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## G = Dic3×C40order 480 = 25·3·5

### Direct product of C40 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C40
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — Dic3×C20 — Dic3×C40
 Lower central C3 — Dic3×C40
 Upper central C1 — C2×C40

Generators and relations for Dic3×C40
G = < a,b,c | a40=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 132 in 88 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, C2×C8, C20, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C4×C8, C40, C40, C2×C20, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C60, C2×C30, C4×C20, C2×C40, C2×C40, C8×Dic3, C5×C3⋊C8, C120, C10×Dic3, C2×C60, C4×C40, C10×C3⋊C8, Dic3×C20, C2×C120, Dic3×C40
Quotients: C1, C2, C4, C22, C5, S3, C8, C2×C4, C10, Dic3, D6, C42, C2×C8, C20, C2×C10, C4×S3, C2×Dic3, C5×S3, C4×C8, C40, C2×C20, S3×C8, C4×Dic3, C5×Dic3, S3×C10, C4×C20, C2×C40, C8×Dic3, S3×C20, C10×Dic3, C4×C40, S3×C40, Dic3×C20, Dic3×C40

Smallest permutation representation of Dic3×C40
Regular action on 480 points
Generators in S480
(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 96 233 283 357 433)(2 97 234 284 358 434)(3 98 235 285 359 435)(4 99 236 286 360 436)(5 100 237 287 321 437)(6 101 238 288 322 438)(7 102 239 289 323 439)(8 103 240 290 324 440)(9 104 201 291 325 401)(10 105 202 292 326 402)(11 106 203 293 327 403)(12 107 204 294 328 404)(13 108 205 295 329 405)(14 109 206 296 330 406)(15 110 207 297 331 407)(16 111 208 298 332 408)(17 112 209 299 333 409)(18 113 210 300 334 410)(19 114 211 301 335 411)(20 115 212 302 336 412)(21 116 213 303 337 413)(22 117 214 304 338 414)(23 118 215 305 339 415)(24 119 216 306 340 416)(25 120 217 307 341 417)(26 81 218 308 342 418)(27 82 219 309 343 419)(28 83 220 310 344 420)(29 84 221 311 345 421)(30 85 222 312 346 422)(31 86 223 313 347 423)(32 87 224 314 348 424)(33 88 225 315 349 425)(34 89 226 316 350 426)(35 90 227 317 351 427)(36 91 228 318 352 428)(37 92 229 319 353 429)(38 93 230 320 354 430)(39 94 231 281 355 431)(40 95 232 282 356 432)(41 159 272 364 169 465)(42 160 273 365 170 466)(43 121 274 366 171 467)(44 122 275 367 172 468)(45 123 276 368 173 469)(46 124 277 369 174 470)(47 125 278 370 175 471)(48 126 279 371 176 472)(49 127 280 372 177 473)(50 128 241 373 178 474)(51 129 242 374 179 475)(52 130 243 375 180 476)(53 131 244 376 181 477)(54 132 245 377 182 478)(55 133 246 378 183 479)(56 134 247 379 184 480)(57 135 248 380 185 441)(58 136 249 381 186 442)(59 137 250 382 187 443)(60 138 251 383 188 444)(61 139 252 384 189 445)(62 140 253 385 190 446)(63 141 254 386 191 447)(64 142 255 387 192 448)(65 143 256 388 193 449)(66 144 257 389 194 450)(67 145 258 390 195 451)(68 146 259 391 196 452)(69 147 260 392 197 453)(70 148 261 393 198 454)(71 149 262 394 199 455)(72 150 263 395 200 456)(73 151 264 396 161 457)(74 152 265 397 162 458)(75 153 266 398 163 459)(76 154 267 399 164 460)(77 155 268 400 165 461)(78 156 269 361 166 462)(79 157 270 362 167 463)(80 158 271 363 168 464)
(1 254 283 447)(2 255 284 448)(3 256 285 449)(4 257 286 450)(5 258 287 451)(6 259 288 452)(7 260 289 453)(8 261 290 454)(9 262 291 455)(10 263 292 456)(11 264 293 457)(12 265 294 458)(13 266 295 459)(14 267 296 460)(15 268 297 461)(16 269 298 462)(17 270 299 463)(18 271 300 464)(19 272 301 465)(20 273 302 466)(21 274 303 467)(22 275 304 468)(23 276 305 469)(24 277 306 470)(25 278 307 471)(26 279 308 472)(27 280 309 473)(28 241 310 474)(29 242 311 475)(30 243 312 476)(31 244 313 477)(32 245 314 478)(33 246 315 479)(34 247 316 480)(35 248 317 441)(36 249 318 442)(37 250 319 443)(38 251 320 444)(39 252 281 445)(40 253 282 446)(41 411 364 211)(42 412 365 212)(43 413 366 213)(44 414 367 214)(45 415 368 215)(46 416 369 216)(47 417 370 217)(48 418 371 218)(49 419 372 219)(50 420 373 220)(51 421 374 221)(52 422 375 222)(53 423 376 223)(54 424 377 224)(55 425 378 225)(56 426 379 226)(57 427 380 227)(58 428 381 228)(59 429 382 229)(60 430 383 230)(61 431 384 231)(62 432 385 232)(63 433 386 233)(64 434 387 234)(65 435 388 235)(66 436 389 236)(67 437 390 237)(68 438 391 238)(69 439 392 239)(70 440 393 240)(71 401 394 201)(72 402 395 202)(73 403 396 203)(74 404 397 204)(75 405 398 205)(76 406 399 206)(77 407 400 207)(78 408 361 208)(79 409 362 209)(80 410 363 210)(81 126 342 176)(82 127 343 177)(83 128 344 178)(84 129 345 179)(85 130 346 180)(86 131 347 181)(87 132 348 182)(88 133 349 183)(89 134 350 184)(90 135 351 185)(91 136 352 186)(92 137 353 187)(93 138 354 188)(94 139 355 189)(95 140 356 190)(96 141 357 191)(97 142 358 192)(98 143 359 193)(99 144 360 194)(100 145 321 195)(101 146 322 196)(102 147 323 197)(103 148 324 198)(104 149 325 199)(105 150 326 200)(106 151 327 161)(107 152 328 162)(108 153 329 163)(109 154 330 164)(110 155 331 165)(111 156 332 166)(112 157 333 167)(113 158 334 168)(114 159 335 169)(115 160 336 170)(116 121 337 171)(117 122 338 172)(118 123 339 173)(119 124 340 174)(120 125 341 175)

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

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

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

240 conjugacy classes

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

240 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 C5 C8 C10 C10 C10 C20 C20 C20 C40 S3 Dic3 D6 C4×S3 C4×S3 C5×S3 S3×C8 C5×Dic3 S3×C10 S3×C20 S3×C20 S3×C40 kernel Dic3×C40 C10×C3⋊C8 Dic3×C20 C2×C120 C5×C3⋊C8 C120 C10×Dic3 C8×Dic3 C5×Dic3 C2×C3⋊C8 C4×Dic3 C2×C24 C3⋊C8 C24 C2×Dic3 Dic3 C2×C40 C40 C2×C20 C20 C2×C10 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 1 1 1 4 4 4 4 16 4 4 4 16 16 16 64 1 2 1 2 2 4 8 8 4 8 8 32

Matrix representation of Dic3×C40 in GL3(𝔽241) generated by

 64 0 0 0 214 0 0 0 214
,
 240 0 0 0 0 1 0 240 240
,
 64 0 0 0 123 139 0 16 118
G:=sub<GL(3,GF(241))| [64,0,0,0,214,0,0,0,214],[240,0,0,0,0,240,0,1,240],[64,0,0,0,123,16,0,139,118] >;

Dic3×C40 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{40}
% in TeX

G:=Group("Dic3xC40");
// GroupNames label

G:=SmallGroup(480,132);
// by ID

G=gap.SmallGroup(480,132);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,288,102,15686]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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