direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12×Dic10, C60⋊12Q8, C5⋊2(Q8×C12), C20⋊3(C3×Q8), C15⋊16(C4×Q8), (C4×C20).5C6, (C4×C12).9D5, C4.9(D5×C12), C10.1(C6×Q8), (C4×C60).11C2, C12.66(C4×D5), C30.73(C2×Q8), C42.3(C3×D5), C60.201(C2×C4), C20.40(C2×C12), C4⋊Dic5.13C6, (C4×Dic5).9C6, C2.1(C6×Dic10), (C2×C12).446D10, Dic5.3(C2×C12), C6.41(C2×Dic10), C10.D4.7C6, C30.180(C4○D4), C6.108(C4○D20), (C2×C30).326C23, (C2×C60).546C22, C10.14(C22×C12), C30.172(C22×C4), (C6×Dic10).22C2, (C12×Dic5).24C2, (C2×Dic10).11C6, (C6×Dic5).231C22, C6.97(C2×C4×D5), C2.4(D5×C2×C12), C22.8(D5×C2×C6), C2.1(C3×C4○D20), C10.1(C3×C4○D4), (C2×C4).73(C6×D5), (C2×C20).72(C2×C6), (C3×C4⋊Dic5).27C2, (C2×C10).9(C22×C6), (C3×Dic5).43(C2×C4), (C2×Dic5).23(C2×C6), (C2×C6).322(C22×D5), (C3×C10.D4).17C2, SmallGroup(480,661)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×Dic10
G = < a,b,c | a12=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 336 in 140 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, C12, C12, C2×C6, C15, C42, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C12, C2×C12, C3×Q8, C30, C4×Q8, Dic10, C2×Dic5, C2×C20, C4×C12, C4×C12, C3×C4⋊C4, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, Q8×C12, C3×Dic10, C6×Dic5, C2×C60, C4×Dic10, C12×Dic5, C3×C10.D4, C3×C4⋊Dic5, C4×C60, C6×Dic10, C12×Dic10
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, Q8, C23, D5, C12, C2×C6, C22×C4, C2×Q8, C4○D4, D10, C2×C12, C3×Q8, C22×C6, C3×D5, C4×Q8, Dic10, C4×D5, C22×D5, C22×C12, C6×Q8, C3×C4○D4, C6×D5, C2×Dic10, C2×C4×D5, C4○D20, Q8×C12, C3×Dic10, D5×C12, D5×C2×C6, C4×Dic10, C6×Dic10, D5×C2×C12, C3×C4○D20, C12×Dic10
►Regular action on 480 points
Generators in S480
(1 109 358 90 136 244 385 265 330 80 443 146)(2 110 359 91 137 245 386 266 331 61 444 147)(3 111 360 92 138 246 387 267 332 62 445 148)(4 112 341 93 139 247 388 268 333 63 446 149)(5 113 342 94 140 248 389 269 334 64 447 150)(6 114 343 95 121 249 390 270 335 65 448 151)(7 115 344 96 122 250 391 271 336 66 449 152)(8 116 345 97 123 251 392 272 337 67 450 153)(9 117 346 98 124 252 393 273 338 68 451 154)(10 118 347 99 125 253 394 274 339 69 452 155)(11 119 348 100 126 254 395 275 340 70 453 156)(12 120 349 81 127 255 396 276 321 71 454 157)(13 101 350 82 128 256 397 277 322 72 455 158)(14 102 351 83 129 257 398 278 323 73 456 159)(15 103 352 84 130 258 399 279 324 74 457 160)(16 104 353 85 131 259 400 280 325 75 458 141)(17 105 354 86 132 260 381 261 326 76 459 142)(18 106 355 87 133 241 382 262 327 77 460 143)(19 107 356 88 134 242 383 263 328 78 441 144)(20 108 357 89 135 243 384 264 329 79 442 145)(21 477 421 162 220 379 185 289 226 406 303 58)(22 478 422 163 201 380 186 290 227 407 304 59)(23 479 423 164 202 361 187 291 228 408 305 60)(24 480 424 165 203 362 188 292 229 409 306 41)(25 461 425 166 204 363 189 293 230 410 307 42)(26 462 426 167 205 364 190 294 231 411 308 43)(27 463 427 168 206 365 191 295 232 412 309 44)(28 464 428 169 207 366 192 296 233 413 310 45)(29 465 429 170 208 367 193 297 234 414 311 46)(30 466 430 171 209 368 194 298 235 415 312 47)(31 467 431 172 210 369 195 299 236 416 313 48)(32 468 432 173 211 370 196 300 237 417 314 49)(33 469 433 174 212 371 197 281 238 418 315 50)(34 470 434 175 213 372 198 282 239 419 316 51)(35 471 435 176 214 373 199 283 240 420 317 52)(36 472 436 177 215 374 200 284 221 401 318 53)(37 473 437 178 216 375 181 285 222 402 319 54)(38 474 438 179 217 376 182 286 223 403 320 55)(39 475 439 180 218 377 183 287 224 404 301 56)(40 476 440 161 219 378 184 288 225 405 302 57)
(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)
(1 367 11 377)(2 366 12 376)(3 365 13 375)(4 364 14 374)(5 363 15 373)(6 362 16 372)(7 361 17 371)(8 380 18 370)(9 379 19 369)(10 378 20 368)(21 263 31 273)(22 262 32 272)(23 261 33 271)(24 280 34 270)(25 279 35 269)(26 278 36 268)(27 277 37 267)(28 276 38 266)(29 275 39 265)(30 274 40 264)(41 400 51 390)(42 399 52 389)(43 398 53 388)(44 397 54 387)(45 396 55 386)(46 395 56 385)(47 394 57 384)(48 393 58 383)(49 392 59 382)(50 391 60 381)(61 428 71 438)(62 427 72 437)(63 426 73 436)(64 425 74 435)(65 424 75 434)(66 423 76 433)(67 422 77 432)(68 421 78 431)(69 440 79 430)(70 439 80 429)(81 223 91 233)(82 222 92 232)(83 221 93 231)(84 240 94 230)(85 239 95 229)(86 238 96 228)(87 237 97 227)(88 236 98 226)(89 235 99 225)(90 234 100 224)(101 181 111 191)(102 200 112 190)(103 199 113 189)(104 198 114 188)(105 197 115 187)(106 196 116 186)(107 195 117 185)(108 194 118 184)(109 193 119 183)(110 192 120 182)(121 409 131 419)(122 408 132 418)(123 407 133 417)(124 406 134 416)(125 405 135 415)(126 404 136 414)(127 403 137 413)(128 402 138 412)(129 401 139 411)(130 420 140 410)(141 213 151 203)(142 212 152 202)(143 211 153 201)(144 210 154 220)(145 209 155 219)(146 208 156 218)(147 207 157 217)(148 206 158 216)(149 205 159 215)(150 204 160 214)(161 442 171 452)(162 441 172 451)(163 460 173 450)(164 459 174 449)(165 458 175 448)(166 457 176 447)(167 456 177 446)(168 455 178 445)(169 454 179 444)(170 453 180 443)(241 314 251 304)(242 313 252 303)(243 312 253 302)(244 311 254 301)(245 310 255 320)(246 309 256 319)(247 308 257 318)(248 307 258 317)(249 306 259 316)(250 305 260 315)(281 344 291 354)(282 343 292 353)(283 342 293 352)(284 341 294 351)(285 360 295 350)(286 359 296 349)(287 358 297 348)(288 357 298 347)(289 356 299 346)(290 355 300 345)(321 474 331 464)(322 473 332 463)(323 472 333 462)(324 471 334 461)(325 470 335 480)(326 469 336 479)(327 468 337 478)(328 467 338 477)(329 466 339 476)(330 465 340 475)
G:=sub<Sym(480)| (1,109,358,90,136,244,385,265,330,80,443,146)(2,110,359,91,137,245,386,266,331,61,444,147)(3,111,360,92,138,246,387,267,332,62,445,148)(4,112,341,93,139,247,388,268,333,63,446,149)(5,113,342,94,140,248,389,269,334,64,447,150)(6,114,343,95,121,249,390,270,335,65,448,151)(7,115,344,96,122,250,391,271,336,66,449,152)(8,116,345,97,123,251,392,272,337,67,450,153)(9,117,346,98,124,252,393,273,338,68,451,154)(10,118,347,99,125,253,394,274,339,69,452,155)(11,119,348,100,126,254,395,275,340,70,453,156)(12,120,349,81,127,255,396,276,321,71,454,157)(13,101,350,82,128,256,397,277,322,72,455,158)(14,102,351,83,129,257,398,278,323,73,456,159)(15,103,352,84,130,258,399,279,324,74,457,160)(16,104,353,85,131,259,400,280,325,75,458,141)(17,105,354,86,132,260,381,261,326,76,459,142)(18,106,355,87,133,241,382,262,327,77,460,143)(19,107,356,88,134,242,383,263,328,78,441,144)(20,108,357,89,135,243,384,264,329,79,442,145)(21,477,421,162,220,379,185,289,226,406,303,58)(22,478,422,163,201,380,186,290,227,407,304,59)(23,479,423,164,202,361,187,291,228,408,305,60)(24,480,424,165,203,362,188,292,229,409,306,41)(25,461,425,166,204,363,189,293,230,410,307,42)(26,462,426,167,205,364,190,294,231,411,308,43)(27,463,427,168,206,365,191,295,232,412,309,44)(28,464,428,169,207,366,192,296,233,413,310,45)(29,465,429,170,208,367,193,297,234,414,311,46)(30,466,430,171,209,368,194,298,235,415,312,47)(31,467,431,172,210,369,195,299,236,416,313,48)(32,468,432,173,211,370,196,300,237,417,314,49)(33,469,433,174,212,371,197,281,238,418,315,50)(34,470,434,175,213,372,198,282,239,419,316,51)(35,471,435,176,214,373,199,283,240,420,317,52)(36,472,436,177,215,374,200,284,221,401,318,53)(37,473,437,178,216,375,181,285,222,402,319,54)(38,474,438,179,217,376,182,286,223,403,320,55)(39,475,439,180,218,377,183,287,224,404,301,56)(40,476,440,161,219,378,184,288,225,405,302,57), (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), (1,367,11,377)(2,366,12,376)(3,365,13,375)(4,364,14,374)(5,363,15,373)(6,362,16,372)(7,361,17,371)(8,380,18,370)(9,379,19,369)(10,378,20,368)(21,263,31,273)(22,262,32,272)(23,261,33,271)(24,280,34,270)(25,279,35,269)(26,278,36,268)(27,277,37,267)(28,276,38,266)(29,275,39,265)(30,274,40,264)(41,400,51,390)(42,399,52,389)(43,398,53,388)(44,397,54,387)(45,396,55,386)(46,395,56,385)(47,394,57,384)(48,393,58,383)(49,392,59,382)(50,391,60,381)(61,428,71,438)(62,427,72,437)(63,426,73,436)(64,425,74,435)(65,424,75,434)(66,423,76,433)(67,422,77,432)(68,421,78,431)(69,440,79,430)(70,439,80,429)(81,223,91,233)(82,222,92,232)(83,221,93,231)(84,240,94,230)(85,239,95,229)(86,238,96,228)(87,237,97,227)(88,236,98,226)(89,235,99,225)(90,234,100,224)(101,181,111,191)(102,200,112,190)(103,199,113,189)(104,198,114,188)(105,197,115,187)(106,196,116,186)(107,195,117,185)(108,194,118,184)(109,193,119,183)(110,192,120,182)(121,409,131,419)(122,408,132,418)(123,407,133,417)(124,406,134,416)(125,405,135,415)(126,404,136,414)(127,403,137,413)(128,402,138,412)(129,401,139,411)(130,420,140,410)(141,213,151,203)(142,212,152,202)(143,211,153,201)(144,210,154,220)(145,209,155,219)(146,208,156,218)(147,207,157,217)(148,206,158,216)(149,205,159,215)(150,204,160,214)(161,442,171,452)(162,441,172,451)(163,460,173,450)(164,459,174,449)(165,458,175,448)(166,457,176,447)(167,456,177,446)(168,455,178,445)(169,454,179,444)(170,453,180,443)(241,314,251,304)(242,313,252,303)(243,312,253,302)(244,311,254,301)(245,310,255,320)(246,309,256,319)(247,308,257,318)(248,307,258,317)(249,306,259,316)(250,305,260,315)(281,344,291,354)(282,343,292,353)(283,342,293,352)(284,341,294,351)(285,360,295,350)(286,359,296,349)(287,358,297,348)(288,357,298,347)(289,356,299,346)(290,355,300,345)(321,474,331,464)(322,473,332,463)(323,472,333,462)(324,471,334,461)(325,470,335,480)(326,469,336,479)(327,468,337,478)(328,467,338,477)(329,466,339,476)(330,465,340,475)>;
G:=Group( (1,109,358,90,136,244,385,265,330,80,443,146)(2,110,359,91,137,245,386,266,331,61,444,147)(3,111,360,92,138,246,387,267,332,62,445,148)(4,112,341,93,139,247,388,268,333,63,446,149)(5,113,342,94,140,248,389,269,334,64,447,150)(6,114,343,95,121,249,390,270,335,65,448,151)(7,115,344,96,122,250,391,271,336,66,449,152)(8,116,345,97,123,251,392,272,337,67,450,153)(9,117,346,98,124,252,393,273,338,68,451,154)(10,118,347,99,125,253,394,274,339,69,452,155)(11,119,348,100,126,254,395,275,340,70,453,156)(12,120,349,81,127,255,396,276,321,71,454,157)(13,101,350,82,128,256,397,277,322,72,455,158)(14,102,351,83,129,257,398,278,323,73,456,159)(15,103,352,84,130,258,399,279,324,74,457,160)(16,104,353,85,131,259,400,280,325,75,458,141)(17,105,354,86,132,260,381,261,326,76,459,142)(18,106,355,87,133,241,382,262,327,77,460,143)(19,107,356,88,134,242,383,263,328,78,441,144)(20,108,357,89,135,243,384,264,329,79,442,145)(21,477,421,162,220,379,185,289,226,406,303,58)(22,478,422,163,201,380,186,290,227,407,304,59)(23,479,423,164,202,361,187,291,228,408,305,60)(24,480,424,165,203,362,188,292,229,409,306,41)(25,461,425,166,204,363,189,293,230,410,307,42)(26,462,426,167,205,364,190,294,231,411,308,43)(27,463,427,168,206,365,191,295,232,412,309,44)(28,464,428,169,207,366,192,296,233,413,310,45)(29,465,429,170,208,367,193,297,234,414,311,46)(30,466,430,171,209,368,194,298,235,415,312,47)(31,467,431,172,210,369,195,299,236,416,313,48)(32,468,432,173,211,370,196,300,237,417,314,49)(33,469,433,174,212,371,197,281,238,418,315,50)(34,470,434,175,213,372,198,282,239,419,316,51)(35,471,435,176,214,373,199,283,240,420,317,52)(36,472,436,177,215,374,200,284,221,401,318,53)(37,473,437,178,216,375,181,285,222,402,319,54)(38,474,438,179,217,376,182,286,223,403,320,55)(39,475,439,180,218,377,183,287,224,404,301,56)(40,476,440,161,219,378,184,288,225,405,302,57), (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), (1,367,11,377)(2,366,12,376)(3,365,13,375)(4,364,14,374)(5,363,15,373)(6,362,16,372)(7,361,17,371)(8,380,18,370)(9,379,19,369)(10,378,20,368)(21,263,31,273)(22,262,32,272)(23,261,33,271)(24,280,34,270)(25,279,35,269)(26,278,36,268)(27,277,37,267)(28,276,38,266)(29,275,39,265)(30,274,40,264)(41,400,51,390)(42,399,52,389)(43,398,53,388)(44,397,54,387)(45,396,55,386)(46,395,56,385)(47,394,57,384)(48,393,58,383)(49,392,59,382)(50,391,60,381)(61,428,71,438)(62,427,72,437)(63,426,73,436)(64,425,74,435)(65,424,75,434)(66,423,76,433)(67,422,77,432)(68,421,78,431)(69,440,79,430)(70,439,80,429)(81,223,91,233)(82,222,92,232)(83,221,93,231)(84,240,94,230)(85,239,95,229)(86,238,96,228)(87,237,97,227)(88,236,98,226)(89,235,99,225)(90,234,100,224)(101,181,111,191)(102,200,112,190)(103,199,113,189)(104,198,114,188)(105,197,115,187)(106,196,116,186)(107,195,117,185)(108,194,118,184)(109,193,119,183)(110,192,120,182)(121,409,131,419)(122,408,132,418)(123,407,133,417)(124,406,134,416)(125,405,135,415)(126,404,136,414)(127,403,137,413)(128,402,138,412)(129,401,139,411)(130,420,140,410)(141,213,151,203)(142,212,152,202)(143,211,153,201)(144,210,154,220)(145,209,155,219)(146,208,156,218)(147,207,157,217)(148,206,158,216)(149,205,159,215)(150,204,160,214)(161,442,171,452)(162,441,172,451)(163,460,173,450)(164,459,174,449)(165,458,175,448)(166,457,176,447)(167,456,177,446)(168,455,178,445)(169,454,179,444)(170,453,180,443)(241,314,251,304)(242,313,252,303)(243,312,253,302)(244,311,254,301)(245,310,255,320)(246,309,256,319)(247,308,257,318)(248,307,258,317)(249,306,259,316)(250,305,260,315)(281,344,291,354)(282,343,292,353)(283,342,293,352)(284,341,294,351)(285,360,295,350)(286,359,296,349)(287,358,297,348)(288,357,298,347)(289,356,299,346)(290,355,300,345)(321,474,331,464)(322,473,332,463)(323,472,333,462)(324,471,334,461)(325,470,335,480)(326,469,336,479)(327,468,337,478)(328,467,338,477)(329,466,339,476)(330,465,340,475) );
G=PermutationGroup([[(1,109,358,90,136,244,385,265,330,80,443,146),(2,110,359,91,137,245,386,266,331,61,444,147),(3,111,360,92,138,246,387,267,332,62,445,148),(4,112,341,93,139,247,388,268,333,63,446,149),(5,113,342,94,140,248,389,269,334,64,447,150),(6,114,343,95,121,249,390,270,335,65,448,151),(7,115,344,96,122,250,391,271,336,66,449,152),(8,116,345,97,123,251,392,272,337,67,450,153),(9,117,346,98,124,252,393,273,338,68,451,154),(10,118,347,99,125,253,394,274,339,69,452,155),(11,119,348,100,126,254,395,275,340,70,453,156),(12,120,349,81,127,255,396,276,321,71,454,157),(13,101,350,82,128,256,397,277,322,72,455,158),(14,102,351,83,129,257,398,278,323,73,456,159),(15,103,352,84,130,258,399,279,324,74,457,160),(16,104,353,85,131,259,400,280,325,75,458,141),(17,105,354,86,132,260,381,261,326,76,459,142),(18,106,355,87,133,241,382,262,327,77,460,143),(19,107,356,88,134,242,383,263,328,78,441,144),(20,108,357,89,135,243,384,264,329,79,442,145),(21,477,421,162,220,379,185,289,226,406,303,58),(22,478,422,163,201,380,186,290,227,407,304,59),(23,479,423,164,202,361,187,291,228,408,305,60),(24,480,424,165,203,362,188,292,229,409,306,41),(25,461,425,166,204,363,189,293,230,410,307,42),(26,462,426,167,205,364,190,294,231,411,308,43),(27,463,427,168,206,365,191,295,232,412,309,44),(28,464,428,169,207,366,192,296,233,413,310,45),(29,465,429,170,208,367,193,297,234,414,311,46),(30,466,430,171,209,368,194,298,235,415,312,47),(31,467,431,172,210,369,195,299,236,416,313,48),(32,468,432,173,211,370,196,300,237,417,314,49),(33,469,433,174,212,371,197,281,238,418,315,50),(34,470,434,175,213,372,198,282,239,419,316,51),(35,471,435,176,214,373,199,283,240,420,317,52),(36,472,436,177,215,374,200,284,221,401,318,53),(37,473,437,178,216,375,181,285,222,402,319,54),(38,474,438,179,217,376,182,286,223,403,320,55),(39,475,439,180,218,377,183,287,224,404,301,56),(40,476,440,161,219,378,184,288,225,405,302,57)], [(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)], [(1,367,11,377),(2,366,12,376),(3,365,13,375),(4,364,14,374),(5,363,15,373),(6,362,16,372),(7,361,17,371),(8,380,18,370),(9,379,19,369),(10,378,20,368),(21,263,31,273),(22,262,32,272),(23,261,33,271),(24,280,34,270),(25,279,35,269),(26,278,36,268),(27,277,37,267),(28,276,38,266),(29,275,39,265),(30,274,40,264),(41,400,51,390),(42,399,52,389),(43,398,53,388),(44,397,54,387),(45,396,55,386),(46,395,56,385),(47,394,57,384),(48,393,58,383),(49,392,59,382),(50,391,60,381),(61,428,71,438),(62,427,72,437),(63,426,73,436),(64,425,74,435),(65,424,75,434),(66,423,76,433),(67,422,77,432),(68,421,78,431),(69,440,79,430),(70,439,80,429),(81,223,91,233),(82,222,92,232),(83,221,93,231),(84,240,94,230),(85,239,95,229),(86,238,96,228),(87,237,97,227),(88,236,98,226),(89,235,99,225),(90,234,100,224),(101,181,111,191),(102,200,112,190),(103,199,113,189),(104,198,114,188),(105,197,115,187),(106,196,116,186),(107,195,117,185),(108,194,118,184),(109,193,119,183),(110,192,120,182),(121,409,131,419),(122,408,132,418),(123,407,133,417),(124,406,134,416),(125,405,135,415),(126,404,136,414),(127,403,137,413),(128,402,138,412),(129,401,139,411),(130,420,140,410),(141,213,151,203),(142,212,152,202),(143,211,153,201),(144,210,154,220),(145,209,155,219),(146,208,156,218),(147,207,157,217),(148,206,158,216),(149,205,159,215),(150,204,160,214),(161,442,171,452),(162,441,172,451),(163,460,173,450),(164,459,174,449),(165,458,175,448),(166,457,176,447),(167,456,177,446),(168,455,178,445),(169,454,179,444),(170,453,180,443),(241,314,251,304),(242,313,252,303),(243,312,253,302),(244,311,254,301),(245,310,255,320),(246,309,256,319),(247,308,257,318),(248,307,258,317),(249,306,259,316),(250,305,260,315),(281,344,291,354),(282,343,292,353),(283,342,293,352),(284,341,294,351),(285,360,295,350),(286,359,296,349),(287,358,297,348),(288,357,298,347),(289,356,299,346),(290,355,300,345),(321,474,331,464),(322,473,332,463),(323,472,333,462),(324,471,334,461),(325,470,335,480),(326,469,336,479),(327,468,337,478),(328,467,338,477),(329,466,339,476),(330,465,340,475)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 6A | ··· | 6F | 10A | ··· | 10F | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AF | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | Q8 | D5 | C4○D4 | D10 | C3×Q8 | C3×D5 | Dic10 | C4×D5 | C3×C4○D4 | C6×D5 | C4○D20 | C3×Dic10 | D5×C12 | C3×C4○D20 |
| kernel | C12×Dic10 | C12×Dic5 | C3×C10.D4 | C3×C4⋊Dic5 | C4×C60 | C6×Dic10 | C4×Dic10 | C3×Dic10 | C4×Dic5 | C10.D4 | C4⋊Dic5 | C4×C20 | C2×Dic10 | Dic10 | C60 | C4×C12 | C30 | C2×C12 | C20 | C42 | C12 | C12 | C10 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 4 | 2 | 2 | 2 | 16 | 2 | 2 | 2 | 6 | 4 | 4 | 8 | 8 | 4 | 12 | 8 | 16 | 16 | 16 |
Matrix representation of C12×Dic10 ►in GL5(𝔽61)
| 11 | 0 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 19 | 43 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 45 | 32 | 0 | 0 |
| 0 | 32 | 16 | 0 | 0 |
| 0 | 0 | 0 | 52 | 37 |
| 0 | 0 | 0 | 44 | 9 |
G:=sub<GL(5,GF(61))| [11,0,0,0,0,0,21,0,0,0,0,0,21,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,60,0,0,0,0,0,0,1,19,0,0,0,60,43],[60,0,0,0,0,0,45,32,0,0,0,32,16,0,0,0,0,0,52,44,0,0,0,37,9] >;
C12×Dic10 in GAP, Magma, Sage, TeX
C_{12}\times {\rm Dic}_{10} % in TeX
G:=Group("C12xDic10"); // GroupNames label
G:=SmallGroup(480,661);
// by ID
G=gap.SmallGroup(480,661);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,701,344,142,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations