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G = C2×C10×Dic6order 480 = 25·3·5

Direct product of C2×C10 and Dic6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C10×Dic6, C30.84C24, C60.273C23, C61(Q8×C10), C307(C2×Q8), (C2×C30)⋊14Q8, C158(C22×Q8), (C2×C20).441D6, C6.1(C23×C10), (C22×C60).24C2, (C22×C12).8C10, (C22×C20).22S3, C10.69(S3×C23), C23.38(S3×C10), (C2×C60).533C22, (C2×C30).441C23, C20.237(C22×S3), C12.34(C22×C10), (C22×C10).152D6, (C5×Dic3).37C23, Dic3.1(C22×C10), (C22×Dic3).6C10, (C22×C30).181C22, (C10×Dic3).234C22, C31(Q8×C2×C10), (C2×C6)⋊4(C5×Q8), C4.34(S3×C2×C10), C2.3(S3×C22×C10), (C2×C4).88(S3×C10), C22.28(S3×C2×C10), (C22×C4).10(C5×S3), (Dic3×C2×C10).14C2, (C2×C12).100(C2×C10), (C2×C6).62(C22×C10), (C22×C6).43(C2×C10), (C2×C10).375(C22×S3), (C2×Dic3).43(C2×C10), SmallGroup(480,1150)

Series: Derived Chief Lower central Upper central

C1C6 — C2×C10×Dic6
C1C3C6C30C5×Dic3C10×Dic3Dic3×C2×C10 — C2×C10×Dic6
C3C6 — C2×C10×Dic6
C1C22×C10C22×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 [×6], C3, C4 [×4], C4 [×8], C22 [×7], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C10, C10 [×6], Dic3 [×8], C12 [×4], C2×C6 [×7], C15, C22×C4, C22×C4 [×2], C2×Q8 [×12], C20 [×4], C20 [×8], C2×C10 [×7], Dic6 [×16], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C30, C30 [×6], C22×Q8, C2×C20 [×6], C2×C20 [×12], C5×Q8 [×16], C22×C10, C2×Dic6 [×12], C22×Dic3 [×2], C22×C12, C5×Dic3 [×8], C60 [×4], C2×C30 [×7], C22×C20, C22×C20 [×2], Q8×C10 [×12], C22×Dic6, C5×Dic6 [×16], C10×Dic3 [×12], C2×C60 [×6], C22×C30, Q8×C2×C10, C10×Dic6 [×12], Dic3×C2×C10 [×2], C22×C60, C2×C10×Dic6
Quotients: C1, C2 [×15], C22 [×35], C5, S3, Q8 [×4], C23 [×15], C10 [×15], D6 [×7], C2×Q8 [×6], C24, C2×C10 [×35], Dic6 [×4], C22×S3 [×7], C5×S3, C22×Q8, C5×Q8 [×4], C22×C10 [×15], C2×Dic6 [×6], S3×C23, S3×C10 [×7], Q8×C10 [×6], C23×C10, C22×Dic6, C5×Dic6 [×4], S3×C2×C10 [×7], Q8×C2×C10, C10×Dic6 [×6], S3×C22×C10, C2×C10×Dic6

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

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

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

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

180 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B5C5D6A···6G10A···10AB12A···12H15A15B15C15D20A···20P20Q···20AV30A···30AB60A···60AF
order12···2344444···455556···610···1012···121515151520···2020···2030···3060···60
size11···1222226···611112···21···12···222222···26···62···22···2

180 irreducible representations

dim111111112222222222
type+++++-++-
imageC1C2C2C2C5C10C10C10S3Q8D6D6Dic6C5×S3C5×Q8S3×C10S3×C10C5×Dic6
kernelC2×C10×Dic6C10×Dic6Dic3×C2×C10C22×C60C22×Dic6C2×Dic6C22×Dic3C22×C12C22×C20C2×C30C2×C20C22×C10C2×C10C22×C4C2×C6C2×C4C23C22
# reps11221448841461841624432

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

10000
060000
006000
00010
00001
,
600000
020000
002000
000580
000058
,
10000
006000
01100
0004623
0003823
,
10000
01100
006000
0005341
000498

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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