direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C40⋊5C4, C40⋊5C12, C120⋊11C4, C24⋊3Dic5, C30.30D8, C6.16D40, C60.28Q8, C30.13Q16, C6.7Dic20, C12.25Dic10, (C2×C40).5C6, C8⋊1(C3×Dic5), C2.1(C3×D40), C10.4(C3×D8), (C2×C24).9D5, C20.5(C3×Q8), (C2×C6).50D20, C15⋊11(C2.D8), C4⋊Dic5.3C6, C30.49(C4⋊C4), C10.2(C3×Q16), C4.7(C6×Dic5), (C2×C120).13C2, C20.56(C2×C12), C60.241(C2×C4), (C2×C30).110D4, C4.5(C3×Dic10), C2.2(C3×Dic20), C22.9(C3×D20), (C2×C12).423D10, C6.13(C4⋊Dic5), C12.46(C2×Dic5), (C2×C60).504C22, C5⋊3(C3×C2.D8), (C2×C8).3(C3×D5), C10.13(C3×C4⋊C4), (C2×C4).70(C6×D5), C2.4(C3×C4⋊Dic5), (C2×C20).87(C2×C6), (C2×C10).14(C3×D4), (C3×C4⋊Dic5).15C2, SmallGroup(480,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C40⋊5C4
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, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C30, C2.D8, C40, C2×Dic5, C2×C20, C3×C4⋊C4, C2×C24, C3×Dic5, C60, C2×C30, C4⋊Dic5, C2×C40, C3×C2.D8, C120, C6×Dic5, C2×C60, C40⋊5C4, C3×C4⋊Dic5, C2×C120, C3×C40⋊5C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, D5, C12, C2×C6, C4⋊C4, D8, Q16, Dic5, D10, C2×C12, C3×D4, C3×Q8, C3×D5, C2.D8, Dic10, D20, C2×Dic5, C3×C4⋊C4, C3×D8, C3×Q16, C3×Dic5, C6×D5, D40, Dic20, C4⋊Dic5, C3×C2.D8, C3×Dic10, C3×D20, C6×Dic5, C40⋊5C4, C3×D40, C3×Dic20, C3×C4⋊Dic5, C3×C40⋊5C4
►Regular action on 480 points
Generators in S480
(1 243 215)(2 244 216)(3 245 217)(4 246 218)(5 247 219)(6 248 220)(7 249 221)(8 250 222)(9 251 223)(10 252 224)(11 253 225)(12 254 226)(13 255 227)(14 256 228)(15 257 229)(16 258 230)(17 259 231)(18 260 232)(19 261 233)(20 262 234)(21 263 235)(22 264 236)(23 265 237)(24 266 238)(25 267 239)(26 268 240)(27 269 201)(28 270 202)(29 271 203)(30 272 204)(31 273 205)(32 274 206)(33 275 207)(34 276 208)(35 277 209)(36 278 210)(37 279 211)(38 280 212)(39 241 213)(40 242 214)(41 376 320)(42 377 281)(43 378 282)(44 379 283)(45 380 284)(46 381 285)(47 382 286)(48 383 287)(49 384 288)(50 385 289)(51 386 290)(52 387 291)(53 388 292)(54 389 293)(55 390 294)(56 391 295)(57 392 296)(58 393 297)(59 394 298)(60 395 299)(61 396 300)(62 397 301)(63 398 302)(64 399 303)(65 400 304)(66 361 305)(67 362 306)(68 363 307)(69 364 308)(70 365 309)(71 366 310)(72 367 311)(73 368 312)(74 369 313)(75 370 314)(76 371 315)(77 372 316)(78 373 317)(79 374 318)(80 375 319)(81 465 125)(82 466 126)(83 467 127)(84 468 128)(85 469 129)(86 470 130)(87 471 131)(88 472 132)(89 473 133)(90 474 134)(91 475 135)(92 476 136)(93 477 137)(94 478 138)(95 479 139)(96 480 140)(97 441 141)(98 442 142)(99 443 143)(100 444 144)(101 445 145)(102 446 146)(103 447 147)(104 448 148)(105 449 149)(106 450 150)(107 451 151)(108 452 152)(109 453 153)(110 454 154)(111 455 155)(112 456 156)(113 457 157)(114 458 158)(115 459 159)(116 460 160)(117 461 121)(118 462 122)(119 463 123)(120 464 124)(161 348 437)(162 349 438)(163 350 439)(164 351 440)(165 352 401)(166 353 402)(167 354 403)(168 355 404)(169 356 405)(170 357 406)(171 358 407)(172 359 408)(173 360 409)(174 321 410)(175 322 411)(176 323 412)(177 324 413)(178 325 414)(179 326 415)(180 327 416)(181 328 417)(182 329 418)(183 330 419)(184 331 420)(185 332 421)(186 333 422)(187 334 423)(188 335 424)(189 336 425)(190 337 426)(191 338 427)(192 339 428)(193 340 429)(194 341 430)(195 342 431)(196 343 432)(197 344 433)(198 345 434)(199 346 435)(200 347 436)
(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)
(1 168 49 119)(2 167 50 118)(3 166 51 117)(4 165 52 116)(5 164 53 115)(6 163 54 114)(7 162 55 113)(8 161 56 112)(9 200 57 111)(10 199 58 110)(11 198 59 109)(12 197 60 108)(13 196 61 107)(14 195 62 106)(15 194 63 105)(16 193 64 104)(17 192 65 103)(18 191 66 102)(19 190 67 101)(20 189 68 100)(21 188 69 99)(22 187 70 98)(23 186 71 97)(24 185 72 96)(25 184 73 95)(26 183 74 94)(27 182 75 93)(28 181 76 92)(29 180 77 91)(30 179 78 90)(31 178 79 89)(32 177 80 88)(33 176 41 87)(34 175 42 86)(35 174 43 85)(36 173 44 84)(37 172 45 83)(38 171 46 82)(39 170 47 81)(40 169 48 120)(121 217 402 290)(122 216 403 289)(123 215 404 288)(124 214 405 287)(125 213 406 286)(126 212 407 285)(127 211 408 284)(128 210 409 283)(129 209 410 282)(130 208 411 281)(131 207 412 320)(132 206 413 319)(133 205 414 318)(134 204 415 317)(135 203 416 316)(136 202 417 315)(137 201 418 314)(138 240 419 313)(139 239 420 312)(140 238 421 311)(141 237 422 310)(142 236 423 309)(143 235 424 308)(144 234 425 307)(145 233 426 306)(146 232 427 305)(147 231 428 304)(148 230 429 303)(149 229 430 302)(150 228 431 301)(151 227 432 300)(152 226 433 299)(153 225 434 298)(154 224 435 297)(155 223 436 296)(156 222 437 295)(157 221 438 294)(158 220 439 293)(159 219 440 292)(160 218 401 291)(241 357 382 465)(242 356 383 464)(243 355 384 463)(244 354 385 462)(245 353 386 461)(246 352 387 460)(247 351 388 459)(248 350 389 458)(249 349 390 457)(250 348 391 456)(251 347 392 455)(252 346 393 454)(253 345 394 453)(254 344 395 452)(255 343 396 451)(256 342 397 450)(257 341 398 449)(258 340 399 448)(259 339 400 447)(260 338 361 446)(261 337 362 445)(262 336 363 444)(263 335 364 443)(264 334 365 442)(265 333 366 441)(266 332 367 480)(267 331 368 479)(268 330 369 478)(269 329 370 477)(270 328 371 476)(271 327 372 475)(272 326 373 474)(273 325 374 473)(274 324 375 472)(275 323 376 471)(276 322 377 470)(277 321 378 469)(278 360 379 468)(279 359 380 467)(280 358 381 466)
G:=sub<Sym(480)| (1,243,215)(2,244,216)(3,245,217)(4,246,218)(5,247,219)(6,248,220)(7,249,221)(8,250,222)(9,251,223)(10,252,224)(11,253,225)(12,254,226)(13,255,227)(14,256,228)(15,257,229)(16,258,230)(17,259,231)(18,260,232)(19,261,233)(20,262,234)(21,263,235)(22,264,236)(23,265,237)(24,266,238)(25,267,239)(26,268,240)(27,269,201)(28,270,202)(29,271,203)(30,272,204)(31,273,205)(32,274,206)(33,275,207)(34,276,208)(35,277,209)(36,278,210)(37,279,211)(38,280,212)(39,241,213)(40,242,214)(41,376,320)(42,377,281)(43,378,282)(44,379,283)(45,380,284)(46,381,285)(47,382,286)(48,383,287)(49,384,288)(50,385,289)(51,386,290)(52,387,291)(53,388,292)(54,389,293)(55,390,294)(56,391,295)(57,392,296)(58,393,297)(59,394,298)(60,395,299)(61,396,300)(62,397,301)(63,398,302)(64,399,303)(65,400,304)(66,361,305)(67,362,306)(68,363,307)(69,364,308)(70,365,309)(71,366,310)(72,367,311)(73,368,312)(74,369,313)(75,370,314)(76,371,315)(77,372,316)(78,373,317)(79,374,318)(80,375,319)(81,465,125)(82,466,126)(83,467,127)(84,468,128)(85,469,129)(86,470,130)(87,471,131)(88,472,132)(89,473,133)(90,474,134)(91,475,135)(92,476,136)(93,477,137)(94,478,138)(95,479,139)(96,480,140)(97,441,141)(98,442,142)(99,443,143)(100,444,144)(101,445,145)(102,446,146)(103,447,147)(104,448,148)(105,449,149)(106,450,150)(107,451,151)(108,452,152)(109,453,153)(110,454,154)(111,455,155)(112,456,156)(113,457,157)(114,458,158)(115,459,159)(116,460,160)(117,461,121)(118,462,122)(119,463,123)(120,464,124)(161,348,437)(162,349,438)(163,350,439)(164,351,440)(165,352,401)(166,353,402)(167,354,403)(168,355,404)(169,356,405)(170,357,406)(171,358,407)(172,359,408)(173,360,409)(174,321,410)(175,322,411)(176,323,412)(177,324,413)(178,325,414)(179,326,415)(180,327,416)(181,328,417)(182,329,418)(183,330,419)(184,331,420)(185,332,421)(186,333,422)(187,334,423)(188,335,424)(189,336,425)(190,337,426)(191,338,427)(192,339,428)(193,340,429)(194,341,430)(195,342,431)(196,343,432)(197,344,433)(198,345,434)(199,346,435)(200,347,436), (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), (1,168,49,119)(2,167,50,118)(3,166,51,117)(4,165,52,116)(5,164,53,115)(6,163,54,114)(7,162,55,113)(8,161,56,112)(9,200,57,111)(10,199,58,110)(11,198,59,109)(12,197,60,108)(13,196,61,107)(14,195,62,106)(15,194,63,105)(16,193,64,104)(17,192,65,103)(18,191,66,102)(19,190,67,101)(20,189,68,100)(21,188,69,99)(22,187,70,98)(23,186,71,97)(24,185,72,96)(25,184,73,95)(26,183,74,94)(27,182,75,93)(28,181,76,92)(29,180,77,91)(30,179,78,90)(31,178,79,89)(32,177,80,88)(33,176,41,87)(34,175,42,86)(35,174,43,85)(36,173,44,84)(37,172,45,83)(38,171,46,82)(39,170,47,81)(40,169,48,120)(121,217,402,290)(122,216,403,289)(123,215,404,288)(124,214,405,287)(125,213,406,286)(126,212,407,285)(127,211,408,284)(128,210,409,283)(129,209,410,282)(130,208,411,281)(131,207,412,320)(132,206,413,319)(133,205,414,318)(134,204,415,317)(135,203,416,316)(136,202,417,315)(137,201,418,314)(138,240,419,313)(139,239,420,312)(140,238,421,311)(141,237,422,310)(142,236,423,309)(143,235,424,308)(144,234,425,307)(145,233,426,306)(146,232,427,305)(147,231,428,304)(148,230,429,303)(149,229,430,302)(150,228,431,301)(151,227,432,300)(152,226,433,299)(153,225,434,298)(154,224,435,297)(155,223,436,296)(156,222,437,295)(157,221,438,294)(158,220,439,293)(159,219,440,292)(160,218,401,291)(241,357,382,465)(242,356,383,464)(243,355,384,463)(244,354,385,462)(245,353,386,461)(246,352,387,460)(247,351,388,459)(248,350,389,458)(249,349,390,457)(250,348,391,456)(251,347,392,455)(252,346,393,454)(253,345,394,453)(254,344,395,452)(255,343,396,451)(256,342,397,450)(257,341,398,449)(258,340,399,448)(259,339,400,447)(260,338,361,446)(261,337,362,445)(262,336,363,444)(263,335,364,443)(264,334,365,442)(265,333,366,441)(266,332,367,480)(267,331,368,479)(268,330,369,478)(269,329,370,477)(270,328,371,476)(271,327,372,475)(272,326,373,474)(273,325,374,473)(274,324,375,472)(275,323,376,471)(276,322,377,470)(277,321,378,469)(278,360,379,468)(279,359,380,467)(280,358,381,466)>;
G:=Group( (1,243,215)(2,244,216)(3,245,217)(4,246,218)(5,247,219)(6,248,220)(7,249,221)(8,250,222)(9,251,223)(10,252,224)(11,253,225)(12,254,226)(13,255,227)(14,256,228)(15,257,229)(16,258,230)(17,259,231)(18,260,232)(19,261,233)(20,262,234)(21,263,235)(22,264,236)(23,265,237)(24,266,238)(25,267,239)(26,268,240)(27,269,201)(28,270,202)(29,271,203)(30,272,204)(31,273,205)(32,274,206)(33,275,207)(34,276,208)(35,277,209)(36,278,210)(37,279,211)(38,280,212)(39,241,213)(40,242,214)(41,376,320)(42,377,281)(43,378,282)(44,379,283)(45,380,284)(46,381,285)(47,382,286)(48,383,287)(49,384,288)(50,385,289)(51,386,290)(52,387,291)(53,388,292)(54,389,293)(55,390,294)(56,391,295)(57,392,296)(58,393,297)(59,394,298)(60,395,299)(61,396,300)(62,397,301)(63,398,302)(64,399,303)(65,400,304)(66,361,305)(67,362,306)(68,363,307)(69,364,308)(70,365,309)(71,366,310)(72,367,311)(73,368,312)(74,369,313)(75,370,314)(76,371,315)(77,372,316)(78,373,317)(79,374,318)(80,375,319)(81,465,125)(82,466,126)(83,467,127)(84,468,128)(85,469,129)(86,470,130)(87,471,131)(88,472,132)(89,473,133)(90,474,134)(91,475,135)(92,476,136)(93,477,137)(94,478,138)(95,479,139)(96,480,140)(97,441,141)(98,442,142)(99,443,143)(100,444,144)(101,445,145)(102,446,146)(103,447,147)(104,448,148)(105,449,149)(106,450,150)(107,451,151)(108,452,152)(109,453,153)(110,454,154)(111,455,155)(112,456,156)(113,457,157)(114,458,158)(115,459,159)(116,460,160)(117,461,121)(118,462,122)(119,463,123)(120,464,124)(161,348,437)(162,349,438)(163,350,439)(164,351,440)(165,352,401)(166,353,402)(167,354,403)(168,355,404)(169,356,405)(170,357,406)(171,358,407)(172,359,408)(173,360,409)(174,321,410)(175,322,411)(176,323,412)(177,324,413)(178,325,414)(179,326,415)(180,327,416)(181,328,417)(182,329,418)(183,330,419)(184,331,420)(185,332,421)(186,333,422)(187,334,423)(188,335,424)(189,336,425)(190,337,426)(191,338,427)(192,339,428)(193,340,429)(194,341,430)(195,342,431)(196,343,432)(197,344,433)(198,345,434)(199,346,435)(200,347,436), (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), (1,168,49,119)(2,167,50,118)(3,166,51,117)(4,165,52,116)(5,164,53,115)(6,163,54,114)(7,162,55,113)(8,161,56,112)(9,200,57,111)(10,199,58,110)(11,198,59,109)(12,197,60,108)(13,196,61,107)(14,195,62,106)(15,194,63,105)(16,193,64,104)(17,192,65,103)(18,191,66,102)(19,190,67,101)(20,189,68,100)(21,188,69,99)(22,187,70,98)(23,186,71,97)(24,185,72,96)(25,184,73,95)(26,183,74,94)(27,182,75,93)(28,181,76,92)(29,180,77,91)(30,179,78,90)(31,178,79,89)(32,177,80,88)(33,176,41,87)(34,175,42,86)(35,174,43,85)(36,173,44,84)(37,172,45,83)(38,171,46,82)(39,170,47,81)(40,169,48,120)(121,217,402,290)(122,216,403,289)(123,215,404,288)(124,214,405,287)(125,213,406,286)(126,212,407,285)(127,211,408,284)(128,210,409,283)(129,209,410,282)(130,208,411,281)(131,207,412,320)(132,206,413,319)(133,205,414,318)(134,204,415,317)(135,203,416,316)(136,202,417,315)(137,201,418,314)(138,240,419,313)(139,239,420,312)(140,238,421,311)(141,237,422,310)(142,236,423,309)(143,235,424,308)(144,234,425,307)(145,233,426,306)(146,232,427,305)(147,231,428,304)(148,230,429,303)(149,229,430,302)(150,228,431,301)(151,227,432,300)(152,226,433,299)(153,225,434,298)(154,224,435,297)(155,223,436,296)(156,222,437,295)(157,221,438,294)(158,220,439,293)(159,219,440,292)(160,218,401,291)(241,357,382,465)(242,356,383,464)(243,355,384,463)(244,354,385,462)(245,353,386,461)(246,352,387,460)(247,351,388,459)(248,350,389,458)(249,349,390,457)(250,348,391,456)(251,347,392,455)(252,346,393,454)(253,345,394,453)(254,344,395,452)(255,343,396,451)(256,342,397,450)(257,341,398,449)(258,340,399,448)(259,339,400,447)(260,338,361,446)(261,337,362,445)(262,336,363,444)(263,335,364,443)(264,334,365,442)(265,333,366,441)(266,332,367,480)(267,331,368,479)(268,330,369,478)(269,329,370,477)(270,328,371,476)(271,327,372,475)(272,326,373,474)(273,325,374,473)(274,324,375,472)(275,323,376,471)(276,322,377,470)(277,321,378,469)(278,360,379,468)(279,359,380,467)(280,358,381,466) );
G=PermutationGroup([[(1,243,215),(2,244,216),(3,245,217),(4,246,218),(5,247,219),(6,248,220),(7,249,221),(8,250,222),(9,251,223),(10,252,224),(11,253,225),(12,254,226),(13,255,227),(14,256,228),(15,257,229),(16,258,230),(17,259,231),(18,260,232),(19,261,233),(20,262,234),(21,263,235),(22,264,236),(23,265,237),(24,266,238),(25,267,239),(26,268,240),(27,269,201),(28,270,202),(29,271,203),(30,272,204),(31,273,205),(32,274,206),(33,275,207),(34,276,208),(35,277,209),(36,278,210),(37,279,211),(38,280,212),(39,241,213),(40,242,214),(41,376,320),(42,377,281),(43,378,282),(44,379,283),(45,380,284),(46,381,285),(47,382,286),(48,383,287),(49,384,288),(50,385,289),(51,386,290),(52,387,291),(53,388,292),(54,389,293),(55,390,294),(56,391,295),(57,392,296),(58,393,297),(59,394,298),(60,395,299),(61,396,300),(62,397,301),(63,398,302),(64,399,303),(65,400,304),(66,361,305),(67,362,306),(68,363,307),(69,364,308),(70,365,309),(71,366,310),(72,367,311),(73,368,312),(74,369,313),(75,370,314),(76,371,315),(77,372,316),(78,373,317),(79,374,318),(80,375,319),(81,465,125),(82,466,126),(83,467,127),(84,468,128),(85,469,129),(86,470,130),(87,471,131),(88,472,132),(89,473,133),(90,474,134),(91,475,135),(92,476,136),(93,477,137),(94,478,138),(95,479,139),(96,480,140),(97,441,141),(98,442,142),(99,443,143),(100,444,144),(101,445,145),(102,446,146),(103,447,147),(104,448,148),(105,449,149),(106,450,150),(107,451,151),(108,452,152),(109,453,153),(110,454,154),(111,455,155),(112,456,156),(113,457,157),(114,458,158),(115,459,159),(116,460,160),(117,461,121),(118,462,122),(119,463,123),(120,464,124),(161,348,437),(162,349,438),(163,350,439),(164,351,440),(165,352,401),(166,353,402),(167,354,403),(168,355,404),(169,356,405),(170,357,406),(171,358,407),(172,359,408),(173,360,409),(174,321,410),(175,322,411),(176,323,412),(177,324,413),(178,325,414),(179,326,415),(180,327,416),(181,328,417),(182,329,418),(183,330,419),(184,331,420),(185,332,421),(186,333,422),(187,334,423),(188,335,424),(189,336,425),(190,337,426),(191,338,427),(192,339,428),(193,340,429),(194,341,430),(195,342,431),(196,343,432),(197,344,433),(198,345,434),(199,346,435),(200,347,436)], [(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)], [(1,168,49,119),(2,167,50,118),(3,166,51,117),(4,165,52,116),(5,164,53,115),(6,163,54,114),(7,162,55,113),(8,161,56,112),(9,200,57,111),(10,199,58,110),(11,198,59,109),(12,197,60,108),(13,196,61,107),(14,195,62,106),(15,194,63,105),(16,193,64,104),(17,192,65,103),(18,191,66,102),(19,190,67,101),(20,189,68,100),(21,188,69,99),(22,187,70,98),(23,186,71,97),(24,185,72,96),(25,184,73,95),(26,183,74,94),(27,182,75,93),(28,181,76,92),(29,180,77,91),(30,179,78,90),(31,178,79,89),(32,177,80,88),(33,176,41,87),(34,175,42,86),(35,174,43,85),(36,173,44,84),(37,172,45,83),(38,171,46,82),(39,170,47,81),(40,169,48,120),(121,217,402,290),(122,216,403,289),(123,215,404,288),(124,214,405,287),(125,213,406,286),(126,212,407,285),(127,211,408,284),(128,210,409,283),(129,209,410,282),(130,208,411,281),(131,207,412,320),(132,206,413,319),(133,205,414,318),(134,204,415,317),(135,203,416,316),(136,202,417,315),(137,201,418,314),(138,240,419,313),(139,239,420,312),(140,238,421,311),(141,237,422,310),(142,236,423,309),(143,235,424,308),(144,234,425,307),(145,233,426,306),(146,232,427,305),(147,231,428,304),(148,230,429,303),(149,229,430,302),(150,228,431,301),(151,227,432,300),(152,226,433,299),(153,225,434,298),(154,224,435,297),(155,223,436,296),(156,222,437,295),(157,221,438,294),(158,220,439,293),(159,219,440,292),(160,218,401,291),(241,357,382,465),(242,356,383,464),(243,355,384,463),(244,354,385,462),(245,353,386,461),(246,352,387,460),(247,351,388,459),(248,350,389,458),(249,349,390,457),(250,348,391,456),(251,347,392,455),(252,346,393,454),(253,345,394,453),(254,344,395,452),(255,343,396,451),(256,342,397,450),(257,341,398,449),(258,340,399,448),(259,339,400,447),(260,338,361,446),(261,337,362,445),(262,336,363,444),(263,335,364,443),(264,334,365,442),(265,333,366,441),(266,332,367,480),(267,331,368,479),(268,330,369,478),(269,329,370,477),(270,328,371,476),(271,327,372,475),(272,326,373,474),(273,325,374,473),(274,324,375,472),(275,323,376,471),(276,322,377,470),(277,321,378,469),(278,360,379,468),(279,359,380,467),(280,358,381,466)]])
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×C40⋊5C4 ►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×C40⋊5C4 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