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## G = C2×C10×Dic6order 480 = 25·3·5

### Direct product of C2×C10 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C10×Dic6
 Chief series C1 — C3 — C6 — C30 — C5×Dic3 — C10×Dic3 — Dic3×C2×C10 — C2×C10×Dic6
 Lower central C3 — C6 — C2×C10×Dic6
 Upper central C1 — C22×C10 — C22×C20

Generators and relations for C2×C10×Dic6
G = < a,b,c,d | a2=b10=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 516 in 312 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C2×C6, C15, C22×C4, C22×C4, C2×Q8, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C22×C6, C30, C30, C22×Q8, C2×C20, C2×C20, C5×Q8, C22×C10, C2×Dic6, C22×Dic3, C22×C12, C5×Dic3, C60, C2×C30, C22×C20, C22×C20, Q8×C10, C22×Dic6, C5×Dic6, C10×Dic3, C2×C60, C22×C30, Q8×C2×C10, C10×Dic6, Dic3×C2×C10, C22×C60, C2×C10×Dic6
Quotients: C1, C2, C22, C5, S3, Q8, C23, C10, D6, C2×Q8, C24, C2×C10, Dic6, C22×S3, C5×S3, C22×Q8, C5×Q8, C22×C10, C2×Dic6, S3×C23, S3×C10, Q8×C10, C23×C10, C22×Dic6, C5×Dic6, S3×C2×C10, Q8×C2×C10, C10×Dic6, S3×C22×C10, C2×C10×Dic6

Smallest permutation representation of C2×C10×Dic6
Regular action on 480 points
Generators in S480
(1 428)(2 429)(3 430)(4 431)(5 432)(6 421)(7 422)(8 423)(9 424)(10 425)(11 426)(12 427)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 109)(22 110)(23 111)(24 112)(25 271)(26 272)(27 273)(28 274)(29 275)(30 276)(31 265)(32 266)(33 267)(34 268)(35 269)(36 270)(37 466)(38 467)(39 468)(40 457)(41 458)(42 459)(43 460)(44 461)(45 462)(46 463)(47 464)(48 465)(49 287)(50 288)(51 277)(52 278)(53 279)(54 280)(55 281)(56 282)(57 283)(58 284)(59 285)(60 286)(61 83)(62 84)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(71 81)(72 82)(85 375)(86 376)(87 377)(88 378)(89 379)(90 380)(91 381)(92 382)(93 383)(94 384)(95 373)(96 374)(97 478)(98 479)(99 480)(100 469)(101 470)(102 471)(103 472)(104 473)(105 474)(106 475)(107 476)(108 477)(121 188)(122 189)(123 190)(124 191)(125 192)(126 181)(127 182)(128 183)(129 184)(130 185)(131 186)(132 187)(133 316)(134 317)(135 318)(136 319)(137 320)(138 321)(139 322)(140 323)(141 324)(142 313)(143 314)(144 315)(145 331)(146 332)(147 333)(148 334)(149 335)(150 336)(151 325)(152 326)(153 327)(154 328)(155 329)(156 330)(157 310)(158 311)(159 312)(160 301)(161 302)(162 303)(163 304)(164 305)(165 306)(166 307)(167 308)(168 309)(169 291)(170 292)(171 293)(172 294)(173 295)(174 296)(175 297)(176 298)(177 299)(178 300)(179 289)(180 290)(193 260)(194 261)(195 262)(196 263)(197 264)(198 253)(199 254)(200 255)(201 256)(202 257)(203 258)(204 259)(205 339)(206 340)(207 341)(208 342)(209 343)(210 344)(211 345)(212 346)(213 347)(214 348)(215 337)(216 338)(217 392)(218 393)(219 394)(220 395)(221 396)(222 385)(223 386)(224 387)(225 388)(226 389)(227 390)(228 391)(229 406)(230 407)(231 408)(232 397)(233 398)(234 399)(235 400)(236 401)(237 402)(238 403)(239 404)(240 405)(241 442)(242 443)(243 444)(244 433)(245 434)(246 435)(247 436)(248 437)(249 438)(250 439)(251 440)(252 441)(349 445)(350 446)(351 447)(352 448)(353 449)(354 450)(355 451)(356 452)(357 453)(358 454)(359 455)(360 456)(361 414)(362 415)(363 416)(364 417)(365 418)(366 419)(367 420)(368 409)(369 410)(370 411)(371 412)(372 413)
(1 445 195 70 37 287 339 134 176 151)(2 446 196 71 38 288 340 135 177 152)(3 447 197 72 39 277 341 136 178 153)(4 448 198 61 40 278 342 137 179 154)(5 449 199 62 41 279 343 138 180 155)(6 450 200 63 42 280 344 139 169 156)(7 451 201 64 43 281 345 140 170 145)(8 452 202 65 44 282 346 141 171 146)(9 453 203 66 45 283 347 142 172 147)(10 454 204 67 46 284 348 143 173 148)(11 455 193 68 47 285 337 144 174 149)(12 456 194 69 48 286 338 133 175 150)(13 250 98 311 127 409 33 396 231 384)(14 251 99 312 128 410 34 385 232 373)(15 252 100 301 129 411 35 386 233 374)(16 241 101 302 130 412 36 387 234 375)(17 242 102 303 131 413 25 388 235 376)(18 243 103 304 132 414 26 389 236 377)(19 244 104 305 121 415 27 390 237 378)(20 245 105 306 122 416 28 391 238 379)(21 246 106 307 123 417 29 392 239 380)(22 247 107 308 124 418 30 393 240 381)(23 248 108 309 125 419 31 394 229 382)(24 249 97 310 126 420 32 395 230 383)(49 205 317 298 325 428 349 262 80 466)(50 206 318 299 326 429 350 263 81 467)(51 207 319 300 327 430 351 264 82 468)(52 208 320 289 328 431 352 253 83 457)(53 209 321 290 329 432 353 254 84 458)(54 210 322 291 330 421 354 255 73 459)(55 211 323 292 331 422 355 256 74 460)(56 212 324 293 332 423 356 257 75 461)(57 213 313 294 333 424 357 258 76 462)(58 214 314 295 334 425 358 259 77 463)(59 215 315 296 335 426 359 260 78 464)(60 216 316 297 336 427 360 261 79 465)(85 116 442 470 161 185 371 270 224 399)(86 117 443 471 162 186 372 271 225 400)(87 118 444 472 163 187 361 272 226 401)(88 119 433 473 164 188 362 273 227 402)(89 120 434 474 165 189 363 274 228 403)(90 109 435 475 166 190 364 275 217 404)(91 110 436 476 167 191 365 276 218 405)(92 111 437 477 168 192 366 265 219 406)(93 112 438 478 157 181 367 266 220 407)(94 113 439 479 158 182 368 267 221 408)(95 114 440 480 159 183 369 268 222 397)(96 115 441 469 160 184 370 269 223 398)
(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)
(1 226 7 220)(2 225 8 219)(3 224 9 218)(4 223 10 217)(5 222 11 228)(6 221 12 227)(13 79 19 73)(14 78 20 84)(15 77 21 83)(16 76 22 82)(17 75 23 81)(18 74 24 80)(25 332 31 326)(26 331 32 325)(27 330 33 336)(28 329 34 335)(29 328 35 334)(30 327 36 333)(37 444 43 438)(38 443 44 437)(39 442 45 436)(40 441 46 435)(41 440 47 434)(42 439 48 433)(49 103 55 97)(50 102 56 108)(51 101 57 107)(52 100 58 106)(53 99 59 105)(54 98 60 104)(61 115 67 109)(62 114 68 120)(63 113 69 119)(64 112 70 118)(65 111 71 117)(66 110 72 116)(85 203 91 197)(86 202 92 196)(87 201 93 195)(88 200 94 194)(89 199 95 193)(90 198 96 204)(121 322 127 316)(122 321 128 315)(123 320 129 314)(124 319 130 313)(125 318 131 324)(126 317 132 323)(133 188 139 182)(134 187 140 181)(135 186 141 192)(136 185 142 191)(137 184 143 190)(138 183 144 189)(145 266 151 272)(146 265 152 271)(147 276 153 270)(148 275 154 269)(149 274 155 268)(150 273 156 267)(157 339 163 345)(158 338 164 344)(159 337 165 343)(160 348 166 342)(161 347 167 341)(162 346 168 340)(169 368 175 362)(170 367 176 361)(171 366 177 372)(172 365 178 371)(173 364 179 370)(174 363 180 369)(205 304 211 310)(206 303 212 309)(207 302 213 308)(208 301 214 307)(209 312 215 306)(210 311 216 305)(229 350 235 356)(230 349 236 355)(231 360 237 354)(232 359 238 353)(233 358 239 352)(234 357 240 351)(241 462 247 468)(242 461 248 467)(243 460 249 466)(244 459 250 465)(245 458 251 464)(246 457 252 463)(253 374 259 380)(254 373 260 379)(255 384 261 378)(256 383 262 377)(257 382 263 376)(258 381 264 375)(277 470 283 476)(278 469 284 475)(279 480 285 474)(280 479 286 473)(281 478 287 472)(282 477 288 471)(289 411 295 417)(290 410 296 416)(291 409 297 415)(292 420 298 414)(293 419 299 413)(294 418 300 412)(385 426 391 432)(386 425 392 431)(387 424 393 430)(388 423 394 429)(389 422 395 428)(390 421 396 427)(397 455 403 449)(398 454 404 448)(399 453 405 447)(400 452 406 446)(401 451 407 445)(402 450 408 456)

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E ··· 4L 5A 5B 5C 5D 6A ··· 6G 10A ··· 10AB 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AV 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 4 4 4 4 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 2 2 2 2 2 6 ··· 6 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C5 C10 C10 C10 S3 Q8 D6 D6 Dic6 C5×S3 C5×Q8 S3×C10 S3×C10 C5×Dic6 kernel C2×C10×Dic6 C10×Dic6 Dic3×C2×C10 C22×C60 C22×Dic6 C2×Dic6 C22×Dic3 C22×C12 C22×C20 C2×C30 C2×C20 C22×C10 C2×C10 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 2 1 4 48 8 4 1 4 6 1 8 4 16 24 4 32

Matrix representation of C2×C10×Dic6 in GL5(𝔽61)

 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 20 0 0 0 0 0 20 0 0 0 0 0 58 0 0 0 0 0 58
,
 1 0 0 0 0 0 0 60 0 0 0 1 1 0 0 0 0 0 46 23 0 0 0 38 23
,
 1 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 53 41 0 0 0 49 8

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,58,0,0,0,0,0,58],[1,0,0,0,0,0,0,1,0,0,0,60,1,0,0,0,0,0,46,38,0,0,0,23,23],[1,0,0,0,0,0,1,0,0,0,0,1,60,0,0,0,0,0,53,49,0,0,0,41,8] >;

C2×C10×Dic6 in GAP, Magma, Sage, TeX

C_2\times C_{10}\times {\rm Dic}_6
% in TeX

G:=Group("C2xC10xDic6");
// GroupNames label

G:=SmallGroup(480,1150);
// by ID

G=gap.SmallGroup(480,1150);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,560,2467,304,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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