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G = C3⋊Dic40order 480 = 25·3·5

The semidirect product of C3 and Dic40 acting via Dic40/Dic20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C152Q32, C6.9D40, C32Dic40, C60.27D4, C12.4D20, C30.11D8, C40.43D6, C24.11D10, C120.5C22, Dic60.2C2, Dic20.1S3, C3⋊C16.D5, C8.10(S3×D5), C51(C3⋊Q32), C10.4(D4⋊S3), C2.7(C3⋊D40), C4.4(C3⋊D20), C20.52(C3⋊D4), (C3×Dic20).1C2, (C5×C3⋊C16).1C2, SmallGroup(480,23)

Series: Derived Chief Lower central Upper central

C1C120 — C3⋊Dic40
C1C5C15C30C60C120C3×Dic20 — C3⋊Dic40
C15C30C60C120 — C3⋊Dic40
C1C2C4C8

Generators and relations for C3⋊Dic40
 G = < a,b,c | a3=b80=1, c2=b40, bab-1=a-1, ac=ca, cbc-1=b-1 >

20C4
60C4
10Q8
30Q8
20Dic3
20C12
4Dic5
12Dic5
3C16
5Q16
15Q16
10C3×Q8
10Dic6
2Dic10
6Dic10
4Dic15
4C3×Dic5
15Q32
5C3×Q16
5Dic12
3Dic20
3C80
2Dic30
2C3×Dic10
5C3⋊Q32
3Dic40

Smallest permutation representation of C3⋊Dic40
Regular action on 480 points
Generators in S480
(1 200 272)(2 273 201)(3 202 274)(4 275 203)(5 204 276)(6 277 205)(7 206 278)(8 279 207)(9 208 280)(10 281 209)(11 210 282)(12 283 211)(13 212 284)(14 285 213)(15 214 286)(16 287 215)(17 216 288)(18 289 217)(19 218 290)(20 291 219)(21 220 292)(22 293 221)(23 222 294)(24 295 223)(25 224 296)(26 297 225)(27 226 298)(28 299 227)(29 228 300)(30 301 229)(31 230 302)(32 303 231)(33 232 304)(34 305 233)(35 234 306)(36 307 235)(37 236 308)(38 309 237)(39 238 310)(40 311 239)(41 240 312)(42 313 161)(43 162 314)(44 315 163)(45 164 316)(46 317 165)(47 166 318)(48 319 167)(49 168 320)(50 241 169)(51 170 242)(52 243 171)(53 172 244)(54 245 173)(55 174 246)(56 247 175)(57 176 248)(58 249 177)(59 178 250)(60 251 179)(61 180 252)(62 253 181)(63 182 254)(64 255 183)(65 184 256)(66 257 185)(67 186 258)(68 259 187)(69 188 260)(70 261 189)(71 190 262)(72 263 191)(73 192 264)(74 265 193)(75 194 266)(76 267 195)(77 196 268)(78 269 197)(79 198 270)(80 271 199)(81 342 445)(82 446 343)(83 344 447)(84 448 345)(85 346 449)(86 450 347)(87 348 451)(88 452 349)(89 350 453)(90 454 351)(91 352 455)(92 456 353)(93 354 457)(94 458 355)(95 356 459)(96 460 357)(97 358 461)(98 462 359)(99 360 463)(100 464 361)(101 362 465)(102 466 363)(103 364 467)(104 468 365)(105 366 469)(106 470 367)(107 368 471)(108 472 369)(109 370 473)(110 474 371)(111 372 475)(112 476 373)(113 374 477)(114 478 375)(115 376 479)(116 480 377)(117 378 401)(118 402 379)(119 380 403)(120 404 381)(121 382 405)(122 406 383)(123 384 407)(124 408 385)(125 386 409)(126 410 387)(127 388 411)(128 412 389)(129 390 413)(130 414 391)(131 392 415)(132 416 393)(133 394 417)(134 418 395)(135 396 419)(136 420 397)(137 398 421)(138 422 399)(139 400 423)(140 424 321)(141 322 425)(142 426 323)(143 324 427)(144 428 325)(145 326 429)(146 430 327)(147 328 431)(148 432 329)(149 330 433)(150 434 331)(151 332 435)(152 436 333)(153 334 437)(154 438 335)(155 336 439)(156 440 337)(157 338 441)(158 442 339)(159 340 443)(160 444 341)
(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)
(1 368 41 328)(2 367 42 327)(3 366 43 326)(4 365 44 325)(5 364 45 324)(6 363 46 323)(7 362 47 322)(8 361 48 321)(9 360 49 400)(10 359 50 399)(11 358 51 398)(12 357 52 397)(13 356 53 396)(14 355 54 395)(15 354 55 394)(16 353 56 393)(17 352 57 392)(18 351 58 391)(19 350 59 390)(20 349 60 389)(21 348 61 388)(22 347 62 387)(23 346 63 386)(24 345 64 385)(25 344 65 384)(26 343 66 383)(27 342 67 382)(28 341 68 381)(29 340 69 380)(30 339 70 379)(31 338 71 378)(32 337 72 377)(33 336 73 376)(34 335 74 375)(35 334 75 374)(36 333 76 373)(37 332 77 372)(38 331 78 371)(39 330 79 370)(40 329 80 369)(81 258 121 298)(82 257 122 297)(83 256 123 296)(84 255 124 295)(85 254 125 294)(86 253 126 293)(87 252 127 292)(88 251 128 291)(89 250 129 290)(90 249 130 289)(91 248 131 288)(92 247 132 287)(93 246 133 286)(94 245 134 285)(95 244 135 284)(96 243 136 283)(97 242 137 282)(98 241 138 281)(99 320 139 280)(100 319 140 279)(101 318 141 278)(102 317 142 277)(103 316 143 276)(104 315 144 275)(105 314 145 274)(106 313 146 273)(107 312 147 272)(108 311 148 271)(109 310 149 270)(110 309 150 269)(111 308 151 268)(112 307 152 267)(113 306 153 266)(114 305 154 265)(115 304 155 264)(116 303 156 263)(117 302 157 262)(118 301 158 261)(119 300 159 260)(120 299 160 259)(161 430 201 470)(162 429 202 469)(163 428 203 468)(164 427 204 467)(165 426 205 466)(166 425 206 465)(167 424 207 464)(168 423 208 463)(169 422 209 462)(170 421 210 461)(171 420 211 460)(172 419 212 459)(173 418 213 458)(174 417 214 457)(175 416 215 456)(176 415 216 455)(177 414 217 454)(178 413 218 453)(179 412 219 452)(180 411 220 451)(181 410 221 450)(182 409 222 449)(183 408 223 448)(184 407 224 447)(185 406 225 446)(186 405 226 445)(187 404 227 444)(188 403 228 443)(189 402 229 442)(190 401 230 441)(191 480 231 440)(192 479 232 439)(193 478 233 438)(194 477 234 437)(195 476 235 436)(196 475 236 435)(197 474 237 434)(198 473 238 433)(199 472 239 432)(200 471 240 431)

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

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

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

66 conjugacy classes

class 1  2  3 4A4B4C5A5B 6 8A8B10A10B12A12B12C15A15B16A16B16C16D20A20B20C20D24A24B30A30B40A···40H60A60B60C60D80A···80P120A···120H
order123444556881010121212151516161616202020202424303040···406060606080···80120···120
size112240120222222244040446666222244442···244446···64···4

66 irreducible representations

dim111122222222222444444
type++++++++++-++-++-++-
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4Q32D20D40Dic40D4⋊S3S3×D5C3⋊Q32C3⋊D20C3⋊D40C3⋊Dic40
kernelC3⋊Dic40C5×C3⋊C16C3×Dic20Dic60Dic20C60C3⋊C16C40C30C24C20C15C12C6C3C10C8C5C4C2C1
# reps1111112122244816122248

Matrix representation of C3⋊Dic40 in GL4(𝔽241) generated by

240100
240000
0010
0001
,
17117100
1017000
0019298
0014314
,
17114000
1017000
00154205
006387
G:=sub<GL(4,GF(241))| [240,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[171,101,0,0,171,70,0,0,0,0,192,143,0,0,98,14],[171,101,0,0,140,70,0,0,0,0,154,63,0,0,205,87] >;

C3⋊Dic40 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_{40}
% in TeX

G:=Group("C3:Dic40");
// GroupNames label

G:=SmallGroup(480,23);
// by ID

G=gap.SmallGroup(480,23);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,92,254,142,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^3=b^80=1,c^2=b^40,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3⋊Dic40 in TeX

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