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## G = Dic5×C2×C12order 480 = 25·3·5

### Direct product of C2×C12 and Dic5

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic5×C2×C12
G = < a,b,c,d | a2=b12=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 432 in 216 conjugacy classes, 162 normal (22 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22, C22 [×6], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], C23, C10, C10 [×6], C12 [×4], C12 [×8], C2×C6, C2×C6 [×6], C15, C42 [×4], C22×C4, C22×C4 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C12 [×6], C2×C12 [×12], C22×C6, C30, C30 [×6], C2×C42, C2×Dic5 [×12], C2×C20 [×6], C22×C10, C4×C12 [×4], C22×C12, C22×C12 [×2], C3×Dic5 [×8], C60 [×4], C2×C30, C2×C30 [×6], C4×Dic5 [×4], C22×Dic5 [×2], C22×C20, C2×C4×C12, C6×Dic5 [×12], C2×C60 [×6], C22×C30, C2×C4×Dic5, C12×Dic5 [×4], C2×C6×Dic5 [×2], C22×C60, Dic5×C2×C12
Quotients: C1, C2 [×7], C3, C4 [×12], C22 [×7], C6 [×7], C2×C4 [×18], C23, D5, C12 [×12], C2×C6 [×7], C42 [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C2×C12 [×18], C22×C6, C3×D5, C2×C42, C4×D5 [×4], C2×Dic5 [×6], C22×D5, C4×C12 [×4], C22×C12 [×3], C3×Dic5 [×4], C6×D5 [×3], C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, C2×C4×C12, D5×C12 [×4], C6×Dic5 [×6], D5×C2×C6, C2×C4×Dic5, C12×Dic5 [×4], D5×C2×C12 [×2], C2×C6×Dic5, Dic5×C2×C12

Smallest permutation representation of Dic5×C2×C12
Regular action on 480 points
Generators in S480
(1 194)(2 195)(3 196)(4 197)(5 198)(6 199)(7 200)(8 201)(9 202)(10 203)(11 204)(12 193)(13 229)(14 230)(15 231)(16 232)(17 233)(18 234)(19 235)(20 236)(21 237)(22 238)(23 239)(24 240)(25 164)(26 165)(27 166)(28 167)(29 168)(30 157)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 284)(38 285)(39 286)(40 287)(41 288)(42 277)(43 278)(44 279)(45 280)(46 281)(47 282)(48 283)(49 410)(50 411)(51 412)(52 413)(53 414)(54 415)(55 416)(56 417)(57 418)(58 419)(59 420)(60 409)(61 364)(62 365)(63 366)(64 367)(65 368)(66 369)(67 370)(68 371)(69 372)(70 361)(71 362)(72 363)(73 253)(74 254)(75 255)(76 256)(77 257)(78 258)(79 259)(80 260)(81 261)(82 262)(83 263)(84 264)(85 294)(86 295)(87 296)(88 297)(89 298)(90 299)(91 300)(92 289)(93 290)(94 291)(95 292)(96 293)(97 447)(98 448)(99 449)(100 450)(101 451)(102 452)(103 453)(104 454)(105 455)(106 456)(107 445)(108 446)(109 378)(110 379)(111 380)(112 381)(113 382)(114 383)(115 384)(116 373)(117 374)(118 375)(119 376)(120 377)(121 399)(122 400)(123 401)(124 402)(125 403)(126 404)(127 405)(128 406)(129 407)(130 408)(131 397)(132 398)(133 424)(134 425)(135 426)(136 427)(137 428)(138 429)(139 430)(140 431)(141 432)(142 421)(143 422)(144 423)(145 477)(146 478)(147 479)(148 480)(149 469)(150 470)(151 471)(152 472)(153 473)(154 474)(155 475)(156 476)(169 192)(170 181)(171 182)(172 183)(173 184)(174 185)(175 186)(176 187)(177 188)(178 189)(179 190)(180 191)(205 433)(206 434)(207 435)(208 436)(209 437)(210 438)(211 439)(212 440)(213 441)(214 442)(215 443)(216 444)(217 302)(218 303)(219 304)(220 305)(221 306)(222 307)(223 308)(224 309)(225 310)(226 311)(227 312)(228 301)(241 385)(242 386)(243 387)(244 388)(245 389)(246 390)(247 391)(248 392)(249 393)(250 394)(251 395)(252 396)(265 352)(266 353)(267 354)(268 355)(269 356)(270 357)(271 358)(272 359)(273 360)(274 349)(275 350)(276 351)(313 343)(314 344)(315 345)(316 346)(317 347)(318 348)(319 337)(320 338)(321 339)(322 340)(323 341)(324 342)(325 460)(326 461)(327 462)(328 463)(329 464)(330 465)(331 466)(332 467)(333 468)(334 457)(335 458)(336 459)
(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)
(1 320 224 330 350 477 425 177 398 77)(2 321 225 331 351 478 426 178 399 78)(3 322 226 332 352 479 427 179 400 79)(4 323 227 333 353 480 428 180 401 80)(5 324 228 334 354 469 429 169 402 81)(6 313 217 335 355 470 430 170 403 82)(7 314 218 336 356 471 431 171 404 83)(8 315 219 325 357 472 432 172 405 84)(9 316 220 326 358 473 421 173 406 73)(10 317 221 327 359 474 422 174 407 74)(11 318 222 328 360 475 423 175 408 75)(12 319 223 329 349 476 424 176 397 76)(13 112 420 453 390 85 27 433 42 370)(14 113 409 454 391 86 28 434 43 371)(15 114 410 455 392 87 29 435 44 372)(16 115 411 456 393 88 30 436 45 361)(17 116 412 445 394 89 31 437 46 362)(18 117 413 446 395 90 32 438 47 363)(19 118 414 447 396 91 33 439 48 364)(20 119 415 448 385 92 34 440 37 365)(21 120 416 449 386 93 35 441 38 366)(22 109 417 450 387 94 36 442 39 367)(23 110 418 451 388 95 25 443 40 368)(24 111 419 452 389 96 26 444 41 369)(49 105 248 296 168 207 279 69 231 383)(50 106 249 297 157 208 280 70 232 384)(51 107 250 298 158 209 281 71 233 373)(52 108 251 299 159 210 282 72 234 374)(53 97 252 300 160 211 283 61 235 375)(54 98 241 289 161 212 284 62 236 376)(55 99 242 290 162 213 285 63 237 377)(56 100 243 291 163 214 286 64 238 378)(57 101 244 292 164 215 287 65 239 379)(58 102 245 293 165 216 288 66 240 380)(59 103 246 294 166 205 277 67 229 381)(60 104 247 295 167 206 278 68 230 382)(121 258 195 339 310 466 276 146 135 189)(122 259 196 340 311 467 265 147 136 190)(123 260 197 341 312 468 266 148 137 191)(124 261 198 342 301 457 267 149 138 192)(125 262 199 343 302 458 268 150 139 181)(126 263 200 344 303 459 269 151 140 182)(127 264 201 345 304 460 270 152 141 183)(128 253 202 346 305 461 271 153 142 184)(129 254 203 347 306 462 272 154 143 185)(130 255 204 348 307 463 273 155 144 186)(131 256 193 337 308 464 274 156 133 187)(132 257 194 338 309 465 275 145 134 188)
(1 236 477 289)(2 237 478 290)(3 238 479 291)(4 239 480 292)(5 240 469 293)(6 229 470 294)(7 230 471 295)(8 231 472 296)(9 232 473 297)(10 233 474 298)(11 234 475 299)(12 235 476 300)(13 150 85 199)(14 151 86 200)(15 152 87 201)(16 153 88 202)(17 154 89 203)(18 155 90 204)(19 156 91 193)(20 145 92 194)(21 146 93 195)(22 147 94 196)(23 148 95 197)(24 149 96 198)(25 260 110 266)(26 261 111 267)(27 262 112 268)(28 263 113 269)(29 264 114 270)(30 253 115 271)(31 254 116 272)(32 255 117 273)(33 256 118 274)(34 257 119 275)(35 258 120 276)(36 259 109 265)(37 188 448 309)(38 189 449 310)(39 190 450 311)(40 191 451 312)(41 192 452 301)(42 181 453 302)(43 182 454 303)(44 183 455 304)(45 184 456 305)(46 185 445 306)(47 186 446 307)(48 187 447 308)(49 325 207 405)(50 326 208 406)(51 327 209 407)(52 328 210 408)(53 329 211 397)(54 330 212 398)(55 331 213 399)(56 332 214 400)(57 333 215 401)(58 334 216 402)(59 335 205 403)(60 336 206 404)(61 424 252 319)(62 425 241 320)(63 426 242 321)(64 427 243 322)(65 428 244 323)(66 429 245 324)(67 430 246 313)(68 431 247 314)(69 432 248 315)(70 421 249 316)(71 422 250 317)(72 423 251 318)(73 384 358 157)(74 373 359 158)(75 374 360 159)(76 375 349 160)(77 376 350 161)(78 377 351 162)(79 378 352 163)(80 379 353 164)(81 380 354 165)(82 381 355 166)(83 382 356 167)(84 383 357 168)(97 223 283 176)(98 224 284 177)(99 225 285 178)(100 226 286 179)(101 227 287 180)(102 228 288 169)(103 217 277 170)(104 218 278 171)(105 219 279 172)(106 220 280 173)(107 221 281 174)(108 222 282 175)(121 416 466 441)(122 417 467 442)(123 418 468 443)(124 419 457 444)(125 420 458 433)(126 409 459 434)(127 410 460 435)(128 411 461 436)(129 412 462 437)(130 413 463 438)(131 414 464 439)(132 415 465 440)(133 396 337 364)(134 385 338 365)(135 386 339 366)(136 387 340 367)(137 388 341 368)(138 389 342 369)(139 390 343 370)(140 391 344 371)(141 392 345 372)(142 393 346 361)(143 394 347 362)(144 395 348 363)

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

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

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

192 conjugacy classes

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

192 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D5 Dic5 D10 D10 C3×D5 C4×D5 C3×Dic5 C6×D5 C6×D5 D5×C12 kernel Dic5×C2×C12 C12×Dic5 C2×C6×Dic5 C22×C60 C2×C4×Dic5 C6×Dic5 C2×C60 C4×Dic5 C22×Dic5 C22×C20 C2×Dic5 C2×C20 C22×C12 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 2 16 8 8 4 2 32 16 2 8 4 2 4 16 16 8 4 32

Matrix representation of Dic5×C2×C12 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 48 0 0 0 0 50 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 60 0 0 0 0 1 60 0 0 19 43
,
 60 0 0 0 0 50 0 0 0 0 52 29 0 0 14 9
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[48,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,60,0,0,0,0,1,19,0,0,60,43],[60,0,0,0,0,50,0,0,0,0,52,14,0,0,29,9] >;

Dic5×C2×C12 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_2\times C_{12}
% in TeX

G:=Group("Dic5xC2xC12");
// GroupNames label

G:=SmallGroup(480,715);
// by ID

G=gap.SmallGroup(480,715);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,268,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^10=1,d^2=c^5,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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