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G = C3×C406C4order 480 = 25·3·5

Direct product of C3 and C406C4

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C406C4, C406C12, C12015C4, C246Dic5, C60.27Q8, C30.20SD16, C12.24Dic10, (C2×C40).8C6, C82(C3×Dic5), C20.4(C3×Q8), (C2×C24).16D5, (C2×C6).49D20, C1511(C4.Q8), C4⋊Dic5.2C6, C30.48(C4⋊C4), C4.6(C6×Dic5), C20.55(C2×C12), C60.240(C2×C4), (C2×C120).23C2, (C2×C30).109D4, C4.4(C3×Dic10), C10.2(C3×SD16), C22.8(C3×D20), (C2×C12).422D10, C6.10(C40⋊C2), C6.12(C4⋊Dic5), C12.45(C2×Dic5), (C2×C60).503C22, C53(C3×C4.Q8), (C2×C8).6(C3×D5), C10.12(C3×C4⋊C4), (C2×C4).69(C6×D5), C2.2(C3×C40⋊C2), C2.3(C3×C4⋊Dic5), (C2×C20).86(C2×C6), (C2×C10).13(C3×D4), (C3×C4⋊Dic5).14C2, SmallGroup(480,95)

Series: Derived Chief Lower central Upper central

C1C20 — C3×C406C4
C1C5C10C20C2×C20C2×C60C3×C4⋊Dic5 — C3×C406C4
C5C10C20 — C3×C406C4
C1C2×C6C2×C12C2×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 [×2], C3, C4 [×2], C4 [×2], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C10, C10 [×2], 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, C30 [×2], C4.Q8, 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×C4.Q8, C120 [×2], C6×Dic5 [×2], C2×C60, C406C4, C3×C4⋊Dic5 [×2], C2×C120, C3×C406C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4, Q8, D5, C12 [×2], C2×C6, C4⋊C4, SD16 [×2], Dic5 [×2], D10, C2×C12, C3×D4, C3×Q8, C3×D5, C4.Q8, Dic10, D20, C2×Dic5, C3×C4⋊C4, C3×SD16 [×2], C3×Dic5 [×2], C6×D5, C40⋊C2 [×2], C4⋊Dic5, C3×C4.Q8, C3×Dic10, C3×D20, C6×Dic5, C406C4, C3×C40⋊C2 [×2], C3×C4⋊Dic5, C3×C406C4

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

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

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

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

138 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F5A5B6A···6F8A8B8C8D10A···10F12A12B12C12D12E···12L15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order122233444444556···6888810···101212121212···121515151520···2024···2430···3040···4060···60120···120
size1111112220202020221···122222···2222220···2022222···22···22···22···22···22···2

138 irreducible representations

dim11111111222222222222222222
type+++-++-+-+
imageC1C2C2C3C4C6C6C12Q8D4D5SD16Dic5D10C3×Q8C3×D4C3×D5Dic10D20C3×SD16C3×Dic5C6×D5C40⋊C2C3×Dic10C3×D20C3×C40⋊C2
kernelC3×C406C4C3×C4⋊Dic5C2×C120C406C4C120C4⋊Dic5C2×C40C40C60C2×C30C2×C24C30C24C2×C12C20C2×C10C2×C8C12C2×C6C10C8C2×C4C6C4C22C2
# reps1212442811244222444884168832

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

1500
010
001
,
100
0173116
0125166
,
17700
042227
0212199
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

׿
×
𝔽