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## G = C3×C40⋊6C4order 480 = 25·3·5

### Direct product of C3 and C40⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C40⋊6C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C3×C4⋊Dic5 — C3×C40⋊6C4
 Lower central C5 — C10 — C20 — C3×C40⋊6C4
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C3×C406C4
G = < a,b,c | a3=b40=c4=1, ab=ba, ac=ca, cbc-1=b19 >

Subgroups: 224 in 72 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C30, C30, C4.Q8, C40, C2×Dic5, C2×C20, C3×C4⋊C4, C2×C24, C3×Dic5, C60, C2×C30, C4⋊Dic5, C2×C40, C3×C4.Q8, C120, C6×Dic5, C2×C60, C406C4, C3×C4⋊Dic5, C2×C120, C3×C406C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, D5, C12, C2×C6, C4⋊C4, SD16, Dic5, D10, C2×C12, C3×D4, C3×Q8, C3×D5, C4.Q8, Dic10, D20, C2×Dic5, C3×C4⋊C4, C3×SD16, C3×Dic5, C6×D5, C40⋊C2, C4⋊Dic5, C3×C4.Q8, C3×Dic10, C3×D20, C6×Dic5, C406C4, C3×C40⋊C2, C3×C4⋊Dic5, C3×C406C4

Smallest permutation representation of C3×C406C4
Regular action on 480 points
Generators in S480
(1 159 229)(2 160 230)(3 121 231)(4 122 232)(5 123 233)(6 124 234)(7 125 235)(8 126 236)(9 127 237)(10 128 238)(11 129 239)(12 130 240)(13 131 201)(14 132 202)(15 133 203)(16 134 204)(17 135 205)(18 136 206)(19 137 207)(20 138 208)(21 139 209)(22 140 210)(23 141 211)(24 142 212)(25 143 213)(26 144 214)(27 145 215)(28 146 216)(29 147 217)(30 148 218)(31 149 219)(32 150 220)(33 151 221)(34 152 222)(35 153 223)(36 154 224)(37 155 225)(38 156 226)(39 157 227)(40 158 228)(41 393 413)(42 394 414)(43 395 415)(44 396 416)(45 397 417)(46 398 418)(47 399 419)(48 400 420)(49 361 421)(50 362 422)(51 363 423)(52 364 424)(53 365 425)(54 366 426)(55 367 427)(56 368 428)(57 369 429)(58 370 430)(59 371 431)(60 372 432)(61 373 433)(62 374 434)(63 375 435)(64 376 436)(65 377 437)(66 378 438)(67 379 439)(68 380 440)(69 381 401)(70 382 402)(71 383 403)(72 384 404)(73 385 405)(74 386 406)(75 387 407)(76 388 408)(77 389 409)(78 390 410)(79 391 411)(80 392 412)(81 352 456)(82 353 457)(83 354 458)(84 355 459)(85 356 460)(86 357 461)(87 358 462)(88 359 463)(89 360 464)(90 321 465)(91 322 466)(92 323 467)(93 324 468)(94 325 469)(95 326 470)(96 327 471)(97 328 472)(98 329 473)(99 330 474)(100 331 475)(101 332 476)(102 333 477)(103 334 478)(104 335 479)(105 336 480)(106 337 441)(107 338 442)(108 339 443)(109 340 444)(110 341 445)(111 342 446)(112 343 447)(113 344 448)(114 345 449)(115 346 450)(116 347 451)(117 348 452)(118 349 453)(119 350 454)(120 351 455)(161 305 270)(162 306 271)(163 307 272)(164 308 273)(165 309 274)(166 310 275)(167 311 276)(168 312 277)(169 313 278)(170 314 279)(171 315 280)(172 316 241)(173 317 242)(174 318 243)(175 319 244)(176 320 245)(177 281 246)(178 282 247)(179 283 248)(180 284 249)(181 285 250)(182 286 251)(183 287 252)(184 288 253)(185 289 254)(186 290 255)(187 291 256)(188 292 257)(189 293 258)(190 294 259)(191 295 260)(192 296 261)(193 297 262)(194 298 263)(195 299 264)(196 300 265)(197 301 266)(198 302 267)(199 303 268)(200 304 269)
(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)
(1 119 436 269)(2 98 437 248)(3 117 438 267)(4 96 439 246)(5 115 440 265)(6 94 401 244)(7 113 402 263)(8 92 403 242)(9 111 404 261)(10 90 405 280)(11 109 406 259)(12 88 407 278)(13 107 408 257)(14 86 409 276)(15 105 410 255)(16 84 411 274)(17 103 412 253)(18 82 413 272)(19 101 414 251)(20 120 415 270)(21 99 416 249)(22 118 417 268)(23 97 418 247)(24 116 419 266)(25 95 420 245)(26 114 421 264)(27 93 422 243)(28 112 423 262)(29 91 424 241)(30 110 425 260)(31 89 426 279)(32 108 427 258)(33 87 428 277)(34 106 429 256)(35 85 430 275)(36 104 431 254)(37 83 432 273)(38 102 433 252)(39 81 434 271)(40 100 435 250)(41 163 136 353)(42 182 137 332)(43 161 138 351)(44 180 139 330)(45 199 140 349)(46 178 141 328)(47 197 142 347)(48 176 143 326)(49 195 144 345)(50 174 145 324)(51 193 146 343)(52 172 147 322)(53 191 148 341)(54 170 149 360)(55 189 150 339)(56 168 151 358)(57 187 152 337)(58 166 153 356)(59 185 154 335)(60 164 155 354)(61 183 156 333)(62 162 157 352)(63 181 158 331)(64 200 159 350)(65 179 160 329)(66 198 121 348)(67 177 122 327)(68 196 123 346)(69 175 124 325)(70 194 125 344)(71 173 126 323)(72 192 127 342)(73 171 128 321)(74 190 129 340)(75 169 130 359)(76 188 131 338)(77 167 132 357)(78 186 133 336)(79 165 134 355)(80 184 135 334)(201 442 388 292)(202 461 389 311)(203 480 390 290)(204 459 391 309)(205 478 392 288)(206 457 393 307)(207 476 394 286)(208 455 395 305)(209 474 396 284)(210 453 397 303)(211 472 398 282)(212 451 399 301)(213 470 400 320)(214 449 361 299)(215 468 362 318)(216 447 363 297)(217 466 364 316)(218 445 365 295)(219 464 366 314)(220 443 367 293)(221 462 368 312)(222 441 369 291)(223 460 370 310)(224 479 371 289)(225 458 372 308)(226 477 373 287)(227 456 374 306)(228 475 375 285)(229 454 376 304)(230 473 377 283)(231 452 378 302)(232 471 379 281)(233 450 380 300)(234 469 381 319)(235 448 382 298)(236 467 383 317)(237 446 384 296)(238 465 385 315)(239 444 386 294)(240 463 387 313)

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

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

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

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 type + + + - + + - + - + image C1 C2 C2 C3 C4 C6 C6 C12 Q8 D4 D5 SD16 Dic5 D10 C3×Q8 C3×D4 C3×D5 Dic10 D20 C3×SD16 C3×Dic5 C6×D5 C40⋊C2 C3×Dic10 C3×D20 C3×C40⋊C2 kernel C3×C40⋊6C4 C3×C4⋊Dic5 C2×C120 C40⋊6C4 C120 C4⋊Dic5 C2×C40 C40 C60 C2×C30 C2×C24 C30 C24 C2×C12 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 4 4 2 8 1 1 2 4 4 2 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C3×C406C4 in GL3(𝔽241) generated by

 15 0 0 0 1 0 0 0 1
,
 1 0 0 0 173 116 0 125 166
,
 177 0 0 0 42 227 0 212 199
G:=sub<GL(3,GF(241))| [15,0,0,0,1,0,0,0,1],[1,0,0,0,173,125,0,116,166],[177,0,0,0,42,212,0,227,199] >;

C3×C406C4 in GAP, Magma, Sage, TeX

C_3\times C_{40}\rtimes_6C_4
% in TeX

G:=Group("C3xC40:6C4");
// GroupNames label

G:=SmallGroup(480,95);
// by ID

G=gap.SmallGroup(480,95);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,2524,102,18822]);
// 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^19>;
// generators/relations

׿
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