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G = C3×C203C8order 480 = 25·3·5

Direct product of C3 and C203C8

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C203C8, C607C8, C203C24, C60.33Q8, C60.185D4, C12.68D20, C12.27Dic10, C30.38M4(2), C1511(C4⋊C8), (C4×C20).4C6, (C4×C12).8D5, C123(C52C8), C20.7(C3×Q8), (C4×C60).10C2, C30.64(C2×C8), (C2×C60).42C4, C20.32(C3×D4), C4.16(C3×D20), C30.44(C4⋊C4), C42.2(C3×D5), (C2×C20).15C12, C10.16(C2×C24), C4.7(C3×Dic10), (C2×C12).440D10, C6.11(C4⋊Dic5), (C2×C12).13Dic5, C6.8(C4.Dic5), C22.9(C6×Dic5), (C2×C60).540C22, C10.10(C3×M4(2)), C4⋊(C3×C52C8), C53(C3×C4⋊C8), C10.8(C3×C4⋊C4), C2.3(C6×C52C8), (C2×C52C8).8C6, C6.12(C2×C52C8), (C2×C4).90(C6×D5), C2.1(C3×C4⋊Dic5), (C6×C52C8).21C2, (C2×C4).3(C3×Dic5), (C2×C30).183(C2×C4), (C2×C10).46(C2×C12), (C2×C20).106(C2×C6), C2.2(C3×C4.Dic5), (C2×C6).40(C2×Dic5), SmallGroup(480,82)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C203C8
C1C5C10C2×C10C2×C20C2×C60C6×C52C8 — C3×C203C8
C5C10 — C3×C203C8
C1C2×C12C4×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 [×3], C3, C4 [×2], C4 [×2], C4, C22, C5, C6 [×3], C8 [×2], C2×C4 [×3], C10 [×3], C12 [×2], C12 [×2], C12, C2×C6, C15, C42, C2×C8 [×2], C20 [×2], C20 [×2], C20, C2×C10, C24 [×2], C2×C12 [×3], C30 [×3], C4⋊C8, C52C8 [×2], C2×C20 [×3], C4×C12, C2×C24 [×2], C60 [×2], C60 [×2], C60, C2×C30, C2×C52C8 [×2], C4×C20, C3×C4⋊C8, C3×C52C8 [×2], C2×C60 [×3], C203C8, C6×C52C8 [×2], C4×C60, C3×C203C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C8 [×2], C2×C4, D4, Q8, D5, C12 [×2], C2×C6, C4⋊C4, C2×C8, M4(2), Dic5 [×2], D10, C24 [×2], C2×C12, C3×D4, C3×Q8, C3×D5, C4⋊C8, C52C8 [×2], Dic10, D20, C2×Dic5, C3×C4⋊C4, C2×C24, C3×M4(2), C3×Dic5 [×2], C6×D5, C2×C52C8, C4.Dic5, C4⋊Dic5, C3×C4⋊C8, C3×C52C8 [×2], C3×Dic10, C3×D20, C6×Dic5, C203C8, C6×C52C8, C3×C4.Dic5, C3×C4⋊Dic5, C3×C203C8

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

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

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

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

156 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H5A5B6A···6F8A···8H10A···10F12A···12H12I···12P15A15B15C15D20A···20X24A···24P30A···30L60A···60AV
order12223344444444556···68···810···1012···1212···121515151520···2024···2430···3060···60
size11111111112222221···110···102···21···12···222222···210···102···22···2

156 irreducible representations

dim111111111122222222222222222222
type++++-+-+-+
imageC1C2C2C3C4C6C6C8C12C24D4Q8D5M4(2)Dic5D10C3×D4C3×Q8C3×D5C52C8Dic10D20C3×M4(2)C3×Dic5C6×D5C4.Dic5C3×C52C8C3×Dic10C3×D20C3×C4.Dic5
kernelC3×C203C8C6×C52C8C4×C60C203C8C2×C60C2×C52C8C4×C20C60C2×C20C20C60C60C4×C12C30C2×C12C2×C12C20C20C42C12C12C12C10C2×C4C2×C4C6C4C4C4C2
# reps121244288161122422248444848168816

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

225000
022500
0010
0001
,
0100
2405100
001192
00123240
,
6022300
15018100
00138157
00203103
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

׿
×
𝔽