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## G = C20×C3⋊C8order 480 = 25·3·5

### Direct product of C20 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C20×C3⋊C8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C10×C3⋊C8 — C20×C3⋊C8
 Lower central C3 — C20×C3⋊C8
 Upper central C1 — C4×C20

Generators and relations for C20×C3⋊C8
G = < a,b,c | a20=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 116 in 88 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C2×C6, C15, C42, C2×C8, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C30, C4×C8, C40, C2×C20, C2×C20, C2×C3⋊C8, C4×C12, C60, C2×C30, C4×C20, C2×C40, C4×C3⋊C8, C5×C3⋊C8, C2×C60, C2×C60, C4×C40, C10×C3⋊C8, C4×C60, C20×C3⋊C8
Quotients: C1, C2, C4, C22, C5, S3, C8, C2×C4, C10, Dic3, D6, C42, C2×C8, C20, C2×C10, C3⋊C8, C4×S3, C2×Dic3, C5×S3, C4×C8, C40, C2×C20, C2×C3⋊C8, C4×Dic3, C5×Dic3, S3×C10, C4×C20, C2×C40, C4×C3⋊C8, C5×C3⋊C8, S3×C20, C10×Dic3, C4×C40, C10×C3⋊C8, Dic3×C20, C20×C3⋊C8

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

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

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

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

240 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4L 5A 5B 5C 5D 6A 6B 6C 8A ··· 8P 10A ··· 10L 12A ··· 12L 15A 15B 15C 15D 20A ··· 20AV 30A ··· 30L 40A ··· 40BL 60A ··· 60AV order 1 2 2 2 3 4 ··· 4 5 5 5 5 6 6 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 1 ··· 1 1 1 1 1 2 2 2 3 ··· 3 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

240 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 S3 Dic3 D6 C3⋊C8 C4×S3 C5×S3 C5×Dic3 S3×C10 C5×C3⋊C8 S3×C20 kernel C20×C3⋊C8 C10×C3⋊C8 C4×C60 C5×C3⋊C8 C2×C60 C4×C3⋊C8 C60 C2×C3⋊C8 C4×C12 C3⋊C8 C2×C12 C12 C4×C20 C2×C20 C2×C20 C20 C20 C42 C2×C4 C2×C4 C4 C4 # reps 1 2 1 8 4 4 16 8 4 32 16 64 1 2 1 8 4 4 8 4 32 16

Matrix representation of C20×C3⋊C8 in GL4(𝔽241) generated by

 91 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 240 0 0 1 240
,
 240 0 0 0 0 177 0 0 0 0 60 195 0 0 14 181
G:=sub<GL(4,GF(241))| [91,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,240],[240,0,0,0,0,177,0,0,0,0,60,14,0,0,195,181] >;

C20×C3⋊C8 in GAP, Magma, Sage, TeX

C_{20}\times C_3\rtimes C_8
% in TeX

G:=Group("C20xC3:C8");
// GroupNames label

G:=SmallGroup(480,121);
// by ID

G=gap.SmallGroup(480,121);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,288,136,15686]);
// Polycyclic

G:=Group<a,b,c|a^20=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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