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## G = C3×C40⋊5C4order 480 = 25·3·5

### Direct product of C3 and C40⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C40⋊5C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C3×C4⋊Dic5 — C3×C40⋊5C4
 Lower central C5 — C10 — C20 — C3×C40⋊5C4
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C3×C405C4
G = < a,b,c | a3=b40=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 224 in 72 conjugacy classes, 50 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], C12 [×2], C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5 [×2], C20 [×2], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C30 [×3], C2.D8, C40 [×2], C2×Dic5 [×2], C2×C20, C3×C4⋊C4 [×2], C2×C24, C3×Dic5 [×2], C60 [×2], C2×C30, C4⋊Dic5 [×2], C2×C40, C3×C2.D8, C120 [×2], C6×Dic5 [×2], C2×C60, C405C4, C3×C4⋊Dic5 [×2], C2×C120, C3×C405C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4, Q8, D5, C12 [×2], C2×C6, C4⋊C4, D8, Q16, Dic5 [×2], D10, C2×C12, C3×D4, C3×Q8, C3×D5, C2.D8, Dic10, D20, C2×Dic5, C3×C4⋊C4, C3×D8, C3×Q16, C3×Dic5 [×2], C6×D5, D40, Dic20, C4⋊Dic5, C3×C2.D8, C3×Dic10, C3×D20, C6×Dic5, C405C4, C3×D40, C3×Dic20, C3×C4⋊Dic5, C3×C405C4

Smallest permutation representation of C3×C405C4
Regular action on 480 points
Generators in S480
(1 453 180)(2 454 181)(3 455 182)(4 456 183)(5 457 184)(6 458 185)(7 459 186)(8 460 187)(9 461 188)(10 462 189)(11 463 190)(12 464 191)(13 465 192)(14 466 193)(15 467 194)(16 468 195)(17 469 196)(18 470 197)(19 471 198)(20 472 199)(21 473 200)(22 474 161)(23 475 162)(24 476 163)(25 477 164)(26 478 165)(27 479 166)(28 480 167)(29 441 168)(30 442 169)(31 443 170)(32 444 171)(33 445 172)(34 446 173)(35 447 174)(36 448 175)(37 449 176)(38 450 177)(39 451 178)(40 452 179)(41 290 100)(42 291 101)(43 292 102)(44 293 103)(45 294 104)(46 295 105)(47 296 106)(48 297 107)(49 298 108)(50 299 109)(51 300 110)(52 301 111)(53 302 112)(54 303 113)(55 304 114)(56 305 115)(57 306 116)(58 307 117)(59 308 118)(60 309 119)(61 310 120)(62 311 81)(63 312 82)(64 313 83)(65 314 84)(66 315 85)(67 316 86)(68 317 87)(69 318 88)(70 319 89)(71 320 90)(72 281 91)(73 282 92)(74 283 93)(75 284 94)(76 285 95)(77 286 96)(78 287 97)(79 288 98)(80 289 99)(121 231 246)(122 232 247)(123 233 248)(124 234 249)(125 235 250)(126 236 251)(127 237 252)(128 238 253)(129 239 254)(130 240 255)(131 201 256)(132 202 257)(133 203 258)(134 204 259)(135 205 260)(136 206 261)(137 207 262)(138 208 263)(139 209 264)(140 210 265)(141 211 266)(142 212 267)(143 213 268)(144 214 269)(145 215 270)(146 216 271)(147 217 272)(148 218 273)(149 219 274)(150 220 275)(151 221 276)(152 222 277)(153 223 278)(154 224 279)(155 225 280)(156 226 241)(157 227 242)(158 228 243)(159 229 244)(160 230 245)(321 371 405)(322 372 406)(323 373 407)(324 374 408)(325 375 409)(326 376 410)(327 377 411)(328 378 412)(329 379 413)(330 380 414)(331 381 415)(332 382 416)(333 383 417)(334 384 418)(335 385 419)(336 386 420)(337 387 421)(338 388 422)(339 389 423)(340 390 424)(341 391 425)(342 392 426)(343 393 427)(344 394 428)(345 395 429)(346 396 430)(347 397 431)(348 398 432)(349 399 433)(350 400 434)(351 361 435)(352 362 436)(353 363 437)(354 364 438)(355 365 439)(356 366 440)(357 367 401)(358 368 402)(359 369 403)(360 370 404)
(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)
(1 333 133 69)(2 332 134 68)(3 331 135 67)(4 330 136 66)(5 329 137 65)(6 328 138 64)(7 327 139 63)(8 326 140 62)(9 325 141 61)(10 324 142 60)(11 323 143 59)(12 322 144 58)(13 321 145 57)(14 360 146 56)(15 359 147 55)(16 358 148 54)(17 357 149 53)(18 356 150 52)(19 355 151 51)(20 354 152 50)(21 353 153 49)(22 352 154 48)(23 351 155 47)(24 350 156 46)(25 349 157 45)(26 348 158 44)(27 347 159 43)(28 346 160 42)(29 345 121 41)(30 344 122 80)(31 343 123 79)(32 342 124 78)(33 341 125 77)(34 340 126 76)(35 339 127 75)(36 338 128 74)(37 337 129 73)(38 336 130 72)(39 335 131 71)(40 334 132 70)(81 187 410 265)(82 186 411 264)(83 185 412 263)(84 184 413 262)(85 183 414 261)(86 182 415 260)(87 181 416 259)(88 180 417 258)(89 179 418 257)(90 178 419 256)(91 177 420 255)(92 176 421 254)(93 175 422 253)(94 174 423 252)(95 173 424 251)(96 172 425 250)(97 171 426 249)(98 170 427 248)(99 169 428 247)(100 168 429 246)(101 167 430 245)(102 166 431 244)(103 165 432 243)(104 164 433 242)(105 163 434 241)(106 162 435 280)(107 161 436 279)(108 200 437 278)(109 199 438 277)(110 198 439 276)(111 197 440 275)(112 196 401 274)(113 195 402 273)(114 194 403 272)(115 193 404 271)(116 192 405 270)(117 191 406 269)(118 190 407 268)(119 189 408 267)(120 188 409 266)(201 320 451 385)(202 319 452 384)(203 318 453 383)(204 317 454 382)(205 316 455 381)(206 315 456 380)(207 314 457 379)(208 313 458 378)(209 312 459 377)(210 311 460 376)(211 310 461 375)(212 309 462 374)(213 308 463 373)(214 307 464 372)(215 306 465 371)(216 305 466 370)(217 304 467 369)(218 303 468 368)(219 302 469 367)(220 301 470 366)(221 300 471 365)(222 299 472 364)(223 298 473 363)(224 297 474 362)(225 296 475 361)(226 295 476 400)(227 294 477 399)(228 293 478 398)(229 292 479 397)(230 291 480 396)(231 290 441 395)(232 289 442 394)(233 288 443 393)(234 287 444 392)(235 286 445 391)(236 285 446 390)(237 284 447 389)(238 283 448 388)(239 282 449 387)(240 281 450 386)

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

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

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

138 conjugacy classes

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

138 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + - + + + - - + - + + - image C1 C2 C2 C3 C4 C6 C6 C12 Q8 D4 D5 D8 Q16 Dic5 D10 C3×Q8 C3×D4 C3×D5 Dic10 D20 C3×D8 C3×Q16 C3×Dic5 C6×D5 D40 Dic20 C3×Dic10 C3×D20 C3×D40 C3×Dic20 kernel C3×C40⋊5C4 C3×C4⋊Dic5 C2×C120 C40⋊5C4 C120 C4⋊Dic5 C2×C40 C40 C60 C2×C30 C2×C24 C30 C30 C24 C2×C12 C20 C2×C10 C2×C8 C12 C2×C6 C10 C10 C8 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 2 2 2 4 2 2 2 4 4 4 4 4 8 4 8 8 8 8 16 16

Matrix representation of C3×C405C4 in GL3(𝔽241) generated by

 225 0 0 0 15 0 0 0 15
,
 240 0 0 0 47 14 0 227 42
,
 177 0 0 0 19 7 0 224 222
G:=sub<GL(3,GF(241))| [225,0,0,0,15,0,0,0,15],[240,0,0,0,47,227,0,14,42],[177,0,0,0,19,224,0,7,222] >;

C3×C405C4 in GAP, Magma, Sage, TeX

C_3\times C_{40}\rtimes_5C_4
% in TeX

G:=Group("C3xC40:5C4");
// GroupNames label

G:=SmallGroup(480,96);
// by ID

G=gap.SmallGroup(480,96);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,512,2524,102,18822]);
// 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^-1>;
// generators/relations

׿
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