metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.12Q8, C20.7Dic6, C12.8Dic10, C4.9(C15⋊Q8), C30.53(C2×Q8), (C2×C20).119D6, C4⋊Dic5.12S3, C4⋊Dic3.12D5, C5⋊4(C4.Dic6), C3⋊4(C4.Dic10), C30.47(C4○D4), (C2×C12).120D10, C6.9(D4⋊2D5), (C2×C30).74C23, (C2×Dic5).25D6, C6.20(C2×Dic10), C10.21(C2×Dic6), C15⋊11(C42.C2), C10.9(D4⋊2S3), (C2×C60).197C22, C6.29(Q8⋊2D5), (C2×Dic3).24D10, (C4×Dic15).14C2, C6.Dic10.13C2, C30.Q8.15C2, C2.13(D20⋊S3), C2.13(D12⋊D5), C10.28(Q8⋊3S3), (C6×Dic5).44C22, (C10×Dic3).43C22, (C2×Dic15).194C22, C2.6(C2×C15⋊Q8), (C2×C4).207(S3×D5), C22.160(C2×S3×D5), (C3×C4⋊Dic5).11C2, (C5×C4⋊Dic3).11C2, (C2×C6).86(C22×D5), (C2×C10).86(C22×S3), SmallGroup(480,460)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.Dic10
G = < a,b,c | a12=b20=1, c2=a6b10, bab-1=a-1, cac-1=a7, cbc-1=a6b-1 >
Subgroups: 460 in 112 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C4.Dic6, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C4.Dic10, C30.Q8, C6.Dic10, C3×C4⋊Dic5, C5×C4⋊Dic3, C4×Dic15, C12.Dic10
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, C42.C2, Dic10, C22×D5, C2×Dic6, D4⋊2S3, Q8⋊3S3, S3×D5, C2×Dic10, D4⋊2D5, Q8⋊2D5, C4.Dic6, C15⋊Q8, C2×S3×D5, C4.Dic10, D20⋊S3, D12⋊D5, C2×C15⋊Q8, C12.Dic10
►Regular action on 480 points
Generators in S480
(1 422 205 305 372 261 182 259 165 463 90 148)(2 149 91 464 166 260 183 262 373 306 206 423)(3 424 207 307 374 263 184 241 167 465 92 150)(4 151 93 466 168 242 185 264 375 308 208 425)(5 426 209 309 376 265 186 243 169 467 94 152)(6 153 95 468 170 244 187 266 377 310 210 427)(7 428 211 311 378 267 188 245 171 469 96 154)(8 155 97 470 172 246 189 268 379 312 212 429)(9 430 213 313 380 269 190 247 173 471 98 156)(10 157 99 472 174 248 191 270 361 314 214 431)(11 432 215 315 362 271 192 249 175 473 100 158)(12 159 81 474 176 250 193 272 363 316 216 433)(13 434 217 317 364 273 194 251 177 475 82 160)(14 141 83 476 178 252 195 274 365 318 218 435)(15 436 219 319 366 275 196 253 179 477 84 142)(16 143 85 478 180 254 197 276 367 320 220 437)(17 438 201 301 368 277 198 255 161 479 86 144)(18 145 87 480 162 256 199 278 369 302 202 439)(19 440 203 303 370 279 200 257 163 461 88 146)(20 147 89 462 164 258 181 280 371 304 204 421)(21 387 341 71 136 330 104 54 231 460 285 403)(22 404 286 441 232 55 105 331 137 72 342 388)(23 389 343 73 138 332 106 56 233 442 287 405)(24 406 288 443 234 57 107 333 139 74 344 390)(25 391 345 75 140 334 108 58 235 444 289 407)(26 408 290 445 236 59 109 335 121 76 346 392)(27 393 347 77 122 336 110 60 237 446 291 409)(28 410 292 447 238 41 111 337 123 78 348 394)(29 395 349 79 124 338 112 42 239 448 293 411)(30 412 294 449 240 43 113 339 125 80 350 396)(31 397 351 61 126 340 114 44 221 450 295 413)(32 414 296 451 222 45 115 321 127 62 352 398)(33 399 353 63 128 322 116 46 223 452 297 415)(34 416 298 453 224 47 117 323 129 64 354 400)(35 381 355 65 130 324 118 48 225 454 299 417)(36 418 300 455 226 49 119 325 131 66 356 382)(37 383 357 67 132 326 120 50 227 456 281 419)(38 420 282 457 228 51 101 327 133 68 358 384)(39 385 359 69 134 328 102 52 229 458 283 401)(40 402 284 459 230 53 103 329 135 70 360 386)
(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)
(1 295 192 136)(2 125 193 284)(3 293 194 134)(4 123 195 282)(5 291 196 132)(6 121 197 300)(7 289 198 130)(8 139 199 298)(9 287 200 128)(10 137 181 296)(11 285 182 126)(12 135 183 294)(13 283 184 124)(14 133 185 292)(15 281 186 122)(16 131 187 290)(17 299 188 140)(18 129 189 288)(19 297 190 138)(20 127 191 286)(21 165 114 215)(22 204 115 174)(23 163 116 213)(24 202 117 172)(25 161 118 211)(26 220 119 170)(27 179 120 209)(28 218 101 168)(29 177 102 207)(30 216 103 166)(31 175 104 205)(32 214 105 164)(33 173 106 203)(34 212 107 162)(35 171 108 201)(36 210 109 180)(37 169 110 219)(38 208 111 178)(39 167 112 217)(40 206 113 176)(41 318 384 466)(42 475 385 307)(43 316 386 464)(44 473 387 305)(45 314 388 462)(46 471 389 303)(47 312 390 480)(48 469 391 301)(49 310 392 478)(50 467 393 319)(51 308 394 476)(52 465 395 317)(53 306 396 474)(54 463 397 315)(55 304 398 472)(56 461 399 313)(57 302 400 470)(58 479 381 311)(59 320 382 468)(60 477 383 309)(61 271 460 148)(62 157 441 280)(63 269 442 146)(64 155 443 278)(65 267 444 144)(66 153 445 276)(67 265 446 142)(68 151 447 274)(69 263 448 160)(70 149 449 272)(71 261 450 158)(72 147 451 270)(73 279 452 156)(74 145 453 268)(75 277 454 154)(76 143 455 266)(77 275 456 152)(78 141 457 264)(79 273 458 150)(80 159 459 262)(81 360 373 240)(82 229 374 349)(83 358 375 238)(84 227 376 347)(85 356 377 236)(86 225 378 345)(87 354 379 234)(88 223 380 343)(89 352 361 232)(90 221 362 341)(91 350 363 230)(92 239 364 359)(93 348 365 228)(94 237 366 357)(95 346 367 226)(96 235 368 355)(97 344 369 224)(98 233 370 353)(99 342 371 222)(100 231 372 351)(241 411 434 328)(242 337 435 420)(243 409 436 326)(244 335 437 418)(245 407 438 324)(246 333 439 416)(247 405 440 322)(248 331 421 414)(249 403 422 340)(250 329 423 412)(251 401 424 338)(252 327 425 410)(253 419 426 336)(254 325 427 408)(255 417 428 334)(256 323 429 406)(257 415 430 332)(258 321 431 404)(259 413 432 330)(260 339 433 402)
G:=sub<Sym(480)| (1,422,205,305,372,261,182,259,165,463,90,148)(2,149,91,464,166,260,183,262,373,306,206,423)(3,424,207,307,374,263,184,241,167,465,92,150)(4,151,93,466,168,242,185,264,375,308,208,425)(5,426,209,309,376,265,186,243,169,467,94,152)(6,153,95,468,170,244,187,266,377,310,210,427)(7,428,211,311,378,267,188,245,171,469,96,154)(8,155,97,470,172,246,189,268,379,312,212,429)(9,430,213,313,380,269,190,247,173,471,98,156)(10,157,99,472,174,248,191,270,361,314,214,431)(11,432,215,315,362,271,192,249,175,473,100,158)(12,159,81,474,176,250,193,272,363,316,216,433)(13,434,217,317,364,273,194,251,177,475,82,160)(14,141,83,476,178,252,195,274,365,318,218,435)(15,436,219,319,366,275,196,253,179,477,84,142)(16,143,85,478,180,254,197,276,367,320,220,437)(17,438,201,301,368,277,198,255,161,479,86,144)(18,145,87,480,162,256,199,278,369,302,202,439)(19,440,203,303,370,279,200,257,163,461,88,146)(20,147,89,462,164,258,181,280,371,304,204,421)(21,387,341,71,136,330,104,54,231,460,285,403)(22,404,286,441,232,55,105,331,137,72,342,388)(23,389,343,73,138,332,106,56,233,442,287,405)(24,406,288,443,234,57,107,333,139,74,344,390)(25,391,345,75,140,334,108,58,235,444,289,407)(26,408,290,445,236,59,109,335,121,76,346,392)(27,393,347,77,122,336,110,60,237,446,291,409)(28,410,292,447,238,41,111,337,123,78,348,394)(29,395,349,79,124,338,112,42,239,448,293,411)(30,412,294,449,240,43,113,339,125,80,350,396)(31,397,351,61,126,340,114,44,221,450,295,413)(32,414,296,451,222,45,115,321,127,62,352,398)(33,399,353,63,128,322,116,46,223,452,297,415)(34,416,298,453,224,47,117,323,129,64,354,400)(35,381,355,65,130,324,118,48,225,454,299,417)(36,418,300,455,226,49,119,325,131,66,356,382)(37,383,357,67,132,326,120,50,227,456,281,419)(38,420,282,457,228,51,101,327,133,68,358,384)(39,385,359,69,134,328,102,52,229,458,283,401)(40,402,284,459,230,53,103,329,135,70,360,386), (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), (1,295,192,136)(2,125,193,284)(3,293,194,134)(4,123,195,282)(5,291,196,132)(6,121,197,300)(7,289,198,130)(8,139,199,298)(9,287,200,128)(10,137,181,296)(11,285,182,126)(12,135,183,294)(13,283,184,124)(14,133,185,292)(15,281,186,122)(16,131,187,290)(17,299,188,140)(18,129,189,288)(19,297,190,138)(20,127,191,286)(21,165,114,215)(22,204,115,174)(23,163,116,213)(24,202,117,172)(25,161,118,211)(26,220,119,170)(27,179,120,209)(28,218,101,168)(29,177,102,207)(30,216,103,166)(31,175,104,205)(32,214,105,164)(33,173,106,203)(34,212,107,162)(35,171,108,201)(36,210,109,180)(37,169,110,219)(38,208,111,178)(39,167,112,217)(40,206,113,176)(41,318,384,466)(42,475,385,307)(43,316,386,464)(44,473,387,305)(45,314,388,462)(46,471,389,303)(47,312,390,480)(48,469,391,301)(49,310,392,478)(50,467,393,319)(51,308,394,476)(52,465,395,317)(53,306,396,474)(54,463,397,315)(55,304,398,472)(56,461,399,313)(57,302,400,470)(58,479,381,311)(59,320,382,468)(60,477,383,309)(61,271,460,148)(62,157,441,280)(63,269,442,146)(64,155,443,278)(65,267,444,144)(66,153,445,276)(67,265,446,142)(68,151,447,274)(69,263,448,160)(70,149,449,272)(71,261,450,158)(72,147,451,270)(73,279,452,156)(74,145,453,268)(75,277,454,154)(76,143,455,266)(77,275,456,152)(78,141,457,264)(79,273,458,150)(80,159,459,262)(81,360,373,240)(82,229,374,349)(83,358,375,238)(84,227,376,347)(85,356,377,236)(86,225,378,345)(87,354,379,234)(88,223,380,343)(89,352,361,232)(90,221,362,341)(91,350,363,230)(92,239,364,359)(93,348,365,228)(94,237,366,357)(95,346,367,226)(96,235,368,355)(97,344,369,224)(98,233,370,353)(99,342,371,222)(100,231,372,351)(241,411,434,328)(242,337,435,420)(243,409,436,326)(244,335,437,418)(245,407,438,324)(246,333,439,416)(247,405,440,322)(248,331,421,414)(249,403,422,340)(250,329,423,412)(251,401,424,338)(252,327,425,410)(253,419,426,336)(254,325,427,408)(255,417,428,334)(256,323,429,406)(257,415,430,332)(258,321,431,404)(259,413,432,330)(260,339,433,402)>;
G:=Group( (1,422,205,305,372,261,182,259,165,463,90,148)(2,149,91,464,166,260,183,262,373,306,206,423)(3,424,207,307,374,263,184,241,167,465,92,150)(4,151,93,466,168,242,185,264,375,308,208,425)(5,426,209,309,376,265,186,243,169,467,94,152)(6,153,95,468,170,244,187,266,377,310,210,427)(7,428,211,311,378,267,188,245,171,469,96,154)(8,155,97,470,172,246,189,268,379,312,212,429)(9,430,213,313,380,269,190,247,173,471,98,156)(10,157,99,472,174,248,191,270,361,314,214,431)(11,432,215,315,362,271,192,249,175,473,100,158)(12,159,81,474,176,250,193,272,363,316,216,433)(13,434,217,317,364,273,194,251,177,475,82,160)(14,141,83,476,178,252,195,274,365,318,218,435)(15,436,219,319,366,275,196,253,179,477,84,142)(16,143,85,478,180,254,197,276,367,320,220,437)(17,438,201,301,368,277,198,255,161,479,86,144)(18,145,87,480,162,256,199,278,369,302,202,439)(19,440,203,303,370,279,200,257,163,461,88,146)(20,147,89,462,164,258,181,280,371,304,204,421)(21,387,341,71,136,330,104,54,231,460,285,403)(22,404,286,441,232,55,105,331,137,72,342,388)(23,389,343,73,138,332,106,56,233,442,287,405)(24,406,288,443,234,57,107,333,139,74,344,390)(25,391,345,75,140,334,108,58,235,444,289,407)(26,408,290,445,236,59,109,335,121,76,346,392)(27,393,347,77,122,336,110,60,237,446,291,409)(28,410,292,447,238,41,111,337,123,78,348,394)(29,395,349,79,124,338,112,42,239,448,293,411)(30,412,294,449,240,43,113,339,125,80,350,396)(31,397,351,61,126,340,114,44,221,450,295,413)(32,414,296,451,222,45,115,321,127,62,352,398)(33,399,353,63,128,322,116,46,223,452,297,415)(34,416,298,453,224,47,117,323,129,64,354,400)(35,381,355,65,130,324,118,48,225,454,299,417)(36,418,300,455,226,49,119,325,131,66,356,382)(37,383,357,67,132,326,120,50,227,456,281,419)(38,420,282,457,228,51,101,327,133,68,358,384)(39,385,359,69,134,328,102,52,229,458,283,401)(40,402,284,459,230,53,103,329,135,70,360,386), (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), (1,295,192,136)(2,125,193,284)(3,293,194,134)(4,123,195,282)(5,291,196,132)(6,121,197,300)(7,289,198,130)(8,139,199,298)(9,287,200,128)(10,137,181,296)(11,285,182,126)(12,135,183,294)(13,283,184,124)(14,133,185,292)(15,281,186,122)(16,131,187,290)(17,299,188,140)(18,129,189,288)(19,297,190,138)(20,127,191,286)(21,165,114,215)(22,204,115,174)(23,163,116,213)(24,202,117,172)(25,161,118,211)(26,220,119,170)(27,179,120,209)(28,218,101,168)(29,177,102,207)(30,216,103,166)(31,175,104,205)(32,214,105,164)(33,173,106,203)(34,212,107,162)(35,171,108,201)(36,210,109,180)(37,169,110,219)(38,208,111,178)(39,167,112,217)(40,206,113,176)(41,318,384,466)(42,475,385,307)(43,316,386,464)(44,473,387,305)(45,314,388,462)(46,471,389,303)(47,312,390,480)(48,469,391,301)(49,310,392,478)(50,467,393,319)(51,308,394,476)(52,465,395,317)(53,306,396,474)(54,463,397,315)(55,304,398,472)(56,461,399,313)(57,302,400,470)(58,479,381,311)(59,320,382,468)(60,477,383,309)(61,271,460,148)(62,157,441,280)(63,269,442,146)(64,155,443,278)(65,267,444,144)(66,153,445,276)(67,265,446,142)(68,151,447,274)(69,263,448,160)(70,149,449,272)(71,261,450,158)(72,147,451,270)(73,279,452,156)(74,145,453,268)(75,277,454,154)(76,143,455,266)(77,275,456,152)(78,141,457,264)(79,273,458,150)(80,159,459,262)(81,360,373,240)(82,229,374,349)(83,358,375,238)(84,227,376,347)(85,356,377,236)(86,225,378,345)(87,354,379,234)(88,223,380,343)(89,352,361,232)(90,221,362,341)(91,350,363,230)(92,239,364,359)(93,348,365,228)(94,237,366,357)(95,346,367,226)(96,235,368,355)(97,344,369,224)(98,233,370,353)(99,342,371,222)(100,231,372,351)(241,411,434,328)(242,337,435,420)(243,409,436,326)(244,335,437,418)(245,407,438,324)(246,333,439,416)(247,405,440,322)(248,331,421,414)(249,403,422,340)(250,329,423,412)(251,401,424,338)(252,327,425,410)(253,419,426,336)(254,325,427,408)(255,417,428,334)(256,323,429,406)(257,415,430,332)(258,321,431,404)(259,413,432,330)(260,339,433,402) );
G=PermutationGroup([[(1,422,205,305,372,261,182,259,165,463,90,148),(2,149,91,464,166,260,183,262,373,306,206,423),(3,424,207,307,374,263,184,241,167,465,92,150),(4,151,93,466,168,242,185,264,375,308,208,425),(5,426,209,309,376,265,186,243,169,467,94,152),(6,153,95,468,170,244,187,266,377,310,210,427),(7,428,211,311,378,267,188,245,171,469,96,154),(8,155,97,470,172,246,189,268,379,312,212,429),(9,430,213,313,380,269,190,247,173,471,98,156),(10,157,99,472,174,248,191,270,361,314,214,431),(11,432,215,315,362,271,192,249,175,473,100,158),(12,159,81,474,176,250,193,272,363,316,216,433),(13,434,217,317,364,273,194,251,177,475,82,160),(14,141,83,476,178,252,195,274,365,318,218,435),(15,436,219,319,366,275,196,253,179,477,84,142),(16,143,85,478,180,254,197,276,367,320,220,437),(17,438,201,301,368,277,198,255,161,479,86,144),(18,145,87,480,162,256,199,278,369,302,202,439),(19,440,203,303,370,279,200,257,163,461,88,146),(20,147,89,462,164,258,181,280,371,304,204,421),(21,387,341,71,136,330,104,54,231,460,285,403),(22,404,286,441,232,55,105,331,137,72,342,388),(23,389,343,73,138,332,106,56,233,442,287,405),(24,406,288,443,234,57,107,333,139,74,344,390),(25,391,345,75,140,334,108,58,235,444,289,407),(26,408,290,445,236,59,109,335,121,76,346,392),(27,393,347,77,122,336,110,60,237,446,291,409),(28,410,292,447,238,41,111,337,123,78,348,394),(29,395,349,79,124,338,112,42,239,448,293,411),(30,412,294,449,240,43,113,339,125,80,350,396),(31,397,351,61,126,340,114,44,221,450,295,413),(32,414,296,451,222,45,115,321,127,62,352,398),(33,399,353,63,128,322,116,46,223,452,297,415),(34,416,298,453,224,47,117,323,129,64,354,400),(35,381,355,65,130,324,118,48,225,454,299,417),(36,418,300,455,226,49,119,325,131,66,356,382),(37,383,357,67,132,326,120,50,227,456,281,419),(38,420,282,457,228,51,101,327,133,68,358,384),(39,385,359,69,134,328,102,52,229,458,283,401),(40,402,284,459,230,53,103,329,135,70,360,386)], [(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)], [(1,295,192,136),(2,125,193,284),(3,293,194,134),(4,123,195,282),(5,291,196,132),(6,121,197,300),(7,289,198,130),(8,139,199,298),(9,287,200,128),(10,137,181,296),(11,285,182,126),(12,135,183,294),(13,283,184,124),(14,133,185,292),(15,281,186,122),(16,131,187,290),(17,299,188,140),(18,129,189,288),(19,297,190,138),(20,127,191,286),(21,165,114,215),(22,204,115,174),(23,163,116,213),(24,202,117,172),(25,161,118,211),(26,220,119,170),(27,179,120,209),(28,218,101,168),(29,177,102,207),(30,216,103,166),(31,175,104,205),(32,214,105,164),(33,173,106,203),(34,212,107,162),(35,171,108,201),(36,210,109,180),(37,169,110,219),(38,208,111,178),(39,167,112,217),(40,206,113,176),(41,318,384,466),(42,475,385,307),(43,316,386,464),(44,473,387,305),(45,314,388,462),(46,471,389,303),(47,312,390,480),(48,469,391,301),(49,310,392,478),(50,467,393,319),(51,308,394,476),(52,465,395,317),(53,306,396,474),(54,463,397,315),(55,304,398,472),(56,461,399,313),(57,302,400,470),(58,479,381,311),(59,320,382,468),(60,477,383,309),(61,271,460,148),(62,157,441,280),(63,269,442,146),(64,155,443,278),(65,267,444,144),(66,153,445,276),(67,265,446,142),(68,151,447,274),(69,263,448,160),(70,149,449,272),(71,261,450,158),(72,147,451,270),(73,279,452,156),(74,145,453,268),(75,277,454,154),(76,143,455,266),(77,275,456,152),(78,141,457,264),(79,273,458,150),(80,159,459,262),(81,360,373,240),(82,229,374,349),(83,358,375,238),(84,227,376,347),(85,356,377,236),(86,225,378,345),(87,354,379,234),(88,223,380,343),(89,352,361,232),(90,221,362,341),(91,350,363,230),(92,239,364,359),(93,348,365,228),(94,237,366,357),(95,346,367,226),(96,235,368,355),(97,344,369,224),(98,233,370,353),(99,342,371,222),(100,231,372,351),(241,411,434,328),(242,337,435,420),(243,409,436,326),(244,335,437,418),(245,407,438,324),(246,333,439,416),(247,405,440,322),(248,331,421,414),(249,403,422,340),(250,329,423,412),(251,401,424,338),(252,327,425,410),(253,419,426,336),(254,325,427,408),(255,417,428,334),(256,323,429,406),(257,415,430,332),(258,321,431,404),(259,413,432,330),(260,339,433,402)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 20 | 20 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | Dic10 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | Q8⋊2D5 | C15⋊Q8 | C2×S3×D5 | D20⋊S3 | D12⋊D5 |
| kernel | C12.Dic10 | C30.Q8 | C6.Dic10 | C3×C4⋊Dic5 | C5×C4⋊Dic3 | C4×Dic15 | C4⋊Dic5 | C60 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C12.Dic10 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 20 | 60 |
| 57 | 27 | 0 | 0 | 0 | 0 |
| 31 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 36 | 0 | 0 |
| 0 | 0 | 27 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 19 |
| 0 | 0 | 0 | 0 | 25 | 21 |
| 38 | 15 | 0 | 0 | 0 | 0 |
| 46 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 4 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 9 |
| 0 | 0 | 0 | 0 | 46 | 45 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,6,60],[57,31,0,0,0,0,27,4,0,0,0,0,0,0,29,27,0,0,0,0,36,2,0,0,0,0,0,0,40,25,0,0,0,0,19,21],[38,46,0,0,0,0,15,23,0,0,0,0,0,0,4,4,0,0,0,0,11,57,0,0,0,0,0,0,16,46,0,0,0,0,9,45] >;
C12.Dic10 in GAP, Magma, Sage, TeX
C_{12}.{\rm Dic}_{10} % in TeX
G:=Group("C12.Dic10"); // GroupNames label
G:=SmallGroup(480,460);
// by ID
G=gap.SmallGroup(480,460);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,176,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=b^20=1,c^2=a^6*b^10,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^6*b^-1>;
// generators/relations