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## G = C2×C6×Dic10order 480 = 25·3·5

### Direct product of C2×C6 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C6×Dic10
 Chief series C1 — C5 — C10 — C30 — C3×Dic5 — C6×Dic5 — C2×C6×Dic5 — C2×C6×Dic10
 Lower central C5 — C10 — C2×C6×Dic10
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C2×C6×Dic10
G = < a,b,c,d | a2=b6=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 720 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, C12, C12, C2×C6, C15, C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C12, C2×C12, C3×Q8, C22×C6, C30, C30, C22×Q8, Dic10, C2×Dic5, C2×C20, C22×C10, C22×C12, C22×C12, C6×Q8, C3×Dic5, C60, C2×C30, C2×Dic10, C22×Dic5, C22×C20, Q8×C2×C6, C3×Dic10, C6×Dic5, C2×C60, C22×C30, C22×Dic10, C6×Dic10, C2×C6×Dic5, C22×C60, C2×C6×Dic10
Quotients: C1, C2, C3, C22, C6, Q8, C23, D5, C2×C6, C2×Q8, C24, D10, C3×Q8, C22×C6, C3×D5, C22×Q8, Dic10, C22×D5, C6×Q8, C23×C6, C6×D5, C2×Dic10, C23×D5, Q8×C2×C6, C3×Dic10, D5×C2×C6, C22×Dic10, C6×Dic10, D5×C22×C6, C2×C6×Dic10

Smallest permutation representation of C2×C6×Dic10
Regular action on 480 points
Generators in S480
(1 379)(2 380)(3 361)(4 362)(5 363)(6 364)(7 365)(8 366)(9 367)(10 368)(11 369)(12 370)(13 371)(14 372)(15 373)(16 374)(17 375)(18 376)(19 377)(20 378)(21 463)(22 464)(23 465)(24 466)(25 467)(26 468)(27 469)(28 470)(29 471)(30 472)(31 473)(32 474)(33 475)(34 476)(35 477)(36 478)(37 479)(38 480)(39 461)(40 462)(41 418)(42 419)(43 420)(44 401)(45 402)(46 403)(47 404)(48 405)(49 406)(50 407)(51 408)(52 409)(53 410)(54 411)(55 412)(56 413)(57 414)(58 415)(59 416)(60 417)(61 295)(62 296)(63 297)(64 298)(65 299)(66 300)(67 281)(68 282)(69 283)(70 284)(71 285)(72 286)(73 287)(74 288)(75 289)(76 290)(77 291)(78 292)(79 293)(80 294)(81 144)(82 145)(83 146)(84 147)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 141)(99 142)(100 143)(101 195)(102 196)(103 197)(104 198)(105 199)(106 200)(107 181)(108 182)(109 183)(110 184)(111 185)(112 186)(113 187)(114 188)(115 189)(116 190)(117 191)(118 192)(119 193)(120 194)(121 251)(122 252)(123 253)(124 254)(125 255)(126 256)(127 257)(128 258)(129 259)(130 260)(131 241)(132 242)(133 243)(134 244)(135 245)(136 246)(137 247)(138 248)(139 249)(140 250)(161 316)(162 317)(163 318)(164 319)(165 320)(166 301)(167 302)(168 303)(169 304)(170 305)(171 306)(172 307)(173 308)(174 309)(175 310)(176 311)(177 312)(178 313)(179 314)(180 315)(201 273)(202 274)(203 275)(204 276)(205 277)(206 278)(207 279)(208 280)(209 261)(210 262)(211 263)(212 264)(213 265)(214 266)(215 267)(216 268)(217 269)(218 270)(219 271)(220 272)(221 384)(222 385)(223 386)(224 387)(225 388)(226 389)(227 390)(228 391)(229 392)(230 393)(231 394)(232 395)(233 396)(234 397)(235 398)(236 399)(237 400)(238 381)(239 382)(240 383)(321 341)(322 342)(323 343)(324 344)(325 345)(326 346)(327 347)(328 348)(329 349)(330 350)(331 351)(332 352)(333 353)(334 354)(335 355)(336 356)(337 357)(338 358)(339 359)(340 360)(421 448)(422 449)(423 450)(424 451)(425 452)(426 453)(427 454)(428 455)(429 456)(430 457)(431 458)(432 459)(433 460)(434 441)(435 442)(436 443)(437 444)(438 445)(439 446)(440 447)
(1 218 61 414 156 325)(2 219 62 415 157 326)(3 220 63 416 158 327)(4 201 64 417 159 328)(5 202 65 418 160 329)(6 203 66 419 141 330)(7 204 67 420 142 331)(8 205 68 401 143 332)(9 206 69 402 144 333)(10 207 70 403 145 334)(11 208 71 404 146 335)(12 209 72 405 147 336)(13 210 73 406 148 337)(14 211 74 407 149 338)(15 212 75 408 150 339)(16 213 76 409 151 340)(17 214 77 410 152 321)(18 215 78 411 153 322)(19 216 79 412 154 323)(20 217 80 413 155 324)(21 192 311 384 121 426)(22 193 312 385 122 427)(23 194 313 386 123 428)(24 195 314 387 124 429)(25 196 315 388 125 430)(26 197 316 389 126 431)(27 198 317 390 127 432)(28 199 318 391 128 433)(29 200 319 392 129 434)(30 181 320 393 130 435)(31 182 301 394 131 436)(32 183 302 395 132 437)(33 184 303 396 133 438)(34 185 304 397 134 439)(35 186 305 398 135 440)(36 187 306 399 136 421)(37 188 307 400 137 422)(38 189 308 381 138 423)(39 190 309 382 139 424)(40 191 310 383 140 425)(41 97 349 363 274 299)(42 98 350 364 275 300)(43 99 351 365 276 281)(44 100 352 366 277 282)(45 81 353 367 278 283)(46 82 354 368 279 284)(47 83 355 369 280 285)(48 84 356 370 261 286)(49 85 357 371 262 287)(50 86 358 372 263 288)(51 87 359 373 264 289)(52 88 360 374 265 290)(53 89 341 375 266 291)(54 90 342 376 267 292)(55 91 343 377 268 293)(56 92 344 378 269 294)(57 93 345 379 270 295)(58 94 346 380 271 296)(59 95 347 361 272 297)(60 96 348 362 273 298)(101 179 224 254 456 466)(102 180 225 255 457 467)(103 161 226 256 458 468)(104 162 227 257 459 469)(105 163 228 258 460 470)(106 164 229 259 441 471)(107 165 230 260 442 472)(108 166 231 241 443 473)(109 167 232 242 444 474)(110 168 233 243 445 475)(111 169 234 244 446 476)(112 170 235 245 447 477)(113 171 236 246 448 478)(114 172 237 247 449 479)(115 173 238 248 450 480)(116 174 239 249 451 461)(117 175 240 250 452 462)(118 176 221 251 453 463)(119 177 222 252 454 464)(120 178 223 253 455 465)
(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)
(1 131 11 121)(2 130 12 140)(3 129 13 139)(4 128 14 138)(5 127 15 137)(6 126 16 136)(7 125 17 135)(8 124 18 134)(9 123 19 133)(10 122 20 132)(21 61 31 71)(22 80 32 70)(23 79 33 69)(24 78 34 68)(25 77 35 67)(26 76 36 66)(27 75 37 65)(28 74 38 64)(29 73 39 63)(30 72 40 62)(41 104 51 114)(42 103 52 113)(43 102 53 112)(44 101 54 111)(45 120 55 110)(46 119 56 109)(47 118 57 108)(48 117 58 107)(49 116 59 106)(50 115 60 105)(81 178 91 168)(82 177 92 167)(83 176 93 166)(84 175 94 165)(85 174 95 164)(86 173 96 163)(87 172 97 162)(88 171 98 161)(89 170 99 180)(90 169 100 179)(141 316 151 306)(142 315 152 305)(143 314 153 304)(144 313 154 303)(145 312 155 302)(146 311 156 301)(147 310 157 320)(148 309 158 319)(149 308 159 318)(150 307 160 317)(181 405 191 415)(182 404 192 414)(183 403 193 413)(184 402 194 412)(185 401 195 411)(186 420 196 410)(187 419 197 409)(188 418 198 408)(189 417 199 407)(190 416 200 406)(201 433 211 423)(202 432 212 422)(203 431 213 421)(204 430 214 440)(205 429 215 439)(206 428 216 438)(207 427 217 437)(208 426 218 436)(209 425 219 435)(210 424 220 434)(221 345 231 355)(222 344 232 354)(223 343 233 353)(224 342 234 352)(225 341 235 351)(226 360 236 350)(227 359 237 349)(228 358 238 348)(229 357 239 347)(230 356 240 346)(241 369 251 379)(242 368 252 378)(243 367 253 377)(244 366 254 376)(245 365 255 375)(246 364 256 374)(247 363 257 373)(248 362 258 372)(249 361 259 371)(250 380 260 370)(261 452 271 442)(262 451 272 441)(263 450 273 460)(264 449 274 459)(265 448 275 458)(266 447 276 457)(267 446 277 456)(268 445 278 455)(269 444 279 454)(270 443 280 453)(281 467 291 477)(282 466 292 476)(283 465 293 475)(284 464 294 474)(285 463 295 473)(286 462 296 472)(287 461 297 471)(288 480 298 470)(289 479 299 469)(290 478 300 468)(321 398 331 388)(322 397 332 387)(323 396 333 386)(324 395 334 385)(325 394 335 384)(326 393 336 383)(327 392 337 382)(328 391 338 381)(329 390 339 400)(330 389 340 399)

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

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

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

156 conjugacy classes

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

156 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 C3 C6 C6 C6 Q8 D5 D10 D10 C3×Q8 C3×D5 Dic10 C6×D5 C6×D5 C3×Dic10 kernel C2×C6×Dic10 C6×Dic10 C2×C6×Dic5 C22×C60 C22×Dic10 C2×Dic10 C22×Dic5 C22×C20 C2×C30 C22×C12 C2×C12 C22×C6 C2×C10 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 2 1 2 24 4 2 4 2 12 2 8 4 16 24 4 32

Matrix representation of C2×C6×Dic10 in GL5(𝔽61)

 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 48 0 0 0 0 0 48
,
 1 0 0 0 0 0 0 1 0 0 0 60 17 0 0 0 0 0 2 36 0 0 0 27 29
,
 60 0 0 0 0 0 0 60 0 0 0 60 0 0 0 0 0 0 53 58 0 0 0 42 8

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,48,0,0,0,0,0,48],[1,0,0,0,0,0,0,60,0,0,0,1,17,0,0,0,0,0,2,27,0,0,0,36,29],[60,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,53,42,0,0,0,58,8] >;

C2×C6×Dic10 in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm Dic}_{10}
% in TeX

G:=Group("C2xC6xDic10");
// GroupNames label

G:=SmallGroup(480,1135);
// by ID

G=gap.SmallGroup(480,1135);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,336,1571,192,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^20=1,d^2=c^10,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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