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## G = C3×C20⋊3C8order 480 = 25·3·5

### Direct product of C3 and C20⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C20⋊3C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C6×C5⋊2C8 — C3×C20⋊3C8
 Lower central C5 — C10 — C3×C20⋊3C8
 Upper central C1 — C2×C12 — C4×C12

Generators and relations for C3×C203C8
G = < a,b,c | a3=b20=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 144 in 76 conjugacy classes, 58 normal (46 characteristic)
C1, C2, C3, C4, C4, C4, C22, C5, C6, C8, C2×C4, C10, C12, C12, C12, C2×C6, C15, C42, C2×C8, C20, C20, C20, C2×C10, C24, C2×C12, C30, C4⋊C8, C52C8, C2×C20, C4×C12, C2×C24, C60, C60, C60, C2×C30, C2×C52C8, C4×C20, C3×C4⋊C8, C3×C52C8, C2×C60, C203C8, C6×C52C8, C4×C60, C3×C203C8
Quotients:

Smallest permutation representation of C3×C203C8
Regular action on 480 points
Generators in S480
(1 200 42)(2 181 43)(3 182 44)(4 183 45)(5 184 46)(6 185 47)(7 186 48)(8 187 49)(9 188 50)(10 189 51)(11 190 52)(12 191 53)(13 192 54)(14 193 55)(15 194 56)(16 195 57)(17 196 58)(18 197 59)(19 198 60)(20 199 41)(21 415 105)(22 416 106)(23 417 107)(24 418 108)(25 419 109)(26 420 110)(27 401 111)(28 402 112)(29 403 113)(30 404 114)(31 405 115)(32 406 116)(33 407 117)(34 408 118)(35 409 119)(36 410 120)(37 411 101)(38 412 102)(39 413 103)(40 414 104)(61 165 329)(62 166 330)(63 167 331)(64 168 332)(65 169 333)(66 170 334)(67 171 335)(68 172 336)(69 173 337)(70 174 338)(71 175 339)(72 176 340)(73 177 321)(74 178 322)(75 179 323)(76 180 324)(77 161 325)(78 162 326)(79 163 327)(80 164 328)(81 359 227)(82 360 228)(83 341 229)(84 342 230)(85 343 231)(86 344 232)(87 345 233)(88 346 234)(89 347 235)(90 348 236)(91 349 237)(92 350 238)(93 351 239)(94 352 240)(95 353 221)(96 354 222)(97 355 223)(98 356 224)(99 357 225)(100 358 226)(121 389 430)(122 390 431)(123 391 432)(124 392 433)(125 393 434)(126 394 435)(127 395 436)(128 396 437)(129 397 438)(130 398 439)(131 399 440)(132 400 421)(133 381 422)(134 382 423)(135 383 424)(136 384 425)(137 385 426)(138 386 427)(139 387 428)(140 388 429)(141 248 361)(142 249 362)(143 250 363)(144 251 364)(145 252 365)(146 253 366)(147 254 367)(148 255 368)(149 256 369)(150 257 370)(151 258 371)(152 259 372)(153 260 373)(154 241 374)(155 242 375)(156 243 376)(157 244 377)(158 245 378)(159 246 379)(160 247 380)(201 289 468)(202 290 469)(203 291 470)(204 292 471)(205 293 472)(206 294 473)(207 295 474)(208 296 475)(209 297 476)(210 298 477)(211 299 478)(212 300 479)(213 281 480)(214 282 461)(215 283 462)(216 284 463)(217 285 464)(218 286 465)(219 287 466)(220 288 467)(261 315 453)(262 316 454)(263 317 455)(264 318 456)(265 319 457)(266 320 458)(267 301 459)(268 302 460)(269 303 441)(270 304 442)(271 305 443)(272 306 444)(273 307 445)(274 308 446)(275 309 447)(276 310 448)(277 311 449)(278 312 450)(279 313 451)(280 314 452)
(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 72 151 221 459 393 407 208)(2 71 152 240 460 392 408 207)(3 70 153 239 441 391 409 206)(4 69 154 238 442 390 410 205)(5 68 155 237 443 389 411 204)(6 67 156 236 444 388 412 203)(7 66 157 235 445 387 413 202)(8 65 158 234 446 386 414 201)(9 64 159 233 447 385 415 220)(10 63 160 232 448 384 416 219)(11 62 141 231 449 383 417 218)(12 61 142 230 450 382 418 217)(13 80 143 229 451 381 419 216)(14 79 144 228 452 400 420 215)(15 78 145 227 453 399 401 214)(16 77 146 226 454 398 402 213)(17 76 147 225 455 397 403 212)(18 75 148 224 456 396 404 211)(19 74 149 223 457 395 405 210)(20 73 150 222 458 394 406 209)(21 467 50 332 379 345 309 137)(22 466 51 331 380 344 310 136)(23 465 52 330 361 343 311 135)(24 464 53 329 362 342 312 134)(25 463 54 328 363 341 313 133)(26 462 55 327 364 360 314 132)(27 461 56 326 365 359 315 131)(28 480 57 325 366 358 316 130)(29 479 58 324 367 357 317 129)(30 478 59 323 368 356 318 128)(31 477 60 322 369 355 319 127)(32 476 41 321 370 354 320 126)(33 475 42 340 371 353 301 125)(34 474 43 339 372 352 302 124)(35 473 44 338 373 351 303 123)(36 472 45 337 374 350 304 122)(37 471 46 336 375 349 305 121)(38 470 47 335 376 348 306 140)(39 469 48 334 377 347 307 139)(40 468 49 333 378 346 308 138)(81 261 440 111 282 194 162 252)(82 280 421 110 283 193 163 251)(83 279 422 109 284 192 164 250)(84 278 423 108 285 191 165 249)(85 277 424 107 286 190 166 248)(86 276 425 106 287 189 167 247)(87 275 426 105 288 188 168 246)(88 274 427 104 289 187 169 245)(89 273 428 103 290 186 170 244)(90 272 429 102 291 185 171 243)(91 271 430 101 292 184 172 242)(92 270 431 120 293 183 173 241)(93 269 432 119 294 182 174 260)(94 268 433 118 295 181 175 259)(95 267 434 117 296 200 176 258)(96 266 435 116 297 199 177 257)(97 265 436 115 298 198 178 256)(98 264 437 114 299 197 179 255)(99 263 438 113 300 196 180 254)(100 262 439 112 281 195 161 253)

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

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

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

156 conjugacy classes

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

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C3 C4 C6 C6 C8 C12 C24 D4 Q8 D5 M4(2) Dic5 D10 C3×D4 C3×Q8 C3×D5 C5⋊2C8 Dic10 D20 C3×M4(2) C3×Dic5 C6×D5 C4.Dic5 C3×C5⋊2C8 C3×Dic10 C3×D20 C3×C4.Dic5 kernel C3×C20⋊3C8 C6×C5⋊2C8 C4×C60 C20⋊3C8 C2×C60 C2×C5⋊2C8 C4×C20 C60 C2×C20 C20 C60 C60 C4×C12 C30 C2×C12 C2×C12 C20 C20 C42 C12 C12 C12 C10 C2×C4 C2×C4 C6 C4 C4 C4 C2 # reps 1 2 1 2 4 4 2 8 8 16 1 1 2 2 4 2 2 2 4 8 4 4 4 8 4 8 16 8 8 16

Matrix representation of C3×C203C8 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 240 51 0 0 0 0 1 192 0 0 123 240
,
 60 223 0 0 150 181 0 0 0 0 138 157 0 0 203 103
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,1,0,0,0,0,1],[0,240,0,0,1,51,0,0,0,0,1,123,0,0,192,240],[60,150,0,0,223,181,0,0,0,0,138,203,0,0,157,103] >;

C3×C203C8 in GAP, Magma, Sage, TeX

C_3\times C_{20}\rtimes_3C_8
% in TeX

G:=Group("C3xC20:3C8");
// GroupNames label

G:=SmallGroup(480,82);
// by ID

G=gap.SmallGroup(480,82);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,136,18822]);
// 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^-1>;
// generators/relations

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