Copied to
clipboard

## G = C3×Dic5⋊3Q8order 480 = 25·3·5

### Direct product of C3 and Dic5⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×Dic5⋊3Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C12×Dic5 — C3×Dic5⋊3Q8
 Lower central C5 — C10 — C3×Dic5⋊3Q8
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×Dic53Q8
G = < a,b,c,d,e | a3=b10=d4=1, c2=b5, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 336 in 140 conjugacy classes, 86 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], C12 [×2], C12 [×9], C2×C6, C15, C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×Q8, Dic5 [×6], Dic5, C20 [×2], C20 [×2], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×4], C3×Q8 [×4], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×C12 [×3], C3×C4⋊C4, C3×C4⋊C4 [×2], C6×Q8, C3×Dic5 [×6], C3×Dic5, C60 [×2], C60 [×2], C2×C30, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4, C2×Dic10, Q8×C12, C3×Dic10 [×4], C6×Dic5 [×2], C6×Dic5 [×2], C2×C60, C2×C60 [×2], Dic53Q8, C12×Dic5, C12×Dic5 [×2], C3×C10.D4 [×2], C15×C4⋊C4, C6×Dic10, C3×Dic53Q8
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], Q8 [×2], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, C2×Q8, C4○D4, D10 [×3], C2×C12 [×6], C3×Q8 [×2], C22×C6, C3×D5, C4×Q8, C4×D5 [×2], C22×D5, C22×C12, C6×Q8, C3×C4○D4, C6×D5 [×3], C2×C4×D5, D42D5, Q8×D5, Q8×C12, D5×C12 [×2], D5×C2×C6, Dic53Q8, D5×C2×C12, C3×D42D5, C3×Q8×D5, C3×Dic53Q8

Smallest permutation representation of C3×Dic53Q8
Regular action on 480 points
Generators in S480
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(121 161 141)(122 162 142)(123 163 143)(124 164 144)(125 165 145)(126 166 146)(127 167 147)(128 168 148)(129 169 149)(130 170 150)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(181 221 201)(182 222 202)(183 223 203)(184 224 204)(185 225 205)(186 226 206)(187 227 207)(188 228 208)(189 229 209)(190 230 210)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)(241 281 261)(242 282 262)(243 283 263)(244 284 264)(245 285 265)(246 286 266)(247 287 267)(248 288 268)(249 289 269)(250 290 270)(251 291 271)(252 292 272)(253 293 273)(254 294 274)(255 295 275)(256 296 276)(257 297 277)(258 298 278)(259 299 279)(260 300 280)(301 341 321)(302 342 322)(303 343 323)(304 344 324)(305 345 325)(306 346 326)(307 347 327)(308 348 328)(309 349 329)(310 350 330)(311 351 331)(312 352 332)(313 353 333)(314 354 334)(315 355 335)(316 356 336)(317 357 337)(318 358 338)(319 359 339)(320 360 340)(361 401 381)(362 402 382)(363 403 383)(364 404 384)(365 405 385)(366 406 386)(367 407 387)(368 408 388)(369 409 389)(370 410 390)(371 411 391)(372 412 392)(373 413 393)(374 414 394)(375 415 395)(376 416 396)(377 417 397)(378 418 398)(379 419 399)(380 420 400)(421 461 441)(422 462 442)(423 463 443)(424 464 444)(425 465 445)(426 466 446)(427 467 447)(428 468 448)(429 469 449)(430 470 450)(431 471 451)(432 472 452)(433 473 453)(434 474 454)(435 475 455)(436 476 456)(437 477 457)(438 478 458)(439 479 459)(440 480 460)
(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)
(1 254 6 259)(2 253 7 258)(3 252 8 257)(4 251 9 256)(5 260 10 255)(11 244 16 249)(12 243 17 248)(13 242 18 247)(14 241 19 246)(15 250 20 245)(21 274 26 279)(22 273 27 278)(23 272 28 277)(24 271 29 276)(25 280 30 275)(31 264 36 269)(32 263 37 268)(33 262 38 267)(34 261 39 266)(35 270 40 265)(41 294 46 299)(42 293 47 298)(43 292 48 297)(44 291 49 296)(45 300 50 295)(51 284 56 289)(52 283 57 288)(53 282 58 287)(54 281 59 286)(55 290 60 285)(61 314 66 319)(62 313 67 318)(63 312 68 317)(64 311 69 316)(65 320 70 315)(71 304 76 309)(72 303 77 308)(73 302 78 307)(74 301 79 306)(75 310 80 305)(81 334 86 339)(82 333 87 338)(83 332 88 337)(84 331 89 336)(85 340 90 335)(91 324 96 329)(92 323 97 328)(93 322 98 327)(94 321 99 326)(95 330 100 325)(101 354 106 359)(102 353 107 358)(103 352 108 357)(104 351 109 356)(105 360 110 355)(111 344 116 349)(112 343 117 348)(113 342 118 347)(114 341 119 346)(115 350 120 345)(121 374 126 379)(122 373 127 378)(123 372 128 377)(124 371 129 376)(125 380 130 375)(131 364 136 369)(132 363 137 368)(133 362 138 367)(134 361 139 366)(135 370 140 365)(141 394 146 399)(142 393 147 398)(143 392 148 397)(144 391 149 396)(145 400 150 395)(151 384 156 389)(152 383 157 388)(153 382 158 387)(154 381 159 386)(155 390 160 385)(161 414 166 419)(162 413 167 418)(163 412 168 417)(164 411 169 416)(165 420 170 415)(171 404 176 409)(172 403 177 408)(173 402 178 407)(174 401 179 406)(175 410 180 405)(181 434 186 439)(182 433 187 438)(183 432 188 437)(184 431 189 436)(185 440 190 435)(191 424 196 429)(192 423 197 428)(193 422 198 427)(194 421 199 426)(195 430 200 425)(201 454 206 459)(202 453 207 458)(203 452 208 457)(204 451 209 456)(205 460 210 455)(211 444 216 449)(212 443 217 448)(213 442 218 447)(214 441 219 446)(215 450 220 445)(221 474 226 479)(222 473 227 478)(223 472 228 477)(224 471 229 476)(225 480 230 475)(231 464 236 469)(232 463 237 468)(233 462 238 467)(234 461 239 466)(235 470 240 465)
(1 309 11 319)(2 308 12 318)(3 307 13 317)(4 306 14 316)(5 305 15 315)(6 304 16 314)(7 303 17 313)(8 302 18 312)(9 301 19 311)(10 310 20 320)(21 329 31 339)(22 328 32 338)(23 327 33 337)(24 326 34 336)(25 325 35 335)(26 324 36 334)(27 323 37 333)(28 322 38 332)(29 321 39 331)(30 330 40 340)(41 349 51 359)(42 348 52 358)(43 347 53 357)(44 346 54 356)(45 345 55 355)(46 344 56 354)(47 343 57 353)(48 342 58 352)(49 341 59 351)(50 350 60 360)(61 254 71 244)(62 253 72 243)(63 252 73 242)(64 251 74 241)(65 260 75 250)(66 259 76 249)(67 258 77 248)(68 257 78 247)(69 256 79 246)(70 255 80 245)(81 274 91 264)(82 273 92 263)(83 272 93 262)(84 271 94 261)(85 280 95 270)(86 279 96 269)(87 278 97 268)(88 277 98 267)(89 276 99 266)(90 275 100 265)(101 294 111 284)(102 293 112 283)(103 292 113 282)(104 291 114 281)(105 300 115 290)(106 299 116 289)(107 298 117 288)(108 297 118 287)(109 296 119 286)(110 295 120 285)(121 439 131 429)(122 438 132 428)(123 437 133 427)(124 436 134 426)(125 435 135 425)(126 434 136 424)(127 433 137 423)(128 432 138 422)(129 431 139 421)(130 440 140 430)(141 459 151 449)(142 458 152 448)(143 457 153 447)(144 456 154 446)(145 455 155 445)(146 454 156 444)(147 453 157 443)(148 452 158 442)(149 451 159 441)(150 460 160 450)(161 479 171 469)(162 478 172 468)(163 477 173 467)(164 476 174 466)(165 475 175 465)(166 474 176 464)(167 473 177 463)(168 472 178 462)(169 471 179 461)(170 480 180 470)(181 364 191 374)(182 363 192 373)(183 362 193 372)(184 361 194 371)(185 370 195 380)(186 369 196 379)(187 368 197 378)(188 367 198 377)(189 366 199 376)(190 365 200 375)(201 384 211 394)(202 383 212 393)(203 382 213 392)(204 381 214 391)(205 390 215 400)(206 389 216 399)(207 388 217 398)(208 387 218 397)(209 386 219 396)(210 385 220 395)(221 404 231 414)(222 403 232 413)(223 402 233 412)(224 401 234 411)(225 410 235 420)(226 409 236 419)(227 408 237 418)(228 407 238 417)(229 406 239 416)(230 405 240 415)
(1 131 11 121)(2 132 12 122)(3 133 13 123)(4 134 14 124)(5 135 15 125)(6 136 16 126)(7 137 17 127)(8 138 18 128)(9 139 19 129)(10 140 20 130)(21 151 31 141)(22 152 32 142)(23 153 33 143)(24 154 34 144)(25 155 35 145)(26 156 36 146)(27 157 37 147)(28 158 38 148)(29 159 39 149)(30 160 40 150)(41 171 51 161)(42 172 52 162)(43 173 53 163)(44 174 54 164)(45 175 55 165)(46 176 56 166)(47 177 57 167)(48 178 58 168)(49 179 59 169)(50 180 60 170)(61 191 71 181)(62 192 72 182)(63 193 73 183)(64 194 74 184)(65 195 75 185)(66 196 76 186)(67 197 77 187)(68 198 78 188)(69 199 79 189)(70 200 80 190)(81 211 91 201)(82 212 92 202)(83 213 93 203)(84 214 94 204)(85 215 95 205)(86 216 96 206)(87 217 97 207)(88 218 98 208)(89 219 99 209)(90 220 100 210)(101 231 111 221)(102 232 112 222)(103 233 113 223)(104 234 114 224)(105 235 115 225)(106 236 116 226)(107 237 117 227)(108 238 118 228)(109 239 119 229)(110 240 120 230)(241 371 251 361)(242 372 252 362)(243 373 253 363)(244 374 254 364)(245 375 255 365)(246 376 256 366)(247 377 257 367)(248 378 258 368)(249 379 259 369)(250 380 260 370)(261 391 271 381)(262 392 272 382)(263 393 273 383)(264 394 274 384)(265 395 275 385)(266 396 276 386)(267 397 277 387)(268 398 278 388)(269 399 279 389)(270 400 280 390)(281 411 291 401)(282 412 292 402)(283 413 293 403)(284 414 294 404)(285 415 295 405)(286 416 296 406)(287 417 297 407)(288 418 298 408)(289 419 299 409)(290 420 300 410)(301 431 311 421)(302 432 312 422)(303 433 313 423)(304 434 314 424)(305 435 315 425)(306 436 316 426)(307 437 317 427)(308 438 318 428)(309 439 319 429)(310 440 320 430)(321 451 331 441)(322 452 332 442)(323 453 333 443)(324 454 334 444)(325 455 335 445)(326 456 336 446)(327 457 337 447)(328 458 338 448)(329 459 339 449)(330 460 340 450)(341 471 351 461)(342 472 352 462)(343 473 353 463)(344 474 354 464)(345 475 355 465)(346 476 356 466)(347 477 357 467)(348 478 358 468)(349 479 359 469)(350 480 360 470)

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

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

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

120 conjugacy classes

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

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - - image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 Q8 D5 C4○D4 D10 C3×Q8 C3×D5 C4×D5 C3×C4○D4 C6×D5 D5×C12 D4⋊2D5 Q8×D5 C3×D4⋊2D5 C3×Q8×D5 kernel C3×Dic5⋊3Q8 C12×Dic5 C3×C10.D4 C15×C4⋊C4 C6×Dic10 Dic5⋊3Q8 C3×Dic10 C4×Dic5 C10.D4 C5×C4⋊C4 C2×Dic10 Dic10 C3×Dic5 C3×C4⋊C4 C30 C2×C12 Dic5 C4⋊C4 C12 C10 C2×C4 C4 C6 C6 C2 C2 # reps 1 3 2 1 1 2 8 6 4 2 2 16 2 2 2 6 4 4 8 4 12 16 2 2 4 4

Matrix representation of C3×Dic53Q8 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 18 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 23 55 0 0 7 38 0 0 0 0 60 0 0 0 0 60
,
 9 56 0 0 16 52 0 0 0 0 60 2 0 0 60 1
,
 1 0 0 0 0 1 0 0 0 0 9 1 0 0 40 52
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[18,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[23,7,0,0,55,38,0,0,0,0,60,0,0,0,0,60],[9,16,0,0,56,52,0,0,0,0,60,60,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,9,40,0,0,1,52] >;

C3×Dic53Q8 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5\rtimes_3Q_8
% in TeX

G:=Group("C3xDic5:3Q8");
// GroupNames label

G:=SmallGroup(480,680);
// by ID

G=gap.SmallGroup(480,680);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,176,303,394,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=d^4=1,c^2=b^5,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽