 — C60 — C5×Dic6 — C10×Dic6 — C10×C3⋊Q16
 Lower central C3 — C6 — C12 — C10×C3⋊Q16
 Upper central C1 — C2×C10 — C2×C20 — Q8×C10

Generators and relations for C10×C3⋊Q16
G = < a,b,c,d | a10=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 228 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], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8, C2×Q8, C20 [×2], C20 [×4], C2×C10, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C30, C30 [×2], C2×Q16, C40 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8 [×4], C2×C3⋊C8, C3⋊Q16 [×4], C2×Dic6, C6×Q8, C5×Dic3 [×2], C60 [×2], C60 [×2], C2×C30, C2×C40, C5×Q16 [×4], Q8×C10, Q8×C10, C2×C3⋊Q16, C5×C3⋊C8 [×2], C5×Dic6 [×2], C5×Dic6, C10×Dic3, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C10×Q16, C10×C3⋊C8, C5×C3⋊Q16 [×4], C10×Dic6, Q8×C30, C10×C3⋊Q16
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], Q16 [×2], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C2×Q16, C5×D4 [×2], C22×C10, C3⋊Q16 [×2], C2×C3⋊D4, S3×C10 [×3], C5×Q16 [×2], D4×C10, C2×C3⋊Q16, C5×C3⋊D4 [×2], S3×C2×C10, C10×Q16, C5×C3⋊Q16 [×2], C10×C3⋊D4, C10×C3⋊Q16

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

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

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

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

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

Matrix representation of C10×C3⋊Q16 in GL4(𝔽241) generated by

 150 0 0 0 0 150 0 0 0 0 87 0 0 0 0 87
,
 240 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 219 18 0 0 40 22 0 0 0 0 0 168 0 0 208 22
,
 70 140 0 0 101 171 0 0 0 0 76 46 0 0 194 165
G:=sub<GL(4,GF(241))| [150,0,0,0,0,150,0,0,0,0,87,0,0,0,0,87],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[219,40,0,0,18,22,0,0,0,0,0,208,0,0,168,22],[70,101,0,0,140,171,0,0,0,0,76,194,0,0,46,165] >;

C10×C3⋊Q16 in GAP, Magma, Sage, TeX

C_{10}\times C_3\rtimes Q_{16}
% in TeX

G:=Group("C10xC3:Q16");
// GroupNames label

G:=SmallGroup(480,822);
// by ID

G=gap.SmallGroup(480,822);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,436,4204,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^3=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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