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## G = C3×C20.6Q8order 480 = 25·3·5

### Direct product of C3 and C20.6Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20.6Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C3×C10.D4 — C3×C20.6Q8
 Lower central C5 — C2×C10 — C3×C20.6Q8
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C20.6Q8
G = < a,b,c,d | a3=b20=c4=1, d2=b10c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 288 in 112 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C20, C2×C10, C2×C12, C2×C12, C2×C12, C30, C30, C42.C2, C2×Dic5, C2×C20, C2×C20, C4×C12, C3×C4⋊C4, C3×Dic5, C60, C60, C2×C30, C10.D4, C4⋊Dic5, C4×C20, C3×C42.C2, C6×Dic5, C2×C60, C2×C60, C20.6Q8, C3×C10.D4, C3×C4⋊Dic5, C4×C60, C3×C20.6Q8
Quotients: C1, C2, C3, C22, C6, Q8, C23, D5, C2×C6, C2×Q8, C4○D4, D10, C3×Q8, C22×C6, C3×D5, C42.C2, Dic10, C22×D5, C6×Q8, C3×C4○D4, C6×D5, C2×Dic10, C4○D20, C3×C42.C2, C3×Dic10, D5×C2×C6, C20.6Q8, C6×Dic10, C3×C4○D20, C3×C20.6Q8

Smallest permutation representation of C3×C20.6Q8
Regular action on 480 points
Generators in S480
(1 457 50)(2 458 51)(3 459 52)(4 460 53)(5 441 54)(6 442 55)(7 443 56)(8 444 57)(9 445 58)(10 446 59)(11 447 60)(12 448 41)(13 449 42)(14 450 43)(15 451 44)(16 452 45)(17 453 46)(18 454 47)(19 455 48)(20 456 49)(21 63 470)(22 64 471)(23 65 472)(24 66 473)(25 67 474)(26 68 475)(27 69 476)(28 70 477)(29 71 478)(30 72 479)(31 73 480)(32 74 461)(33 75 462)(34 76 463)(35 77 464)(36 78 465)(37 79 466)(38 80 467)(39 61 468)(40 62 469)(81 213 331)(82 214 332)(83 215 333)(84 216 334)(85 217 335)(86 218 336)(87 219 337)(88 220 338)(89 201 339)(90 202 340)(91 203 321)(92 204 322)(93 205 323)(94 206 324)(95 207 325)(96 208 326)(97 209 327)(98 210 328)(99 211 329)(100 212 330)(101 148 238)(102 149 239)(103 150 240)(104 151 221)(105 152 222)(106 153 223)(107 154 224)(108 155 225)(109 156 226)(110 157 227)(111 158 228)(112 159 229)(113 160 230)(114 141 231)(115 142 232)(116 143 233)(117 144 234)(118 145 235)(119 146 236)(120 147 237)(121 271 245)(122 272 246)(123 273 247)(124 274 248)(125 275 249)(126 276 250)(127 277 251)(128 278 252)(129 279 253)(130 280 254)(131 261 255)(132 262 256)(133 263 257)(134 264 258)(135 265 259)(136 266 260)(137 267 241)(138 268 242)(139 269 243)(140 270 244)(161 317 405)(162 318 406)(163 319 407)(164 320 408)(165 301 409)(166 302 410)(167 303 411)(168 304 412)(169 305 413)(170 306 414)(171 307 415)(172 308 416)(173 309 417)(174 310 418)(175 311 419)(176 312 420)(177 313 401)(178 314 402)(179 315 403)(180 316 404)(181 346 375)(182 347 376)(183 348 377)(184 349 378)(185 350 379)(186 351 380)(187 352 361)(188 353 362)(189 354 363)(190 355 364)(191 356 365)(192 357 366)(193 358 367)(194 359 368)(195 360 369)(196 341 370)(197 342 371)(198 343 372)(199 344 373)(200 345 374)(281 430 381)(282 431 382)(283 432 383)(284 433 384)(285 434 385)(286 435 386)(287 436 387)(288 437 388)(289 438 389)(290 439 390)(291 440 391)(292 421 392)(293 422 393)(294 423 394)(295 424 395)(296 425 396)(297 426 397)(298 427 398)(299 428 399)(300 429 400)
(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)
(1 301 477 112)(2 302 478 113)(3 303 479 114)(4 304 480 115)(5 305 461 116)(6 306 462 117)(7 307 463 118)(8 308 464 119)(9 309 465 120)(10 310 466 101)(11 311 467 102)(12 312 468 103)(13 313 469 104)(14 314 470 105)(15 315 471 106)(16 316 472 107)(17 317 473 108)(18 318 474 109)(19 319 475 110)(20 320 476 111)(21 152 450 402)(22 153 451 403)(23 154 452 404)(24 155 453 405)(25 156 454 406)(26 157 455 407)(27 158 456 408)(28 159 457 409)(29 160 458 410)(30 141 459 411)(31 142 460 412)(32 143 441 413)(33 144 442 414)(34 145 443 415)(35 146 444 416)(36 147 445 417)(37 148 446 418)(38 149 447 419)(39 150 448 420)(40 151 449 401)(41 176 61 240)(42 177 62 221)(43 178 63 222)(44 179 64 223)(45 180 65 224)(46 161 66 225)(47 162 67 226)(48 163 68 227)(49 164 69 228)(50 165 70 229)(51 166 71 230)(52 167 72 231)(53 168 73 232)(54 169 74 233)(55 170 75 234)(56 171 76 235)(57 172 77 236)(58 173 78 237)(59 174 79 238)(60 175 80 239)(81 373 399 251)(82 374 400 252)(83 375 381 253)(84 376 382 254)(85 377 383 255)(86 378 384 256)(87 379 385 257)(88 380 386 258)(89 361 387 259)(90 362 388 260)(91 363 389 241)(92 364 390 242)(93 365 391 243)(94 366 392 244)(95 367 393 245)(96 368 394 246)(97 369 395 247)(98 370 396 248)(99 371 397 249)(100 372 398 250)(121 207 193 293)(122 208 194 294)(123 209 195 295)(124 210 196 296)(125 211 197 297)(126 212 198 298)(127 213 199 299)(128 214 200 300)(129 215 181 281)(130 216 182 282)(131 217 183 283)(132 218 184 284)(133 219 185 285)(134 220 186 286)(135 201 187 287)(136 202 188 288)(137 203 189 289)(138 204 190 290)(139 205 191 291)(140 206 192 292)(261 335 348 432)(262 336 349 433)(263 337 350 434)(264 338 351 435)(265 339 352 436)(266 340 353 437)(267 321 354 438)(268 322 355 439)(269 323 356 440)(270 324 357 421)(271 325 358 422)(272 326 359 423)(273 327 360 424)(274 328 341 425)(275 329 342 426)(276 330 343 427)(277 331 344 428)(278 332 345 429)(279 333 346 430)(280 334 347 431)
(1 284 467 208)(2 283 468 207)(3 282 469 206)(4 281 470 205)(5 300 471 204)(6 299 472 203)(7 298 473 202)(8 297 474 201)(9 296 475 220)(10 295 476 219)(11 294 477 218)(12 293 478 217)(13 292 479 216)(14 291 480 215)(15 290 461 214)(16 289 462 213)(17 288 463 212)(18 287 464 211)(19 286 465 210)(20 285 466 209)(21 323 460 430)(22 322 441 429)(23 321 442 428)(24 340 443 427)(25 339 444 426)(26 338 445 425)(27 337 446 424)(28 336 447 423)(29 335 448 422)(30 334 449 421)(31 333 450 440)(32 332 451 439)(33 331 452 438)(34 330 453 437)(35 329 454 436)(36 328 455 435)(37 327 456 434)(38 326 457 433)(39 325 458 432)(40 324 459 431)(41 393 71 85)(42 392 72 84)(43 391 73 83)(44 390 74 82)(45 389 75 81)(46 388 76 100)(47 387 77 99)(48 386 78 98)(49 385 79 97)(50 384 80 96)(51 383 61 95)(52 382 62 94)(53 381 63 93)(54 400 64 92)(55 399 65 91)(56 398 66 90)(57 397 67 89)(58 396 68 88)(59 395 69 87)(60 394 70 86)(101 133 320 195)(102 132 301 194)(103 131 302 193)(104 130 303 192)(105 129 304 191)(106 128 305 190)(107 127 306 189)(108 126 307 188)(109 125 308 187)(110 124 309 186)(111 123 310 185)(112 122 311 184)(113 121 312 183)(114 140 313 182)(115 139 314 181)(116 138 315 200)(117 137 316 199)(118 136 317 198)(119 135 318 197)(120 134 319 196)(141 270 401 347)(142 269 402 346)(143 268 403 345)(144 267 404 344)(145 266 405 343)(146 265 406 342)(147 264 407 341)(148 263 408 360)(149 262 409 359)(150 261 410 358)(151 280 411 357)(152 279 412 356)(153 278 413 355)(154 277 414 354)(155 276 415 353)(156 275 416 352)(157 274 417 351)(158 273 418 350)(159 272 419 349)(160 271 420 348)(161 372 235 260)(162 371 236 259)(163 370 237 258)(164 369 238 257)(165 368 239 256)(166 367 240 255)(167 366 221 254)(168 365 222 253)(169 364 223 252)(170 363 224 251)(171 362 225 250)(172 361 226 249)(173 380 227 248)(174 379 228 247)(175 378 229 246)(176 377 230 245)(177 376 231 244)(178 375 232 243)(179 374 233 242)(180 373 234 241)

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

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

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

138 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4F 4G 4H 4I 4J 5A 5B 6A ··· 6F 10A ··· 10F 12A ··· 12L 12M ··· 12T 15A 15B 15C 15D 20A ··· 20X 30A ··· 30L 60A ··· 60AV order 1 2 2 2 3 3 4 ··· 4 4 4 4 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 ··· 2 20 20 20 20 2 2 1 ··· 1 2 ··· 2 2 ··· 2 20 ··· 20 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

138 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + - image C1 C2 C2 C2 C3 C6 C6 C6 Q8 D5 C4○D4 D10 C3×Q8 C3×D5 Dic10 C3×C4○D4 C6×D5 C4○D20 C3×Dic10 C3×C4○D20 kernel C3×C20.6Q8 C3×C10.D4 C3×C4⋊Dic5 C4×C60 C20.6Q8 C10.D4 C4⋊Dic5 C4×C20 C60 C4×C12 C30 C2×C12 C20 C42 C12 C10 C2×C4 C6 C4 C2 # reps 1 4 2 1 2 8 4 2 2 2 4 6 4 4 8 8 12 16 16 32

Matrix representation of C3×C20.6Q8 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 1 60 0 0 19 43 0 0 0 0 57 25 0 0 36 34
,
 25 7 0 0 50 36 0 0 0 0 31 17 0 0 44 30
,
 38 20 0 0 4 23 0 0 0 0 34 9 0 0 7 27
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[1,19,0,0,60,43,0,0,0,0,57,36,0,0,25,34],[25,50,0,0,7,36,0,0,0,0,31,44,0,0,17,30],[38,4,0,0,20,23,0,0,0,0,34,7,0,0,9,27] >;

C3×C20.6Q8 in GAP, Magma, Sage, TeX

C_3\times C_{20}._6Q_8
% in TeX

G:=Group("C3xC20.6Q8");
// GroupNames label

G:=SmallGroup(480,663);
// by ID

G=gap.SmallGroup(480,663);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,701,176,590,268,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^4=1,d^2=b^10*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

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