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G = C12.Dic10order 480 = 25·3·5

8th non-split extension by C12 of Dic10 acting via Dic10/C10=C22

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, C54(C4.Dic6), C34(C4.Dic10), C30.47(C4○D4), (C2×C12).120D10, C6.9(D42D5), (C2×C30).74C23, (C2×Dic5).25D6, C6.20(C2×Dic10), C10.21(C2×Dic6), C1511(C42.C2), C10.9(D42S3), (C2×C60).197C22, C6.29(Q82D5), (C2×Dic3).24D10, (C4×Dic15).14C2, C6.Dic10.13C2, C30.Q8.15C2, C2.13(D20⋊S3), C2.13(D12⋊D5), C10.28(Q83S3), (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

C1C2×C30 — C12.Dic10
C1C5C15C30C2×C30C6×Dic5C30.Q8 — C12.Dic10
C15C2×C30 — C12.Dic10
C1C22C2×C4

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 [×3], C3, C4 [×2], C4 [×6], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×6], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C42.C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], C3×C4⋊C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], C5×C4⋊C4, C4.Dic6, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C4.Dic10, C30.Q8 [×2], C6.Dic10 [×2], C3×C4⋊Dic5, C5×C4⋊Dic3, C4×Dic15, C12.Dic10
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, C4○D4 [×2], D10 [×3], Dic6 [×2], C22×S3, C42.C2, Dic10 [×2], C22×D5, C2×Dic6, D42S3, Q83S3, S3×D5, C2×Dic10, D42D5, Q82D5, C4.Dic6, C15⋊Q8 [×2], C2×S3×D5, C4.Dic10, D20⋊S3, D12⋊D5, C2×C15⋊Q8, C12.Dic10

Smallest permutation representation of C12.Dic10
Regular action on 480 points
Generators in S480
(1 243 175 84 221 331 78 158 467 305 394 425)(2 426 395 306 468 159 79 332 222 85 176 244)(3 245 177 86 223 333 80 160 469 307 396 427)(4 428 397 308 470 141 61 334 224 87 178 246)(5 247 179 88 225 335 62 142 471 309 398 429)(6 430 399 310 472 143 63 336 226 89 180 248)(7 249 161 90 227 337 64 144 473 311 400 431)(8 432 381 312 474 145 65 338 228 91 162 250)(9 251 163 92 229 339 66 146 475 313 382 433)(10 434 383 314 476 147 67 340 230 93 164 252)(11 253 165 94 231 321 68 148 477 315 384 435)(12 436 385 316 478 149 69 322 232 95 166 254)(13 255 167 96 233 323 70 150 479 317 386 437)(14 438 387 318 480 151 71 324 234 97 168 256)(15 257 169 98 235 325 72 152 461 319 388 439)(16 440 389 320 462 153 73 326 236 99 170 258)(17 259 171 100 237 327 74 154 463 301 390 421)(18 422 391 302 464 155 75 328 238 81 172 260)(19 241 173 82 239 329 76 156 465 303 392 423)(20 424 393 304 466 157 77 330 240 83 174 242)(21 445 198 299 346 55 204 268 104 409 372 136)(22 137 373 410 105 269 205 56 347 300 199 446)(23 447 200 281 348 57 206 270 106 411 374 138)(24 139 375 412 107 271 207 58 349 282 181 448)(25 449 182 283 350 59 208 272 108 413 376 140)(26 121 377 414 109 273 209 60 351 284 183 450)(27 451 184 285 352 41 210 274 110 415 378 122)(28 123 379 416 111 275 211 42 353 286 185 452)(29 453 186 287 354 43 212 276 112 417 380 124)(30 125 361 418 113 277 213 44 355 288 187 454)(31 455 188 289 356 45 214 278 114 419 362 126)(32 127 363 420 115 279 215 46 357 290 189 456)(33 457 190 291 358 47 216 280 116 401 364 128)(34 129 365 402 117 261 217 48 359 292 191 458)(35 459 192 293 360 49 218 262 118 403 366 130)(36 131 367 404 119 263 219 50 341 294 193 460)(37 441 194 295 342 51 220 264 120 405 368 132)(38 133 369 406 101 265 201 52 343 296 195 442)(39 443 196 297 344 53 202 266 102 407 370 134)(40 135 371 408 103 267 203 54 345 298 197 444)
(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)
(1 355 68 371)(2 380 69 344)(3 353 70 369)(4 378 71 342)(5 351 72 367)(6 376 73 360)(7 349 74 365)(8 374 75 358)(9 347 76 363)(10 372 77 356)(11 345 78 361)(12 370 79 354)(13 343 80 379)(14 368 61 352)(15 341 62 377)(16 366 63 350)(17 359 64 375)(18 364 65 348)(19 357 66 373)(20 362 67 346)(21 240 214 383)(22 392 215 229)(23 238 216 381)(24 390 217 227)(25 236 218 399)(26 388 219 225)(27 234 220 397)(28 386 201 223)(29 232 202 395)(30 384 203 221)(31 230 204 393)(32 382 205 239)(33 228 206 391)(34 400 207 237)(35 226 208 389)(36 398 209 235)(37 224 210 387)(38 396 211 233)(39 222 212 385)(40 394 213 231)(41 324 132 428)(42 437 133 333)(43 322 134 426)(44 435 135 331)(45 340 136 424)(46 433 137 329)(47 338 138 422)(48 431 139 327)(49 336 140 440)(50 429 121 325)(51 334 122 438)(52 427 123 323)(53 332 124 436)(54 425 125 321)(55 330 126 434)(56 423 127 339)(57 328 128 432)(58 421 129 337)(59 326 130 430)(60 439 131 335)(81 457 312 270)(82 279 313 446)(83 455 314 268)(84 277 315 444)(85 453 316 266)(86 275 317 442)(87 451 318 264)(88 273 319 460)(89 449 320 262)(90 271 301 458)(91 447 302 280)(92 269 303 456)(93 445 304 278)(94 267 305 454)(95 443 306 276)(96 265 307 452)(97 441 308 274)(98 263 309 450)(99 459 310 272)(100 261 311 448)(101 177 185 479)(102 468 186 166)(103 175 187 477)(104 466 188 164)(105 173 189 475)(106 464 190 162)(107 171 191 473)(108 462 192 180)(109 169 193 471)(110 480 194 178)(111 167 195 469)(112 478 196 176)(113 165 197 467)(114 476 198 174)(115 163 199 465)(116 474 200 172)(117 161 181 463)(118 472 182 170)(119 179 183 461)(120 470 184 168)(141 415 256 295)(142 284 257 404)(143 413 258 293)(144 282 259 402)(145 411 260 291)(146 300 241 420)(147 409 242 289)(148 298 243 418)(149 407 244 287)(150 296 245 416)(151 405 246 285)(152 294 247 414)(153 403 248 283)(154 292 249 412)(155 401 250 281)(156 290 251 410)(157 419 252 299)(158 288 253 408)(159 417 254 297)(160 286 255 406)

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···1012121212121215152020202020···2030···3060···60
size11112221212202030303030222222···2442020202044444412···124···44···4

60 irreducible representations

dim1111112222222222444444444
type+++++++-+++++---++-+-+
imageC1C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10Dic6Dic10D42S3Q83S3S3×D5D42D5Q82D5C15⋊Q8C2×S3×D5D20⋊S3D12⋊D5
kernelC12.Dic10C30.Q8C6.Dic10C3×C4⋊Dic5C5×C4⋊Dic3C4×Dic15C4⋊Dic5C60C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12C20C12C10C10C2×C4C6C6C4C22C2C2
# reps1221111222144248112224244

Matrix representation of C12.Dic10 in GL6(𝔽61)

0600000
110000
001000
000100
000016
00002060
,
57270000
3140000
00293600
0027200
00004019
00002521
,
38150000
46230000
0041100
0045700
0000169
00004645

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

׿
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