metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.57D4, C30.4Q16, Dic6⋊1Dic5, C10.3Dic12, C30.10SD16, C20.37(C4×S3), (C5×Dic6)⋊7C4, (C2×C20).52D6, (C2×C30).20D4, C6.2(Q8⋊D5), C4.2(S3×Dic5), C60.107(C2×C4), C15⋊6(Q8⋊C4), (C2×C10).31D12, C3⋊1(Q8⋊Dic5), (C2×Dic6).1D5, C60⋊5C4.21C2, C6.1(C5⋊Q16), C5⋊4(C2.Dic12), C10.5(C24⋊C2), C10.44(D6⋊C4), (C2×C12).285D10, C4.15(C15⋊D4), C12.81(C5⋊D4), C20.13(C3⋊D4), C2.8(D6⋊Dic5), (C10×Dic6).2C2, C12.17(C2×Dic5), C2.1(C5⋊Dic12), C6.7(C23.D5), C30.56(C22⋊C4), (C2×C60).129C22, C2.2(Dic6⋊D5), C22.14(C5⋊D12), (C6×C5⋊2C8).2C2, (C2×C5⋊2C8).1S3, (C2×C4).135(S3×D5), (C2×C6).26(C5⋊D4), SmallGroup(480,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.Dic12
G = < a,b,c | a10=b24=1, c2=b12, bab-1=a-1, ac=ca, cbc-1=a5b-1 >
Subgroups: 348 in 84 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C24, Dic6, Dic6, C2×Dic3, C2×C12, C30, Q8⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4⋊Dic3, C2×C24, C2×Dic6, C5×Dic3, Dic15, C60, C2×C30, C2×C5⋊2C8, C4⋊Dic5, Q8×C10, C2.Dic12, C3×C5⋊2C8, C5×Dic6, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, Q8⋊Dic5, C6×C5⋊2C8, C60⋊5C4, C10×Dic6, C10.Dic12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, SD16, Q16, Dic5, D10, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×Dic5, C5⋊D4, C24⋊C2, Dic12, D6⋊C4, S3×D5, Q8⋊D5, C5⋊Q16, C23.D5, C2.Dic12, S3×Dic5, C15⋊D4, C5⋊D12, Q8⋊Dic5, Dic6⋊D5, C5⋊Dic12, D6⋊Dic5, C10.Dic12
►Regular action on 480 points
Generators in S480
(1 317 71 283 240 477 445 198 348 170)(2 171 349 199 446 478 217 284 72 318)(3 319 49 285 218 479 447 200 350 172)(4 173 351 201 448 480 219 286 50 320)(5 321 51 287 220 457 449 202 352 174)(6 175 353 203 450 458 221 288 52 322)(7 323 53 265 222 459 451 204 354 176)(8 177 355 205 452 460 223 266 54 324)(9 325 55 267 224 461 453 206 356 178)(10 179 357 207 454 462 225 268 56 326)(11 327 57 269 226 463 455 208 358 180)(12 181 359 209 456 464 227 270 58 328)(13 329 59 271 228 465 433 210 360 182)(14 183 337 211 434 466 229 272 60 330)(15 331 61 273 230 467 435 212 338 184)(16 185 339 213 436 468 231 274 62 332)(17 333 63 275 232 469 437 214 340 186)(18 187 341 215 438 470 233 276 64 334)(19 335 65 277 234 471 439 216 342 188)(20 189 343 193 440 472 235 278 66 336)(21 313 67 279 236 473 441 194 344 190)(22 191 345 195 442 474 237 280 68 314)(23 315 69 281 238 475 443 196 346 192)(24 169 347 197 444 476 239 282 70 316)(25 368 301 90 163 414 109 386 130 260)(26 261 131 387 110 415 164 91 302 369)(27 370 303 92 165 416 111 388 132 262)(28 263 133 389 112 417 166 93 304 371)(29 372 305 94 167 418 113 390 134 264)(30 241 135 391 114 419 168 95 306 373)(31 374 307 96 145 420 115 392 136 242)(32 243 137 393 116 421 146 73 308 375)(33 376 309 74 147 422 117 394 138 244)(34 245 139 395 118 423 148 75 310 377)(35 378 311 76 149 424 119 396 140 246)(36 247 141 397 120 425 150 77 312 379)(37 380 289 78 151 426 97 398 142 248)(38 249 143 399 98 427 152 79 290 381)(39 382 291 80 153 428 99 400 144 250)(40 251 121 401 100 429 154 81 292 383)(41 384 293 82 155 430 101 402 122 252)(42 253 123 403 102 431 156 83 294 361)(43 362 295 84 157 432 103 404 124 254)(44 255 125 405 104 409 158 85 296 363)(45 364 297 86 159 410 105 406 126 256)(46 257 127 407 106 411 160 87 298 365)(47 366 299 88 161 412 107 408 128 258)(48 259 129 385 108 413 162 89 300 367)
(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)
(1 255 13 243)(2 157 14 145)(3 253 15 241)(4 155 16 167)(5 251 17 263)(6 153 18 165)(7 249 19 261)(8 151 20 163)(9 247 21 259)(10 149 22 161)(11 245 23 257)(12 147 24 159)(25 223 37 235)(26 176 38 188)(27 221 39 233)(28 174 40 186)(29 219 41 231)(30 172 42 184)(31 217 43 229)(32 170 44 182)(33 239 45 227)(34 192 46 180)(35 237 47 225)(36 190 48 178)(49 403 61 391)(50 293 62 305)(51 401 63 389)(52 291 64 303)(53 399 65 387)(54 289 66 301)(55 397 67 385)(56 311 68 299)(57 395 69 407)(58 309 70 297)(59 393 71 405)(60 307 72 295)(73 445 85 433)(74 316 86 328)(75 443 87 455)(76 314 88 326)(77 441 89 453)(78 336 90 324)(79 439 91 451)(80 334 92 322)(81 437 93 449)(82 332 94 320)(83 435 95 447)(84 330 96 318)(97 343 109 355)(98 277 110 265)(99 341 111 353)(100 275 112 287)(101 339 113 351)(102 273 114 285)(103 337 115 349)(104 271 116 283)(105 359 117 347)(106 269 118 281)(107 357 119 345)(108 267 120 279)(121 333 133 321)(122 436 134 448)(123 331 135 319)(124 434 136 446)(125 329 137 317)(126 456 138 444)(127 327 139 315)(128 454 140 442)(129 325 141 313)(130 452 142 440)(131 323 143 335)(132 450 144 438)(146 477 158 465)(148 475 160 463)(150 473 162 461)(152 471 164 459)(154 469 166 457)(156 467 168 479)(169 410 181 422)(171 432 183 420)(173 430 185 418)(175 428 187 416)(177 426 189 414)(179 424 191 412)(193 386 205 398)(194 300 206 312)(195 408 207 396)(196 298 208 310)(197 406 209 394)(198 296 210 308)(199 404 211 392)(200 294 212 306)(201 402 213 390)(202 292 214 304)(203 400 215 388)(204 290 216 302)(218 431 230 419)(220 429 232 417)(222 427 234 415)(224 425 236 413)(226 423 238 411)(228 421 240 409)(242 478 254 466)(244 476 256 464)(246 474 258 462)(248 472 260 460)(250 470 262 458)(252 468 264 480)(266 380 278 368)(268 378 280 366)(270 376 282 364)(272 374 284 362)(274 372 286 384)(276 370 288 382)(338 373 350 361)(340 371 352 383)(342 369 354 381)(344 367 356 379)(346 365 358 377)(348 363 360 375)
G:=sub<Sym(480)| (1,317,71,283,240,477,445,198,348,170)(2,171,349,199,446,478,217,284,72,318)(3,319,49,285,218,479,447,200,350,172)(4,173,351,201,448,480,219,286,50,320)(5,321,51,287,220,457,449,202,352,174)(6,175,353,203,450,458,221,288,52,322)(7,323,53,265,222,459,451,204,354,176)(8,177,355,205,452,460,223,266,54,324)(9,325,55,267,224,461,453,206,356,178)(10,179,357,207,454,462,225,268,56,326)(11,327,57,269,226,463,455,208,358,180)(12,181,359,209,456,464,227,270,58,328)(13,329,59,271,228,465,433,210,360,182)(14,183,337,211,434,466,229,272,60,330)(15,331,61,273,230,467,435,212,338,184)(16,185,339,213,436,468,231,274,62,332)(17,333,63,275,232,469,437,214,340,186)(18,187,341,215,438,470,233,276,64,334)(19,335,65,277,234,471,439,216,342,188)(20,189,343,193,440,472,235,278,66,336)(21,313,67,279,236,473,441,194,344,190)(22,191,345,195,442,474,237,280,68,314)(23,315,69,281,238,475,443,196,346,192)(24,169,347,197,444,476,239,282,70,316)(25,368,301,90,163,414,109,386,130,260)(26,261,131,387,110,415,164,91,302,369)(27,370,303,92,165,416,111,388,132,262)(28,263,133,389,112,417,166,93,304,371)(29,372,305,94,167,418,113,390,134,264)(30,241,135,391,114,419,168,95,306,373)(31,374,307,96,145,420,115,392,136,242)(32,243,137,393,116,421,146,73,308,375)(33,376,309,74,147,422,117,394,138,244)(34,245,139,395,118,423,148,75,310,377)(35,378,311,76,149,424,119,396,140,246)(36,247,141,397,120,425,150,77,312,379)(37,380,289,78,151,426,97,398,142,248)(38,249,143,399,98,427,152,79,290,381)(39,382,291,80,153,428,99,400,144,250)(40,251,121,401,100,429,154,81,292,383)(41,384,293,82,155,430,101,402,122,252)(42,253,123,403,102,431,156,83,294,361)(43,362,295,84,157,432,103,404,124,254)(44,255,125,405,104,409,158,85,296,363)(45,364,297,86,159,410,105,406,126,256)(46,257,127,407,106,411,160,87,298,365)(47,366,299,88,161,412,107,408,128,258)(48,259,129,385,108,413,162,89,300,367), (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), (1,255,13,243)(2,157,14,145)(3,253,15,241)(4,155,16,167)(5,251,17,263)(6,153,18,165)(7,249,19,261)(8,151,20,163)(9,247,21,259)(10,149,22,161)(11,245,23,257)(12,147,24,159)(25,223,37,235)(26,176,38,188)(27,221,39,233)(28,174,40,186)(29,219,41,231)(30,172,42,184)(31,217,43,229)(32,170,44,182)(33,239,45,227)(34,192,46,180)(35,237,47,225)(36,190,48,178)(49,403,61,391)(50,293,62,305)(51,401,63,389)(52,291,64,303)(53,399,65,387)(54,289,66,301)(55,397,67,385)(56,311,68,299)(57,395,69,407)(58,309,70,297)(59,393,71,405)(60,307,72,295)(73,445,85,433)(74,316,86,328)(75,443,87,455)(76,314,88,326)(77,441,89,453)(78,336,90,324)(79,439,91,451)(80,334,92,322)(81,437,93,449)(82,332,94,320)(83,435,95,447)(84,330,96,318)(97,343,109,355)(98,277,110,265)(99,341,111,353)(100,275,112,287)(101,339,113,351)(102,273,114,285)(103,337,115,349)(104,271,116,283)(105,359,117,347)(106,269,118,281)(107,357,119,345)(108,267,120,279)(121,333,133,321)(122,436,134,448)(123,331,135,319)(124,434,136,446)(125,329,137,317)(126,456,138,444)(127,327,139,315)(128,454,140,442)(129,325,141,313)(130,452,142,440)(131,323,143,335)(132,450,144,438)(146,477,158,465)(148,475,160,463)(150,473,162,461)(152,471,164,459)(154,469,166,457)(156,467,168,479)(169,410,181,422)(171,432,183,420)(173,430,185,418)(175,428,187,416)(177,426,189,414)(179,424,191,412)(193,386,205,398)(194,300,206,312)(195,408,207,396)(196,298,208,310)(197,406,209,394)(198,296,210,308)(199,404,211,392)(200,294,212,306)(201,402,213,390)(202,292,214,304)(203,400,215,388)(204,290,216,302)(218,431,230,419)(220,429,232,417)(222,427,234,415)(224,425,236,413)(226,423,238,411)(228,421,240,409)(242,478,254,466)(244,476,256,464)(246,474,258,462)(248,472,260,460)(250,470,262,458)(252,468,264,480)(266,380,278,368)(268,378,280,366)(270,376,282,364)(272,374,284,362)(274,372,286,384)(276,370,288,382)(338,373,350,361)(340,371,352,383)(342,369,354,381)(344,367,356,379)(346,365,358,377)(348,363,360,375)>;
G:=Group( (1,317,71,283,240,477,445,198,348,170)(2,171,349,199,446,478,217,284,72,318)(3,319,49,285,218,479,447,200,350,172)(4,173,351,201,448,480,219,286,50,320)(5,321,51,287,220,457,449,202,352,174)(6,175,353,203,450,458,221,288,52,322)(7,323,53,265,222,459,451,204,354,176)(8,177,355,205,452,460,223,266,54,324)(9,325,55,267,224,461,453,206,356,178)(10,179,357,207,454,462,225,268,56,326)(11,327,57,269,226,463,455,208,358,180)(12,181,359,209,456,464,227,270,58,328)(13,329,59,271,228,465,433,210,360,182)(14,183,337,211,434,466,229,272,60,330)(15,331,61,273,230,467,435,212,338,184)(16,185,339,213,436,468,231,274,62,332)(17,333,63,275,232,469,437,214,340,186)(18,187,341,215,438,470,233,276,64,334)(19,335,65,277,234,471,439,216,342,188)(20,189,343,193,440,472,235,278,66,336)(21,313,67,279,236,473,441,194,344,190)(22,191,345,195,442,474,237,280,68,314)(23,315,69,281,238,475,443,196,346,192)(24,169,347,197,444,476,239,282,70,316)(25,368,301,90,163,414,109,386,130,260)(26,261,131,387,110,415,164,91,302,369)(27,370,303,92,165,416,111,388,132,262)(28,263,133,389,112,417,166,93,304,371)(29,372,305,94,167,418,113,390,134,264)(30,241,135,391,114,419,168,95,306,373)(31,374,307,96,145,420,115,392,136,242)(32,243,137,393,116,421,146,73,308,375)(33,376,309,74,147,422,117,394,138,244)(34,245,139,395,118,423,148,75,310,377)(35,378,311,76,149,424,119,396,140,246)(36,247,141,397,120,425,150,77,312,379)(37,380,289,78,151,426,97,398,142,248)(38,249,143,399,98,427,152,79,290,381)(39,382,291,80,153,428,99,400,144,250)(40,251,121,401,100,429,154,81,292,383)(41,384,293,82,155,430,101,402,122,252)(42,253,123,403,102,431,156,83,294,361)(43,362,295,84,157,432,103,404,124,254)(44,255,125,405,104,409,158,85,296,363)(45,364,297,86,159,410,105,406,126,256)(46,257,127,407,106,411,160,87,298,365)(47,366,299,88,161,412,107,408,128,258)(48,259,129,385,108,413,162,89,300,367), (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), (1,255,13,243)(2,157,14,145)(3,253,15,241)(4,155,16,167)(5,251,17,263)(6,153,18,165)(7,249,19,261)(8,151,20,163)(9,247,21,259)(10,149,22,161)(11,245,23,257)(12,147,24,159)(25,223,37,235)(26,176,38,188)(27,221,39,233)(28,174,40,186)(29,219,41,231)(30,172,42,184)(31,217,43,229)(32,170,44,182)(33,239,45,227)(34,192,46,180)(35,237,47,225)(36,190,48,178)(49,403,61,391)(50,293,62,305)(51,401,63,389)(52,291,64,303)(53,399,65,387)(54,289,66,301)(55,397,67,385)(56,311,68,299)(57,395,69,407)(58,309,70,297)(59,393,71,405)(60,307,72,295)(73,445,85,433)(74,316,86,328)(75,443,87,455)(76,314,88,326)(77,441,89,453)(78,336,90,324)(79,439,91,451)(80,334,92,322)(81,437,93,449)(82,332,94,320)(83,435,95,447)(84,330,96,318)(97,343,109,355)(98,277,110,265)(99,341,111,353)(100,275,112,287)(101,339,113,351)(102,273,114,285)(103,337,115,349)(104,271,116,283)(105,359,117,347)(106,269,118,281)(107,357,119,345)(108,267,120,279)(121,333,133,321)(122,436,134,448)(123,331,135,319)(124,434,136,446)(125,329,137,317)(126,456,138,444)(127,327,139,315)(128,454,140,442)(129,325,141,313)(130,452,142,440)(131,323,143,335)(132,450,144,438)(146,477,158,465)(148,475,160,463)(150,473,162,461)(152,471,164,459)(154,469,166,457)(156,467,168,479)(169,410,181,422)(171,432,183,420)(173,430,185,418)(175,428,187,416)(177,426,189,414)(179,424,191,412)(193,386,205,398)(194,300,206,312)(195,408,207,396)(196,298,208,310)(197,406,209,394)(198,296,210,308)(199,404,211,392)(200,294,212,306)(201,402,213,390)(202,292,214,304)(203,400,215,388)(204,290,216,302)(218,431,230,419)(220,429,232,417)(222,427,234,415)(224,425,236,413)(226,423,238,411)(228,421,240,409)(242,478,254,466)(244,476,256,464)(246,474,258,462)(248,472,260,460)(250,470,262,458)(252,468,264,480)(266,380,278,368)(268,378,280,366)(270,376,282,364)(272,374,284,362)(274,372,286,384)(276,370,288,382)(338,373,350,361)(340,371,352,383)(342,369,354,381)(344,367,356,379)(346,365,358,377)(348,363,360,375) );
G=PermutationGroup([[(1,317,71,283,240,477,445,198,348,170),(2,171,349,199,446,478,217,284,72,318),(3,319,49,285,218,479,447,200,350,172),(4,173,351,201,448,480,219,286,50,320),(5,321,51,287,220,457,449,202,352,174),(6,175,353,203,450,458,221,288,52,322),(7,323,53,265,222,459,451,204,354,176),(8,177,355,205,452,460,223,266,54,324),(9,325,55,267,224,461,453,206,356,178),(10,179,357,207,454,462,225,268,56,326),(11,327,57,269,226,463,455,208,358,180),(12,181,359,209,456,464,227,270,58,328),(13,329,59,271,228,465,433,210,360,182),(14,183,337,211,434,466,229,272,60,330),(15,331,61,273,230,467,435,212,338,184),(16,185,339,213,436,468,231,274,62,332),(17,333,63,275,232,469,437,214,340,186),(18,187,341,215,438,470,233,276,64,334),(19,335,65,277,234,471,439,216,342,188),(20,189,343,193,440,472,235,278,66,336),(21,313,67,279,236,473,441,194,344,190),(22,191,345,195,442,474,237,280,68,314),(23,315,69,281,238,475,443,196,346,192),(24,169,347,197,444,476,239,282,70,316),(25,368,301,90,163,414,109,386,130,260),(26,261,131,387,110,415,164,91,302,369),(27,370,303,92,165,416,111,388,132,262),(28,263,133,389,112,417,166,93,304,371),(29,372,305,94,167,418,113,390,134,264),(30,241,135,391,114,419,168,95,306,373),(31,374,307,96,145,420,115,392,136,242),(32,243,137,393,116,421,146,73,308,375),(33,376,309,74,147,422,117,394,138,244),(34,245,139,395,118,423,148,75,310,377),(35,378,311,76,149,424,119,396,140,246),(36,247,141,397,120,425,150,77,312,379),(37,380,289,78,151,426,97,398,142,248),(38,249,143,399,98,427,152,79,290,381),(39,382,291,80,153,428,99,400,144,250),(40,251,121,401,100,429,154,81,292,383),(41,384,293,82,155,430,101,402,122,252),(42,253,123,403,102,431,156,83,294,361),(43,362,295,84,157,432,103,404,124,254),(44,255,125,405,104,409,158,85,296,363),(45,364,297,86,159,410,105,406,126,256),(46,257,127,407,106,411,160,87,298,365),(47,366,299,88,161,412,107,408,128,258),(48,259,129,385,108,413,162,89,300,367)], [(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)], [(1,255,13,243),(2,157,14,145),(3,253,15,241),(4,155,16,167),(5,251,17,263),(6,153,18,165),(7,249,19,261),(8,151,20,163),(9,247,21,259),(10,149,22,161),(11,245,23,257),(12,147,24,159),(25,223,37,235),(26,176,38,188),(27,221,39,233),(28,174,40,186),(29,219,41,231),(30,172,42,184),(31,217,43,229),(32,170,44,182),(33,239,45,227),(34,192,46,180),(35,237,47,225),(36,190,48,178),(49,403,61,391),(50,293,62,305),(51,401,63,389),(52,291,64,303),(53,399,65,387),(54,289,66,301),(55,397,67,385),(56,311,68,299),(57,395,69,407),(58,309,70,297),(59,393,71,405),(60,307,72,295),(73,445,85,433),(74,316,86,328),(75,443,87,455),(76,314,88,326),(77,441,89,453),(78,336,90,324),(79,439,91,451),(80,334,92,322),(81,437,93,449),(82,332,94,320),(83,435,95,447),(84,330,96,318),(97,343,109,355),(98,277,110,265),(99,341,111,353),(100,275,112,287),(101,339,113,351),(102,273,114,285),(103,337,115,349),(104,271,116,283),(105,359,117,347),(106,269,118,281),(107,357,119,345),(108,267,120,279),(121,333,133,321),(122,436,134,448),(123,331,135,319),(124,434,136,446),(125,329,137,317),(126,456,138,444),(127,327,139,315),(128,454,140,442),(129,325,141,313),(130,452,142,440),(131,323,143,335),(132,450,144,438),(146,477,158,465),(148,475,160,463),(150,473,162,461),(152,471,164,459),(154,469,166,457),(156,467,168,479),(169,410,181,422),(171,432,183,420),(173,430,185,418),(175,428,187,416),(177,426,189,414),(179,424,191,412),(193,386,205,398),(194,300,206,312),(195,408,207,396),(196,298,208,310),(197,406,209,394),(198,296,210,308),(199,404,211,392),(200,294,212,306),(201,402,213,390),(202,292,214,304),(203,400,215,388),(204,290,216,302),(218,431,230,419),(220,429,232,417),(222,427,234,415),(224,425,236,413),(226,423,238,411),(228,421,240,409),(242,478,254,466),(244,476,256,464),(246,474,258,462),(248,472,260,460),(250,470,262,458),(252,468,264,480),(266,380,278,368),(268,378,280,366),(270,376,282,364),(272,374,284,362),(274,372,286,384),(276,370,288,382),(338,373,350,361),(340,371,352,383),(342,369,354,381),(344,367,356,379),(346,365,358,377),(348,363,360,375)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | + | - | + | + | - | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | SD16 | Q16 | Dic5 | D10 | C4×S3 | C3⋊D4 | D12 | C5⋊D4 | C5⋊D4 | C24⋊C2 | Dic12 | S3×D5 | Q8⋊D5 | C5⋊Q16 | S3×Dic5 | C15⋊D4 | C5⋊D12 | Dic6⋊D5 | C5⋊Dic12 |
| kernel | C10.Dic12 | C6×C5⋊2C8 | C60⋊5C4 | C10×Dic6 | C5×Dic6 | C2×C5⋊2C8 | C60 | C2×C30 | C2×Dic6 | C2×C20 | C30 | C30 | Dic6 | C2×C12 | C20 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C10.Dic12 ►in GL4(𝔽241) generated by
| 1 | 51 | 0 | 0 |
| 190 | 51 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 128 | 183 | 0 | 0 |
| 162 | 113 | 0 | 0 |
| 0 | 0 | 66 | 28 |
| 0 | 0 | 213 | 94 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 210 | 67 |
| 0 | 0 | 36 | 31 |
G:=sub<GL(4,GF(241))| [1,190,0,0,51,51,0,0,0,0,240,0,0,0,0,240],[128,162,0,0,183,113,0,0,0,0,66,213,0,0,28,94],[240,0,0,0,0,240,0,0,0,0,210,36,0,0,67,31] >;
C10.Dic12 in GAP, Magma, Sage, TeX
C_{10}.{\rm Dic}_{12} % in TeX
G:=Group("C10.Dic12"); // GroupNames label
G:=SmallGroup(480,49);
// by ID
G=gap.SmallGroup(480,49);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,197,120,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^10=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;
// generators/relations