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## G = C3×C5⋊Q32order 480 = 25·3·5

### Direct product of C3 and C5⋊Q32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C3×C5⋊Q32
 Chief series C1 — C5 — C10 — C20 — C40 — C120 — C3×Dic20 — C3×C5⋊Q32
 Lower central C5 — C10 — C20 — C40 — C3×C5⋊Q32
 Upper central C1 — C6 — C12 — C24 — C3×Q16

Generators and relations for C3×C5⋊Q32
G = < a,b,c,d | a3=b5=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

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

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

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

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

75 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 5A 5B 6A 6B 8A 8B 10A 10B 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 30C 30D 40A 40B 40C 40D 48A ··· 48H 60A 60B 60C 60D 60E ··· 60L 120A ··· 120H order 1 2 3 3 4 4 4 5 5 6 6 8 8 10 10 12 12 12 12 12 12 15 15 15 15 16 16 16 16 20 20 20 20 20 20 24 24 24 24 30 30 30 30 40 40 40 40 48 ··· 48 60 60 60 60 60 ··· 60 120 ··· 120 size 1 1 1 1 2 8 40 2 2 1 1 2 2 2 2 2 2 8 8 40 40 2 2 2 2 10 10 10 10 4 4 8 8 8 8 2 2 2 2 2 2 2 2 4 4 4 4 10 ··· 10 4 4 4 4 8 ··· 8 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 D8 D10 C3×D4 C3×D5 Q32 C5⋊D4 C3×D8 C6×D5 C3×Q32 C3×C5⋊D4 D4⋊D5 C5⋊Q32 C3×D4⋊D5 C3×C5⋊Q32 kernel C3×C5⋊Q32 C3×C5⋊2C16 C3×Dic20 C15×Q16 C5⋊Q32 C5⋊2C16 Dic20 C5×Q16 C60 C3×Q16 C30 C24 C20 Q16 C15 C12 C10 C8 C5 C4 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 2 4 4 4 4 4 8 8 2 4 4 8

Matrix representation of C3×C5⋊Q32 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 1 0 0 0 0 1
,
 0 240 0 0 1 189 0 0 0 0 1 0 0 0 0 1
,
 132 130 0 0 5 109 0 0 0 0 214 156 0 0 85 214
,
 214 103 0 0 138 27 0 0 0 0 238 183 0 0 183 3
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,240,189,0,0,0,0,1,0,0,0,0,1],[132,5,0,0,130,109,0,0,0,0,214,85,0,0,156,214],[214,138,0,0,103,27,0,0,0,0,238,183,0,0,183,3] >;

C3×C5⋊Q32 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes Q_{32}
% in TeX

G:=Group("C3xC5:Q32");
// GroupNames label

G:=SmallGroup(480,107);
// by ID

G=gap.SmallGroup(480,107);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,336,197,344,1011,514,192,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^16=1,d^2=c^8,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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