direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×Dic12, C30⋊6Q16, C40.79D6, C20.46D12, C60.184D4, C120.97C22, C60.270C23, C6⋊1(C5×Q16), C3⋊1(C10×Q16), C15⋊12(C2×Q16), C4.8(C5×D12), C8.16(S3×C10), (C2×C40).14S3, (C2×C24).6C10, C6.12(D4×C10), C12.31(C5×D4), (C2×C120).21C2, C24.18(C2×C10), C30.299(C2×D4), C10.83(C2×D12), C2.14(C10×D12), (C2×C10).55D12, (C2×C30).126D4, (C2×C20).435D6, Dic6.7(C2×C10), (C2×Dic6).4C10, C22.14(C5×D12), C12.31(C22×C10), (C2×C60).529C22, C20.234(C22×S3), (C10×Dic6).14C2, (C5×Dic6).49C22, C4.31(S3×C2×C10), (C2×C8).4(C5×S3), (C2×C6).19(C5×D4), (C2×C4).83(S3×C10), (C2×C12).95(C2×C10), SmallGroup(480,784)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×Dic12
G = < a,b,c | a10=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 260 in 120 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, C20, C20, C2×C10, C24, Dic6, Dic6, C2×Dic3, C2×C12, C30, C30, C2×Q16, C40, C2×C20, C2×C20, C5×Q8, Dic12, C2×C24, C2×Dic6, C5×Dic3, C60, C2×C30, C2×C40, C5×Q16, Q8×C10, C2×Dic12, C120, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, C10×Q16, C5×Dic12, C2×C120, C10×Dic6, C10×Dic12
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, Q16, C2×D4, C2×C10, D12, C22×S3, C5×S3, C2×Q16, C5×D4, C22×C10, Dic12, C2×D12, S3×C10, C5×Q16, D4×C10, C2×Dic12, C5×D12, S3×C2×C10, C10×Q16, C5×Dic12, C10×D12, C10×Dic12
►Regular action on 480 points
Generators in S480
(1 314 467 413 455 287 121 237 371 312)(2 315 468 414 456 288 122 238 372 289)(3 316 469 415 433 265 123 239 373 290)(4 317 470 416 434 266 124 240 374 291)(5 318 471 417 435 267 125 217 375 292)(6 319 472 418 436 268 126 218 376 293)(7 320 473 419 437 269 127 219 377 294)(8 321 474 420 438 270 128 220 378 295)(9 322 475 421 439 271 129 221 379 296)(10 323 476 422 440 272 130 222 380 297)(11 324 477 423 441 273 131 223 381 298)(12 325 478 424 442 274 132 224 382 299)(13 326 479 425 443 275 133 225 383 300)(14 327 480 426 444 276 134 226 384 301)(15 328 457 427 445 277 135 227 361 302)(16 329 458 428 446 278 136 228 362 303)(17 330 459 429 447 279 137 229 363 304)(18 331 460 430 448 280 138 230 364 305)(19 332 461 431 449 281 139 231 365 306)(20 333 462 432 450 282 140 232 366 307)(21 334 463 409 451 283 141 233 367 308)(22 335 464 410 452 284 142 234 368 309)(23 336 465 411 453 285 143 235 369 310)(24 313 466 412 454 286 144 236 370 311)(25 406 260 186 201 61 153 111 345 74)(26 407 261 187 202 62 154 112 346 75)(27 408 262 188 203 63 155 113 347 76)(28 385 263 189 204 64 156 114 348 77)(29 386 264 190 205 65 157 115 349 78)(30 387 241 191 206 66 158 116 350 79)(31 388 242 192 207 67 159 117 351 80)(32 389 243 169 208 68 160 118 352 81)(33 390 244 170 209 69 161 119 353 82)(34 391 245 171 210 70 162 120 354 83)(35 392 246 172 211 71 163 97 355 84)(36 393 247 173 212 72 164 98 356 85)(37 394 248 174 213 49 165 99 357 86)(38 395 249 175 214 50 166 100 358 87)(39 396 250 176 215 51 167 101 359 88)(40 397 251 177 216 52 168 102 360 89)(41 398 252 178 193 53 145 103 337 90)(42 399 253 179 194 54 146 104 338 91)(43 400 254 180 195 55 147 105 339 92)(44 401 255 181 196 56 148 106 340 93)(45 402 256 182 197 57 149 107 341 94)(46 403 257 183 198 58 150 108 342 95)(47 404 258 184 199 59 151 109 343 96)(48 405 259 185 200 60 152 110 344 73)
(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)
(1 256 13 244)(2 255 14 243)(3 254 15 242)(4 253 16 241)(5 252 17 264)(6 251 18 263)(7 250 19 262)(8 249 20 261)(9 248 21 260)(10 247 22 259)(11 246 23 258)(12 245 24 257)(25 379 37 367)(26 378 38 366)(27 377 39 365)(28 376 40 364)(29 375 41 363)(30 374 42 362)(31 373 43 361)(32 372 44 384)(33 371 45 383)(34 370 46 382)(35 369 47 381)(36 368 48 380)(49 409 61 421)(50 432 62 420)(51 431 63 419)(52 430 64 418)(53 429 65 417)(54 428 66 416)(55 427 67 415)(56 426 68 414)(57 425 69 413)(58 424 70 412)(59 423 71 411)(60 422 72 410)(73 222 85 234)(74 221 86 233)(75 220 87 232)(76 219 88 231)(77 218 89 230)(78 217 90 229)(79 240 91 228)(80 239 92 227)(81 238 93 226)(82 237 94 225)(83 236 95 224)(84 235 96 223)(97 285 109 273)(98 284 110 272)(99 283 111 271)(100 282 112 270)(101 281 113 269)(102 280 114 268)(103 279 115 267)(104 278 116 266)(105 277 117 265)(106 276 118 288)(107 275 119 287)(108 274 120 286)(121 341 133 353)(122 340 134 352)(123 339 135 351)(124 338 136 350)(125 337 137 349)(126 360 138 348)(127 359 139 347)(128 358 140 346)(129 357 141 345)(130 356 142 344)(131 355 143 343)(132 354 144 342)(145 447 157 435)(146 446 158 434)(147 445 159 433)(148 444 160 456)(149 443 161 455)(150 442 162 454)(151 441 163 453)(152 440 164 452)(153 439 165 451)(154 438 166 450)(155 437 167 449)(156 436 168 448)(169 315 181 327)(170 314 182 326)(171 313 183 325)(172 336 184 324)(173 335 185 323)(174 334 186 322)(175 333 187 321)(176 332 188 320)(177 331 189 319)(178 330 190 318)(179 329 191 317)(180 328 192 316)(193 459 205 471)(194 458 206 470)(195 457 207 469)(196 480 208 468)(197 479 209 467)(198 478 210 466)(199 477 211 465)(200 476 212 464)(201 475 213 463)(202 474 214 462)(203 473 215 461)(204 472 216 460)(289 401 301 389)(290 400 302 388)(291 399 303 387)(292 398 304 386)(293 397 305 385)(294 396 306 408)(295 395 307 407)(296 394 308 406)(297 393 309 405)(298 392 310 404)(299 391 311 403)(300 390 312 402)
G:=sub<Sym(480)| (1,314,467,413,455,287,121,237,371,312)(2,315,468,414,456,288,122,238,372,289)(3,316,469,415,433,265,123,239,373,290)(4,317,470,416,434,266,124,240,374,291)(5,318,471,417,435,267,125,217,375,292)(6,319,472,418,436,268,126,218,376,293)(7,320,473,419,437,269,127,219,377,294)(8,321,474,420,438,270,128,220,378,295)(9,322,475,421,439,271,129,221,379,296)(10,323,476,422,440,272,130,222,380,297)(11,324,477,423,441,273,131,223,381,298)(12,325,478,424,442,274,132,224,382,299)(13,326,479,425,443,275,133,225,383,300)(14,327,480,426,444,276,134,226,384,301)(15,328,457,427,445,277,135,227,361,302)(16,329,458,428,446,278,136,228,362,303)(17,330,459,429,447,279,137,229,363,304)(18,331,460,430,448,280,138,230,364,305)(19,332,461,431,449,281,139,231,365,306)(20,333,462,432,450,282,140,232,366,307)(21,334,463,409,451,283,141,233,367,308)(22,335,464,410,452,284,142,234,368,309)(23,336,465,411,453,285,143,235,369,310)(24,313,466,412,454,286,144,236,370,311)(25,406,260,186,201,61,153,111,345,74)(26,407,261,187,202,62,154,112,346,75)(27,408,262,188,203,63,155,113,347,76)(28,385,263,189,204,64,156,114,348,77)(29,386,264,190,205,65,157,115,349,78)(30,387,241,191,206,66,158,116,350,79)(31,388,242,192,207,67,159,117,351,80)(32,389,243,169,208,68,160,118,352,81)(33,390,244,170,209,69,161,119,353,82)(34,391,245,171,210,70,162,120,354,83)(35,392,246,172,211,71,163,97,355,84)(36,393,247,173,212,72,164,98,356,85)(37,394,248,174,213,49,165,99,357,86)(38,395,249,175,214,50,166,100,358,87)(39,396,250,176,215,51,167,101,359,88)(40,397,251,177,216,52,168,102,360,89)(41,398,252,178,193,53,145,103,337,90)(42,399,253,179,194,54,146,104,338,91)(43,400,254,180,195,55,147,105,339,92)(44,401,255,181,196,56,148,106,340,93)(45,402,256,182,197,57,149,107,341,94)(46,403,257,183,198,58,150,108,342,95)(47,404,258,184,199,59,151,109,343,96)(48,405,259,185,200,60,152,110,344,73), (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), (1,256,13,244)(2,255,14,243)(3,254,15,242)(4,253,16,241)(5,252,17,264)(6,251,18,263)(7,250,19,262)(8,249,20,261)(9,248,21,260)(10,247,22,259)(11,246,23,258)(12,245,24,257)(25,379,37,367)(26,378,38,366)(27,377,39,365)(28,376,40,364)(29,375,41,363)(30,374,42,362)(31,373,43,361)(32,372,44,384)(33,371,45,383)(34,370,46,382)(35,369,47,381)(36,368,48,380)(49,409,61,421)(50,432,62,420)(51,431,63,419)(52,430,64,418)(53,429,65,417)(54,428,66,416)(55,427,67,415)(56,426,68,414)(57,425,69,413)(58,424,70,412)(59,423,71,411)(60,422,72,410)(73,222,85,234)(74,221,86,233)(75,220,87,232)(76,219,88,231)(77,218,89,230)(78,217,90,229)(79,240,91,228)(80,239,92,227)(81,238,93,226)(82,237,94,225)(83,236,95,224)(84,235,96,223)(97,285,109,273)(98,284,110,272)(99,283,111,271)(100,282,112,270)(101,281,113,269)(102,280,114,268)(103,279,115,267)(104,278,116,266)(105,277,117,265)(106,276,118,288)(107,275,119,287)(108,274,120,286)(121,341,133,353)(122,340,134,352)(123,339,135,351)(124,338,136,350)(125,337,137,349)(126,360,138,348)(127,359,139,347)(128,358,140,346)(129,357,141,345)(130,356,142,344)(131,355,143,343)(132,354,144,342)(145,447,157,435)(146,446,158,434)(147,445,159,433)(148,444,160,456)(149,443,161,455)(150,442,162,454)(151,441,163,453)(152,440,164,452)(153,439,165,451)(154,438,166,450)(155,437,167,449)(156,436,168,448)(169,315,181,327)(170,314,182,326)(171,313,183,325)(172,336,184,324)(173,335,185,323)(174,334,186,322)(175,333,187,321)(176,332,188,320)(177,331,189,319)(178,330,190,318)(179,329,191,317)(180,328,192,316)(193,459,205,471)(194,458,206,470)(195,457,207,469)(196,480,208,468)(197,479,209,467)(198,478,210,466)(199,477,211,465)(200,476,212,464)(201,475,213,463)(202,474,214,462)(203,473,215,461)(204,472,216,460)(289,401,301,389)(290,400,302,388)(291,399,303,387)(292,398,304,386)(293,397,305,385)(294,396,306,408)(295,395,307,407)(296,394,308,406)(297,393,309,405)(298,392,310,404)(299,391,311,403)(300,390,312,402)>;
G:=Group( (1,314,467,413,455,287,121,237,371,312)(2,315,468,414,456,288,122,238,372,289)(3,316,469,415,433,265,123,239,373,290)(4,317,470,416,434,266,124,240,374,291)(5,318,471,417,435,267,125,217,375,292)(6,319,472,418,436,268,126,218,376,293)(7,320,473,419,437,269,127,219,377,294)(8,321,474,420,438,270,128,220,378,295)(9,322,475,421,439,271,129,221,379,296)(10,323,476,422,440,272,130,222,380,297)(11,324,477,423,441,273,131,223,381,298)(12,325,478,424,442,274,132,224,382,299)(13,326,479,425,443,275,133,225,383,300)(14,327,480,426,444,276,134,226,384,301)(15,328,457,427,445,277,135,227,361,302)(16,329,458,428,446,278,136,228,362,303)(17,330,459,429,447,279,137,229,363,304)(18,331,460,430,448,280,138,230,364,305)(19,332,461,431,449,281,139,231,365,306)(20,333,462,432,450,282,140,232,366,307)(21,334,463,409,451,283,141,233,367,308)(22,335,464,410,452,284,142,234,368,309)(23,336,465,411,453,285,143,235,369,310)(24,313,466,412,454,286,144,236,370,311)(25,406,260,186,201,61,153,111,345,74)(26,407,261,187,202,62,154,112,346,75)(27,408,262,188,203,63,155,113,347,76)(28,385,263,189,204,64,156,114,348,77)(29,386,264,190,205,65,157,115,349,78)(30,387,241,191,206,66,158,116,350,79)(31,388,242,192,207,67,159,117,351,80)(32,389,243,169,208,68,160,118,352,81)(33,390,244,170,209,69,161,119,353,82)(34,391,245,171,210,70,162,120,354,83)(35,392,246,172,211,71,163,97,355,84)(36,393,247,173,212,72,164,98,356,85)(37,394,248,174,213,49,165,99,357,86)(38,395,249,175,214,50,166,100,358,87)(39,396,250,176,215,51,167,101,359,88)(40,397,251,177,216,52,168,102,360,89)(41,398,252,178,193,53,145,103,337,90)(42,399,253,179,194,54,146,104,338,91)(43,400,254,180,195,55,147,105,339,92)(44,401,255,181,196,56,148,106,340,93)(45,402,256,182,197,57,149,107,341,94)(46,403,257,183,198,58,150,108,342,95)(47,404,258,184,199,59,151,109,343,96)(48,405,259,185,200,60,152,110,344,73), (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), (1,256,13,244)(2,255,14,243)(3,254,15,242)(4,253,16,241)(5,252,17,264)(6,251,18,263)(7,250,19,262)(8,249,20,261)(9,248,21,260)(10,247,22,259)(11,246,23,258)(12,245,24,257)(25,379,37,367)(26,378,38,366)(27,377,39,365)(28,376,40,364)(29,375,41,363)(30,374,42,362)(31,373,43,361)(32,372,44,384)(33,371,45,383)(34,370,46,382)(35,369,47,381)(36,368,48,380)(49,409,61,421)(50,432,62,420)(51,431,63,419)(52,430,64,418)(53,429,65,417)(54,428,66,416)(55,427,67,415)(56,426,68,414)(57,425,69,413)(58,424,70,412)(59,423,71,411)(60,422,72,410)(73,222,85,234)(74,221,86,233)(75,220,87,232)(76,219,88,231)(77,218,89,230)(78,217,90,229)(79,240,91,228)(80,239,92,227)(81,238,93,226)(82,237,94,225)(83,236,95,224)(84,235,96,223)(97,285,109,273)(98,284,110,272)(99,283,111,271)(100,282,112,270)(101,281,113,269)(102,280,114,268)(103,279,115,267)(104,278,116,266)(105,277,117,265)(106,276,118,288)(107,275,119,287)(108,274,120,286)(121,341,133,353)(122,340,134,352)(123,339,135,351)(124,338,136,350)(125,337,137,349)(126,360,138,348)(127,359,139,347)(128,358,140,346)(129,357,141,345)(130,356,142,344)(131,355,143,343)(132,354,144,342)(145,447,157,435)(146,446,158,434)(147,445,159,433)(148,444,160,456)(149,443,161,455)(150,442,162,454)(151,441,163,453)(152,440,164,452)(153,439,165,451)(154,438,166,450)(155,437,167,449)(156,436,168,448)(169,315,181,327)(170,314,182,326)(171,313,183,325)(172,336,184,324)(173,335,185,323)(174,334,186,322)(175,333,187,321)(176,332,188,320)(177,331,189,319)(178,330,190,318)(179,329,191,317)(180,328,192,316)(193,459,205,471)(194,458,206,470)(195,457,207,469)(196,480,208,468)(197,479,209,467)(198,478,210,466)(199,477,211,465)(200,476,212,464)(201,475,213,463)(202,474,214,462)(203,473,215,461)(204,472,216,460)(289,401,301,389)(290,400,302,388)(291,399,303,387)(292,398,304,386)(293,397,305,385)(294,396,306,408)(295,395,307,407)(296,394,308,406)(297,393,309,405)(298,392,310,404)(299,391,311,403)(300,390,312,402) );
G=PermutationGroup([[(1,314,467,413,455,287,121,237,371,312),(2,315,468,414,456,288,122,238,372,289),(3,316,469,415,433,265,123,239,373,290),(4,317,470,416,434,266,124,240,374,291),(5,318,471,417,435,267,125,217,375,292),(6,319,472,418,436,268,126,218,376,293),(7,320,473,419,437,269,127,219,377,294),(8,321,474,420,438,270,128,220,378,295),(9,322,475,421,439,271,129,221,379,296),(10,323,476,422,440,272,130,222,380,297),(11,324,477,423,441,273,131,223,381,298),(12,325,478,424,442,274,132,224,382,299),(13,326,479,425,443,275,133,225,383,300),(14,327,480,426,444,276,134,226,384,301),(15,328,457,427,445,277,135,227,361,302),(16,329,458,428,446,278,136,228,362,303),(17,330,459,429,447,279,137,229,363,304),(18,331,460,430,448,280,138,230,364,305),(19,332,461,431,449,281,139,231,365,306),(20,333,462,432,450,282,140,232,366,307),(21,334,463,409,451,283,141,233,367,308),(22,335,464,410,452,284,142,234,368,309),(23,336,465,411,453,285,143,235,369,310),(24,313,466,412,454,286,144,236,370,311),(25,406,260,186,201,61,153,111,345,74),(26,407,261,187,202,62,154,112,346,75),(27,408,262,188,203,63,155,113,347,76),(28,385,263,189,204,64,156,114,348,77),(29,386,264,190,205,65,157,115,349,78),(30,387,241,191,206,66,158,116,350,79),(31,388,242,192,207,67,159,117,351,80),(32,389,243,169,208,68,160,118,352,81),(33,390,244,170,209,69,161,119,353,82),(34,391,245,171,210,70,162,120,354,83),(35,392,246,172,211,71,163,97,355,84),(36,393,247,173,212,72,164,98,356,85),(37,394,248,174,213,49,165,99,357,86),(38,395,249,175,214,50,166,100,358,87),(39,396,250,176,215,51,167,101,359,88),(40,397,251,177,216,52,168,102,360,89),(41,398,252,178,193,53,145,103,337,90),(42,399,253,179,194,54,146,104,338,91),(43,400,254,180,195,55,147,105,339,92),(44,401,255,181,196,56,148,106,340,93),(45,402,256,182,197,57,149,107,341,94),(46,403,257,183,198,58,150,108,342,95),(47,404,258,184,199,59,151,109,343,96),(48,405,259,185,200,60,152,110,344,73)], [(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)], [(1,256,13,244),(2,255,14,243),(3,254,15,242),(4,253,16,241),(5,252,17,264),(6,251,18,263),(7,250,19,262),(8,249,20,261),(9,248,21,260),(10,247,22,259),(11,246,23,258),(12,245,24,257),(25,379,37,367),(26,378,38,366),(27,377,39,365),(28,376,40,364),(29,375,41,363),(30,374,42,362),(31,373,43,361),(32,372,44,384),(33,371,45,383),(34,370,46,382),(35,369,47,381),(36,368,48,380),(49,409,61,421),(50,432,62,420),(51,431,63,419),(52,430,64,418),(53,429,65,417),(54,428,66,416),(55,427,67,415),(56,426,68,414),(57,425,69,413),(58,424,70,412),(59,423,71,411),(60,422,72,410),(73,222,85,234),(74,221,86,233),(75,220,87,232),(76,219,88,231),(77,218,89,230),(78,217,90,229),(79,240,91,228),(80,239,92,227),(81,238,93,226),(82,237,94,225),(83,236,95,224),(84,235,96,223),(97,285,109,273),(98,284,110,272),(99,283,111,271),(100,282,112,270),(101,281,113,269),(102,280,114,268),(103,279,115,267),(104,278,116,266),(105,277,117,265),(106,276,118,288),(107,275,119,287),(108,274,120,286),(121,341,133,353),(122,340,134,352),(123,339,135,351),(124,338,136,350),(125,337,137,349),(126,360,138,348),(127,359,139,347),(128,358,140,346),(129,357,141,345),(130,356,142,344),(131,355,143,343),(132,354,144,342),(145,447,157,435),(146,446,158,434),(147,445,159,433),(148,444,160,456),(149,443,161,455),(150,442,162,454),(151,441,163,453),(152,440,164,452),(153,439,165,451),(154,438,166,450),(155,437,167,449),(156,436,168,448),(169,315,181,327),(170,314,182,326),(171,313,183,325),(172,336,184,324),(173,335,185,323),(174,334,186,322),(175,333,187,321),(176,332,188,320),(177,331,189,319),(178,330,190,318),(179,329,191,317),(180,328,192,316),(193,459,205,471),(194,458,206,470),(195,457,207,469),(196,480,208,468),(197,479,209,467),(198,478,210,466),(199,477,211,465),(200,476,212,464),(201,475,213,463),(202,474,214,462),(203,473,215,461),(204,472,216,460),(289,401,301,389),(290,400,302,388),(291,399,303,387),(292,398,304,386),(293,397,305,385),(294,396,306,408),(295,395,307,407),(296,394,308,406),(297,393,309,405),(298,392,310,404),(299,391,311,403),(300,390,312,402)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20X | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | Q16 | D12 | D12 | C5×S3 | C5×D4 | C5×D4 | Dic12 | S3×C10 | S3×C10 | C5×Q16 | C5×D12 | C5×D12 | C5×Dic12 |
| kernel | C10×Dic12 | C5×Dic12 | C2×C120 | C10×Dic6 | C2×Dic12 | Dic12 | C2×C24 | C2×Dic6 | C2×C40 | C60 | C2×C30 | C40 | C2×C20 | C30 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C10×Dic12 ►in GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 240 | 0 | 0 |
| 0 | 232 | 105 |
| 0 | 136 | 127 |
| 1 | 0 | 0 |
| 0 | 20 | 131 |
| 0 | 111 | 221 |
G:=sub<GL(3,GF(241))| [240,0,0,0,36,0,0,0,36],[240,0,0,0,232,136,0,105,127],[1,0,0,0,20,111,0,131,221] >;
C10×Dic12 in GAP, Magma, Sage, TeX
C_{10}\times {\rm Dic}_{12} % in TeX
G:=Group("C10xDic12"); // GroupNames label
G:=SmallGroup(480,784);
// by ID
G=gap.SmallGroup(480,784);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,926,646,4204,102,15686]);
// Polycyclic
G:=Group<a,b,c|a^10=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations