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## G = C3×C40⋊8C4order 480 = 25·3·5

### Direct product of C3 and C40⋊8C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C40⋊8C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C12×Dic5 — C3×C40⋊8C4
 Lower central C5 — C10 — C3×C40⋊8C4
 Upper central C1 — C2×C12 — C2×C24

Generators and relations for C3×C408C4
G = < a,b,c | a3=b40=c4=1, ab=ba, ac=ca, cbc-1=b29 >

Subgroups: 176 in 80 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C2×C8, C2×C8, Dic5, C20, C2×C10, C24, C24, C2×C12, C2×C12, C30, C30, C8⋊C4, C52C8, C40, C2×Dic5, C2×C20, C4×C12, C2×C24, C2×C24, C3×Dic5, C60, C2×C30, C2×C52C8, C4×Dic5, C2×C40, C3×C8⋊C4, C3×C52C8, C120, C6×Dic5, C2×C60, C408C4, C6×C52C8, C12×Dic5, C2×C120, C3×C408C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D5, C12, C2×C6, C42, M4(2), Dic5, D10, C2×C12, C3×D5, C8⋊C4, C4×D5, C2×Dic5, C4×C12, C3×M4(2), C3×Dic5, C6×D5, C8⋊D5, C4×Dic5, C3×C8⋊C4, D5×C12, C6×Dic5, C408C4, C3×C8⋊D5, C12×Dic5, C3×C408C4

Smallest permutation representation of C3×C408C4
Regular action on 480 points
Generators in S480
(1 65 470)(2 66 471)(3 67 472)(4 68 473)(5 69 474)(6 70 475)(7 71 476)(8 72 477)(9 73 478)(10 74 479)(11 75 480)(12 76 441)(13 77 442)(14 78 443)(15 79 444)(16 80 445)(17 41 446)(18 42 447)(19 43 448)(20 44 449)(21 45 450)(22 46 451)(23 47 452)(24 48 453)(25 49 454)(26 50 455)(27 51 456)(28 52 457)(29 53 458)(30 54 459)(31 55 460)(32 56 461)(33 57 462)(34 58 463)(35 59 464)(36 60 465)(37 61 466)(38 62 467)(39 63 468)(40 64 469)(81 333 301)(82 334 302)(83 335 303)(84 336 304)(85 337 305)(86 338 306)(87 339 307)(88 340 308)(89 341 309)(90 342 310)(91 343 311)(92 344 312)(93 345 313)(94 346 314)(95 347 315)(96 348 316)(97 349 317)(98 350 318)(99 351 319)(100 352 320)(101 353 281)(102 354 282)(103 355 283)(104 356 284)(105 357 285)(106 358 286)(107 359 287)(108 360 288)(109 321 289)(110 322 290)(111 323 291)(112 324 292)(113 325 293)(114 326 294)(115 327 295)(116 328 296)(117 329 297)(118 330 298)(119 331 299)(120 332 300)(121 383 195)(122 384 196)(123 385 197)(124 386 198)(125 387 199)(126 388 200)(127 389 161)(128 390 162)(129 391 163)(130 392 164)(131 393 165)(132 394 166)(133 395 167)(134 396 168)(135 397 169)(136 398 170)(137 399 171)(138 400 172)(139 361 173)(140 362 174)(141 363 175)(142 364 176)(143 365 177)(144 366 178)(145 367 179)(146 368 180)(147 369 181)(148 370 182)(149 371 183)(150 372 184)(151 373 185)(152 374 186)(153 375 187)(154 376 188)(155 377 189)(156 378 190)(157 379 191)(158 380 192)(159 381 193)(160 382 194)(201 242 440)(202 243 401)(203 244 402)(204 245 403)(205 246 404)(206 247 405)(207 248 406)(208 249 407)(209 250 408)(210 251 409)(211 252 410)(212 253 411)(213 254 412)(214 255 413)(215 256 414)(216 257 415)(217 258 416)(218 259 417)(219 260 418)(220 261 419)(221 262 420)(222 263 421)(223 264 422)(224 265 423)(225 266 424)(226 267 425)(227 268 426)(228 269 427)(229 270 428)(230 271 429)(231 272 430)(232 273 431)(233 274 432)(234 275 433)(235 276 434)(236 277 435)(237 278 436)(238 279 437)(239 280 438)(240 241 439)
(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)
(1 164 260 295)(2 193 261 284)(3 182 262 313)(4 171 263 302)(5 200 264 291)(6 189 265 320)(7 178 266 309)(8 167 267 298)(9 196 268 287)(10 185 269 316)(11 174 270 305)(12 163 271 294)(13 192 272 283)(14 181 273 312)(15 170 274 301)(16 199 275 290)(17 188 276 319)(18 177 277 308)(19 166 278 297)(20 195 279 286)(21 184 280 315)(22 173 241 304)(23 162 242 293)(24 191 243 282)(25 180 244 311)(26 169 245 300)(27 198 246 289)(28 187 247 318)(29 176 248 307)(30 165 249 296)(31 194 250 285)(32 183 251 314)(33 172 252 303)(34 161 253 292)(35 190 254 281)(36 179 255 310)(37 168 256 299)(38 197 257 288)(39 186 258 317)(40 175 259 306)(41 154 434 99)(42 143 435 88)(43 132 436 117)(44 121 437 106)(45 150 438 95)(46 139 439 84)(47 128 440 113)(48 157 401 102)(49 146 402 91)(50 135 403 120)(51 124 404 109)(52 153 405 98)(53 142 406 87)(54 131 407 116)(55 160 408 105)(56 149 409 94)(57 138 410 83)(58 127 411 112)(59 156 412 101)(60 145 413 90)(61 134 414 119)(62 123 415 108)(63 152 416 97)(64 141 417 86)(65 130 418 115)(66 159 419 104)(67 148 420 93)(68 137 421 82)(69 126 422 111)(70 155 423 100)(71 144 424 89)(72 133 425 118)(73 122 426 107)(74 151 427 96)(75 140 428 85)(76 129 429 114)(77 158 430 103)(78 147 431 92)(79 136 432 81)(80 125 433 110)(201 325 452 390)(202 354 453 379)(203 343 454 368)(204 332 455 397)(205 321 456 386)(206 350 457 375)(207 339 458 364)(208 328 459 393)(209 357 460 382)(210 346 461 371)(211 335 462 400)(212 324 463 389)(213 353 464 378)(214 342 465 367)(215 331 466 396)(216 360 467 385)(217 349 468 374)(218 338 469 363)(219 327 470 392)(220 356 471 381)(221 345 472 370)(222 334 473 399)(223 323 474 388)(224 352 475 377)(225 341 476 366)(226 330 477 395)(227 359 478 384)(228 348 479 373)(229 337 480 362)(230 326 441 391)(231 355 442 380)(232 344 443 369)(233 333 444 398)(234 322 445 387)(235 351 446 376)(236 340 447 365)(237 329 448 394)(238 358 449 383)(239 347 450 372)(240 336 451 361)

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

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

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

156 conjugacy classes

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

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C12 C12 C12 D5 M4(2) Dic5 D10 C3×D5 C4×D5 C4×D5 C3×M4(2) C3×Dic5 C6×D5 C8⋊D5 D5×C12 D5×C12 C3×C8⋊D5 kernel C3×C40⋊8C4 C6×C5⋊2C8 C12×Dic5 C2×C120 C40⋊8C4 C3×C5⋊2C8 C120 C6×Dic5 C2×C5⋊2C8 C4×Dic5 C2×C40 C5⋊2C8 C40 C2×Dic5 C2×C24 C30 C24 C2×C12 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 1 1 1 2 4 4 4 2 2 2 8 8 8 2 4 4 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C3×C408C4 in GL5(𝔽241)

 1 0 0 0 0 0 15 0 0 0 0 0 15 0 0 0 0 0 1 0 0 0 0 0 1
,
 240 0 0 0 0 0 189 189 0 0 0 52 1 0 0 0 0 0 1 239 0 0 0 89 240
,
 177 0 0 0 0 0 123 184 0 0 0 54 118 0 0 0 0 0 5 185 0 0 0 164 236

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,189,52,0,0,0,189,1,0,0,0,0,0,1,89,0,0,0,239,240],[177,0,0,0,0,0,123,54,0,0,0,184,118,0,0,0,0,0,5,164,0,0,0,185,236] >;

C3×C408C4 in GAP, Magma, Sage, TeX

C_3\times C_{40}\rtimes_8C_4
% in TeX

G:=Group("C3xC40:8C4");
// GroupNames label

G:=SmallGroup(480,93);
// by ID

G=gap.SmallGroup(480,93);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^3=b^40=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^29>;
// generators/relations

׿
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