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## G = C3×C20⋊C8order 480 = 25·3·5

### Direct product of C3 and C20⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C20⋊C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C6×Dic5 — C6×C5⋊C8 — C3×C20⋊C8
 Lower central C5 — C10 — C3×C20⋊C8
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×C20⋊C8
G = < a,b,c | a3=b20=c8=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 200 in 76 conjugacy classes, 48 normal (36 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C30, C4⋊C8, C5⋊C8, C2×Dic5, C2×C20, C4×C12, C2×C24, C3×Dic5, C3×Dic5, C60, C2×C30, C4×Dic5, C2×C5⋊C8, C3×C4⋊C8, C3×C5⋊C8, C6×Dic5, C2×C60, C20⋊C8, C12×Dic5, C6×C5⋊C8, C3×C20⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, C2×C8, M4(2), F5, C24, C2×C12, C3×D4, C3×Q8, C4⋊C8, C5⋊C8, C2×F5, C3×C4⋊C4, C2×C24, C3×M4(2), C3×F5, C4.F5, C4⋊F5, C2×C5⋊C8, C3×C4⋊C8, C3×C5⋊C8, C6×F5, C20⋊C8, C3×C4.F5, C3×C4⋊F5, C6×C5⋊C8, C3×C20⋊C8

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5 6A ··· 6F 8A ··· 8H 10A 10B 10C 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P 15A 15B 20A 20B 20C 20D 24A ··· 24P 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 6 ··· 6 8 ··· 8 10 10 10 12 12 12 12 12 ··· 12 12 12 12 12 15 15 20 20 20 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 2 5 5 5 5 10 10 4 1 ··· 1 10 ··· 10 4 4 4 2 2 2 2 5 ··· 5 10 10 10 10 4 4 4 4 4 4 10 ··· 10 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + - + - + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 Q8 M4(2) C3×D4 C3×Q8 C3×M4(2) F5 C5⋊C8 C2×F5 C3×F5 C4.F5 C4⋊F5 C3×C5⋊C8 C6×F5 C3×C4.F5 C3×C4⋊F5 kernel C3×C20⋊C8 C12×Dic5 C6×C5⋊C8 C20⋊C8 C6×Dic5 C2×C60 C4×Dic5 C2×C5⋊C8 C60 C2×Dic5 C2×C20 C20 C3×Dic5 C3×Dic5 C30 Dic5 Dic5 C10 C2×C12 C12 C2×C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 1 2 2 2 2 2 4 8 4 4 16 1 1 2 2 2 4 1 2 1 2 2 2 4 2 4 4

Matrix representation of C3×C20⋊C8 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 0 1 0 0 0 0 240 0 0 0 0 0 0 0 224 17 207 0 0 0 17 0 224 207 0 0 17 34 207 224 0 0 224 34 0 207
,
 205 93 0 0 0 0 93 36 0 0 0 0 0 0 27 163 186 201 0 0 213 123 91 228 0 0 118 150 13 173 0 0 40 95 214 78

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,224,17,17,224,0,0,17,0,34,34,0,0,207,224,207,0,0,0,0,207,224,207],[205,93,0,0,0,0,93,36,0,0,0,0,0,0,27,213,118,40,0,0,163,123,150,95,0,0,186,91,13,214,0,0,201,228,173,78] >;

C3×C20⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC20:C8");
// GroupNames label

G:=SmallGroup(480,281);
// by ID

G=gap.SmallGroup(480,281);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,136,9414,1595]);
// Polycyclic

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

׿
×
𝔽