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## G = C3×C20.Q8order 480 = 25·3·5

### Direct product of C3 and C20.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C20.Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C6×C5⋊2C8 — C3×C20.Q8
 Lower central C5 — C10 — C20 — C3×C20.Q8
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×C20.Q8
G = < a,b,c,d | a3=b20=c4=1, d2=b5c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b9, dcd-1=b15c-1 >

Subgroups: 192 in 72 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C24, C2×C12, C2×C12, C30, C4.Q8, C52C8, C2×Dic5, C2×C20, C2×C20, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×Dic5, C60, C60, C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C3×C4.Q8, C3×C52C8, C6×Dic5, C2×C60, C2×C60, C20.Q8, C6×C52C8, C3×C4⋊Dic5, C15×C4⋊C4, C3×C20.Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, D5, C12, C2×C6, C4⋊C4, SD16, D10, C2×C12, C3×D4, C3×Q8, C3×D5, C4.Q8, Dic10, C4×D5, C5⋊D4, C3×C4⋊C4, C3×SD16, C6×D5, C10.D4, D4.D5, Q8⋊D5, C3×C4.Q8, C3×Dic10, D5×C12, C3×C5⋊D4, C20.Q8, C3×C10.D4, C3×D4.D5, C3×Q8⋊D5, C3×C20.Q8

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

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

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

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

102 conjugacy classes

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

102 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 4 4 4 4 type + + + + - + + + - - + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 Q8 D4 D5 SD16 D10 C3×Q8 C3×D4 C3×D5 Dic10 C4×D5 C5⋊D4 C3×SD16 C6×D5 C3×Dic10 D5×C12 C3×C5⋊D4 D4.D5 Q8⋊D5 C3×D4.D5 C3×Q8⋊D5 kernel C3×C20.Q8 C6×C5⋊2C8 C3×C4⋊Dic5 C15×C4⋊C4 C20.Q8 C3×C5⋊2C8 C2×C5⋊2C8 C4⋊Dic5 C5×C4⋊C4 C5⋊2C8 C60 C2×C30 C3×C4⋊C4 C30 C2×C12 C20 C2×C10 C4⋊C4 C12 C12 C2×C6 C10 C2×C4 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 4 2 2 2 4 4 4 4 8 4 8 8 8 2 2 4 4

Matrix representation of C3×C20.Q8 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 1 0 0 0 0 1
,
 190 240 0 0 1 0 0 0 0 0 1 192 0 0 123 240
,
 177 0 0 0 0 177 0 0 0 0 216 6 0 0 137 25
,
 125 97 0 0 229 116 0 0 0 0 0 208 0 0 73 38
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[190,1,0,0,240,0,0,0,0,0,1,123,0,0,192,240],[177,0,0,0,0,177,0,0,0,0,216,137,0,0,6,25],[125,229,0,0,97,116,0,0,0,0,0,73,0,0,208,38] >;

C3×C20.Q8 in GAP, Magma, Sage, TeX

C_3\times C_{20}.Q_8
% in TeX

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

G:=SmallGroup(480,86);
// by ID

G=gap.SmallGroup(480,86);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,168,1037,92,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^4=1,d^2=b^5*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^11,d*b*d^-1=b^9,d*c*d^-1=b^15*c^-1>;
// generators/relations

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