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G = C10.Dic12order 480 = 25·3·5

1st non-split extension by C10 of Dic12 acting via Dic12/Dic6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.57D4, C30.4Q16, Dic61Dic5, 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), C156(Q8⋊C4), (C2×C10).31D12, C31(Q8⋊Dic5), (C2×Dic6).1D5, C605C4.21C2, C6.1(C5⋊Q16), C54(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×C52C8).2C2, (C2×C52C8).1S3, (C2×C4).135(S3×D5), (C2×C6).26(C5⋊D4), SmallGroup(480,49)

Series: Derived Chief Lower central Upper central

C1C60 — C10.Dic12
C1C5C15C30C60C2×C60C6×C52C8 — C10.Dic12
C15C30C60 — C10.Dic12
C1C22C2×C4

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 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3 [×3], C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20 [×2], C20 [×2], C2×C10, C24, Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12, C30 [×3], Q8⋊C4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×3], C4⋊Dic3, C2×C24, C2×Dic6, C5×Dic3 [×2], Dic15, C60 [×2], C2×C30, C2×C52C8, C4⋊Dic5, Q8×C10, C2.Dic12, C3×C52C8, C5×Dic6 [×2], C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, Q8⋊Dic5, C6×C52C8, C605C4, C10×Dic6, C10.Dic12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, SD16, Q16, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×Dic5, C5⋊D4 [×2], 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

Smallest permutation representation of C10.Dic12
Regular action on 480 points
Generators in S480
(1 89 117 252 393 379 189 412 221 479)(2 480 222 413 190 380 394 253 118 90)(3 91 119 254 395 381 191 414 223 457)(4 458 224 415 192 382 396 255 120 92)(5 93 97 256 397 383 169 416 225 459)(6 460 226 417 170 384 398 257 98 94)(7 95 99 258 399 361 171 418 227 461)(8 462 228 419 172 362 400 259 100 96)(9 73 101 260 401 363 173 420 229 463)(10 464 230 421 174 364 402 261 102 74)(11 75 103 262 403 365 175 422 231 465)(12 466 232 423 176 366 404 263 104 76)(13 77 105 264 405 367 177 424 233 467)(14 468 234 425 178 368 406 241 106 78)(15 79 107 242 407 369 179 426 235 469)(16 470 236 427 180 370 408 243 108 80)(17 81 109 244 385 371 181 428 237 471)(18 472 238 429 182 372 386 245 110 82)(19 83 111 246 387 373 183 430 239 473)(20 474 240 431 184 374 388 247 112 84)(21 85 113 248 389 375 185 432 217 475)(22 476 218 409 186 376 390 249 114 86)(23 87 115 250 391 377 187 410 219 477)(24 478 220 411 188 378 392 251 116 88)(25 72 283 341 453 317 154 214 144 307)(26 308 121 215 155 318 454 342 284 49)(27 50 285 343 455 319 156 216 122 309)(28 310 123 193 157 320 456 344 286 51)(29 52 287 345 433 321 158 194 124 311)(30 312 125 195 159 322 434 346 288 53)(31 54 265 347 435 323 160 196 126 289)(32 290 127 197 161 324 436 348 266 55)(33 56 267 349 437 325 162 198 128 291)(34 292 129 199 163 326 438 350 268 57)(35 58 269 351 439 327 164 200 130 293)(36 294 131 201 165 328 440 352 270 59)(37 60 271 353 441 329 166 202 132 295)(38 296 133 203 167 330 442 354 272 61)(39 62 273 355 443 331 168 204 134 297)(40 298 135 205 145 332 444 356 274 63)(41 64 275 357 445 333 146 206 136 299)(42 300 137 207 147 334 446 358 276 65)(43 66 277 359 447 335 148 208 138 301)(44 302 139 209 149 336 448 360 278 67)(45 68 279 337 449 313 150 210 140 303)(46 304 141 211 151 314 450 338 280 69)(47 70 281 339 451 315 152 212 142 305)(48 306 143 213 153 316 452 340 282 71)
(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)
(1 316 13 328)(2 47 14 35)(3 314 15 326)(4 45 16 33)(5 336 17 324)(6 43 18 31)(7 334 19 322)(8 41 20 29)(9 332 21 320)(10 39 22 27)(11 330 23 318)(12 37 24 25)(26 365 38 377)(28 363 40 375)(30 361 42 373)(32 383 44 371)(34 381 46 369)(36 379 48 367)(49 403 61 391)(50 464 62 476)(51 401 63 389)(52 462 64 474)(53 399 65 387)(54 460 66 472)(55 397 67 385)(56 458 68 470)(57 395 69 407)(58 480 70 468)(59 393 71 405)(60 478 72 466)(73 444 85 456)(74 297 86 309)(75 442 87 454)(76 295 88 307)(77 440 89 452)(78 293 90 305)(79 438 91 450)(80 291 92 303)(81 436 93 448)(82 289 94 301)(83 434 95 446)(84 311 96 299)(97 360 109 348)(98 138 110 126)(99 358 111 346)(100 136 112 124)(101 356 113 344)(102 134 114 122)(103 354 115 342)(104 132 116 144)(105 352 117 340)(106 130 118 142)(107 350 119 338)(108 128 120 140)(121 422 133 410)(123 420 135 432)(125 418 137 430)(127 416 139 428)(129 414 141 426)(131 412 143 424)(145 475 157 463)(146 388 158 400)(147 473 159 461)(148 386 160 398)(149 471 161 459)(150 408 162 396)(151 469 163 457)(152 406 164 394)(153 467 165 479)(154 404 166 392)(155 465 167 477)(156 402 168 390)(169 302 181 290)(170 447 182 435)(171 300 183 312)(172 445 184 433)(173 298 185 310)(174 443 186 455)(175 296 187 308)(176 441 188 453)(177 294 189 306)(178 439 190 451)(179 292 191 304)(180 437 192 449)(193 229 205 217)(194 259 206 247)(195 227 207 239)(196 257 208 245)(197 225 209 237)(198 255 210 243)(199 223 211 235)(200 253 212 241)(201 221 213 233)(202 251 214 263)(203 219 215 231)(204 249 216 261)(218 285 230 273)(220 283 232 271)(222 281 234 269)(224 279 236 267)(226 277 238 265)(228 275 240 287)(242 268 254 280)(244 266 256 278)(246 288 258 276)(248 286 260 274)(250 284 262 272)(252 282 264 270)(313 370 325 382)(315 368 327 380)(317 366 329 378)(319 364 331 376)(321 362 333 374)(323 384 335 372)(337 427 349 415)(339 425 351 413)(341 423 353 411)(343 421 355 409)(345 419 357 431)(347 417 359 429)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L24A···24H30A···30F60A···60H
order1222344444455666888810···101212121215152020202020···2024···2430···3060···60
size11112221212606022222101010102···2222244444412···1210···104···44···4

66 irreducible representations

dim11111222222222222222244444444
type+++++++++--++-++---++-
imageC1C2C2C2C4S3D4D4D5D6SD16Q16Dic5D10C4×S3C3⋊D4D12C5⋊D4C5⋊D4C24⋊C2Dic12S3×D5Q8⋊D5C5⋊Q16S3×Dic5C15⋊D4C5⋊D12Dic6⋊D5C5⋊Dic12
kernelC10.Dic12C6×C52C8C605C4C10×Dic6C5×Dic6C2×C52C8C60C2×C30C2×Dic6C2×C20C30C30Dic6C2×C12C20C20C2×C10C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111212242222444422222244

Matrix representation of C10.Dic12 in GL4(𝔽241) generated by

15100
1905100
002400
000240
,
12818300
16211300
006628
0021394
,
240000
024000
0021067
003631
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

׿
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