direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C12×C5⋊C8, C60⋊4C8, C20⋊2C24, Dic5⋊2C24, C30.15C42, C5⋊2(C4×C24), C15⋊7(C4×C8), C2.2(C12×F5), C6.15(C4×F5), (C2×C60).21C4, C30.16(C2×C8), C10.7(C2×C24), C10.3(C4×C12), (C3×Dic5)⋊6C8, (C2×C12).21F5, (C2×C20).10C12, C6.13(D5⋊C8), C22.8(C6×F5), (C4×Dic5).13C6, (C2×Dic5).8C12, (C6×Dic5).24C4, Dic5.9(C2×C12), (C12×Dic5).28C2, (C6×Dic5).270C22, C2.2(C6×C5⋊C8), (C2×C5⋊C8).4C6, (C6×C5⋊C8).8C2, C6.13(C2×C5⋊C8), C2.2(C3×D5⋊C8), (C2×C4).10(C3×F5), (C2×C6).51(C2×F5), (C2×C10).3(C2×C12), (C2×C30).46(C2×C4), (C3×Dic5).53(C2×C4), (C2×Dic5).47(C2×C6), SmallGroup(480,280)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C12×C5⋊C8 |
Generators and relations for C12×C5⋊C8
G = < a,b,c | a12=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Subgroups: 200 in 88 conjugacy classes, 60 normal (32 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C30, C4×C8, C5⋊C8, C2×Dic5, C2×C20, C4×C12, C2×C24, C3×Dic5, C60, C2×C30, C4×Dic5, C2×C5⋊C8, C4×C24, C3×C5⋊C8, C6×Dic5, C2×C60, C4×C5⋊C8, C12×Dic5, C6×C5⋊C8, C12×C5⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C42, C2×C8, F5, C24, C2×C12, C4×C8, C5⋊C8, C2×F5, C4×C12, C2×C24, C3×F5, D5⋊C8, C4×F5, C2×C5⋊C8, C4×C24, C3×C5⋊C8, C6×F5, C4×C5⋊C8, C3×D5⋊C8, C12×F5, C6×C5⋊C8, C12×C5⋊C8
►Regular action on 480 points
Generators in S480
(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 52 121 273 159)(2 53 122 274 160)(3 54 123 275 161)(4 55 124 276 162)(5 56 125 265 163)(6 57 126 266 164)(7 58 127 267 165)(8 59 128 268 166)(9 60 129 269 167)(10 49 130 270 168)(11 50 131 271 157)(12 51 132 272 158)(13 235 439 155 359)(14 236 440 156 360)(15 237 441 145 349)(16 238 442 146 350)(17 239 443 147 351)(18 240 444 148 352)(19 229 433 149 353)(20 230 434 150 354)(21 231 435 151 355)(22 232 436 152 356)(23 233 437 153 357)(24 234 438 154 358)(25 141 214 244 73)(26 142 215 245 74)(27 143 216 246 75)(28 144 205 247 76)(29 133 206 248 77)(30 134 207 249 78)(31 135 208 250 79)(32 136 209 251 80)(33 137 210 252 81)(34 138 211 241 82)(35 139 212 242 83)(36 140 213 243 84)(37 254 414 107 315)(38 255 415 108 316)(39 256 416 97 317)(40 257 417 98 318)(41 258 418 99 319)(42 259 419 100 320)(43 260 420 101 321)(44 261 409 102 322)(45 262 410 103 323)(46 263 411 104 324)(47 264 412 105 313)(48 253 413 106 314)(61 459 334 345 421)(62 460 335 346 422)(63 461 336 347 423)(64 462 325 348 424)(65 463 326 337 425)(66 464 327 338 426)(67 465 328 339 427)(68 466 329 340 428)(69 467 330 341 429)(70 468 331 342 430)(71 457 332 343 431)(72 458 333 344 432)(85 288 298 304 476)(86 277 299 305 477)(87 278 300 306 478)(88 279 289 307 479)(89 280 290 308 480)(90 281 291 309 469)(91 282 292 310 470)(92 283 293 311 471)(93 284 294 312 472)(94 285 295 301 473)(95 286 296 302 474)(96 287 297 303 475)(109 454 371 385 200)(110 455 372 386 201)(111 456 361 387 202)(112 445 362 388 203)(113 446 363 389 204)(114 447 364 390 193)(115 448 365 391 194)(116 449 366 392 195)(117 450 367 393 196)(118 451 368 394 197)(119 452 369 395 198)(120 453 370 396 199)(169 401 384 187 226)(170 402 373 188 227)(171 403 374 189 228)(172 404 375 190 217)(173 405 376 191 218)(174 406 377 192 219)(175 407 378 181 220)(176 408 379 182 221)(177 397 380 183 222)(178 398 381 184 223)(179 399 382 185 224)(180 400 383 186 225)
(1 409 376 455 248 330 154 284)(2 410 377 456 249 331 155 285)(3 411 378 445 250 332 156 286)(4 412 379 446 251 333 145 287)(5 413 380 447 252 334 146 288)(6 414 381 448 241 335 147 277)(7 415 382 449 242 336 148 278)(8 416 383 450 243 325 149 279)(9 417 384 451 244 326 150 280)(10 418 373 452 245 327 151 281)(11 419 374 453 246 328 152 282)(12 420 375 454 247 329 153 283)(13 94 274 103 219 111 134 342)(14 95 275 104 220 112 135 343)(15 96 276 105 221 113 136 344)(16 85 265 106 222 114 137 345)(17 86 266 107 223 115 138 346)(18 87 267 108 224 116 139 347)(19 88 268 97 225 117 140 348)(20 89 269 98 226 118 141 337)(21 90 270 99 227 119 142 338)(22 91 271 100 228 120 143 339)(23 92 272 101 217 109 144 340)(24 93 273 102 218 110 133 341)(25 463 230 290 129 257 169 368)(26 464 231 291 130 258 170 369)(27 465 232 292 131 259 171 370)(28 466 233 293 132 260 172 371)(29 467 234 294 121 261 173 372)(30 468 235 295 122 262 174 361)(31 457 236 296 123 263 175 362)(32 458 237 297 124 264 176 363)(33 459 238 298 125 253 177 364)(34 460 239 299 126 254 178 365)(35 461 240 300 127 255 179 366)(36 462 229 289 128 256 180 367)(37 184 391 211 62 351 305 164)(38 185 392 212 63 352 306 165)(39 186 393 213 64 353 307 166)(40 187 394 214 65 354 308 167)(41 188 395 215 66 355 309 168)(42 189 396 216 67 356 310 157)(43 190 385 205 68 357 311 158)(44 191 386 206 69 358 312 159)(45 192 387 207 70 359 301 160)(46 181 388 208 71 360 302 161)(47 182 389 209 72 349 303 162)(48 183 390 210 61 350 304 163)(49 319 402 198 74 426 435 469)(50 320 403 199 75 427 436 470)(51 321 404 200 76 428 437 471)(52 322 405 201 77 429 438 472)(53 323 406 202 78 430 439 473)(54 324 407 203 79 431 440 474)(55 313 408 204 80 432 441 475)(56 314 397 193 81 421 442 476)(57 315 398 194 82 422 443 477)(58 316 399 195 83 423 444 478)(59 317 400 196 84 424 433 479)(60 318 401 197 73 425 434 480)
G:=sub<Sym(480)| (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), (1,52,121,273,159)(2,53,122,274,160)(3,54,123,275,161)(4,55,124,276,162)(5,56,125,265,163)(6,57,126,266,164)(7,58,127,267,165)(8,59,128,268,166)(9,60,129,269,167)(10,49,130,270,168)(11,50,131,271,157)(12,51,132,272,158)(13,235,439,155,359)(14,236,440,156,360)(15,237,441,145,349)(16,238,442,146,350)(17,239,443,147,351)(18,240,444,148,352)(19,229,433,149,353)(20,230,434,150,354)(21,231,435,151,355)(22,232,436,152,356)(23,233,437,153,357)(24,234,438,154,358)(25,141,214,244,73)(26,142,215,245,74)(27,143,216,246,75)(28,144,205,247,76)(29,133,206,248,77)(30,134,207,249,78)(31,135,208,250,79)(32,136,209,251,80)(33,137,210,252,81)(34,138,211,241,82)(35,139,212,242,83)(36,140,213,243,84)(37,254,414,107,315)(38,255,415,108,316)(39,256,416,97,317)(40,257,417,98,318)(41,258,418,99,319)(42,259,419,100,320)(43,260,420,101,321)(44,261,409,102,322)(45,262,410,103,323)(46,263,411,104,324)(47,264,412,105,313)(48,253,413,106,314)(61,459,334,345,421)(62,460,335,346,422)(63,461,336,347,423)(64,462,325,348,424)(65,463,326,337,425)(66,464,327,338,426)(67,465,328,339,427)(68,466,329,340,428)(69,467,330,341,429)(70,468,331,342,430)(71,457,332,343,431)(72,458,333,344,432)(85,288,298,304,476)(86,277,299,305,477)(87,278,300,306,478)(88,279,289,307,479)(89,280,290,308,480)(90,281,291,309,469)(91,282,292,310,470)(92,283,293,311,471)(93,284,294,312,472)(94,285,295,301,473)(95,286,296,302,474)(96,287,297,303,475)(109,454,371,385,200)(110,455,372,386,201)(111,456,361,387,202)(112,445,362,388,203)(113,446,363,389,204)(114,447,364,390,193)(115,448,365,391,194)(116,449,366,392,195)(117,450,367,393,196)(118,451,368,394,197)(119,452,369,395,198)(120,453,370,396,199)(169,401,384,187,226)(170,402,373,188,227)(171,403,374,189,228)(172,404,375,190,217)(173,405,376,191,218)(174,406,377,192,219)(175,407,378,181,220)(176,408,379,182,221)(177,397,380,183,222)(178,398,381,184,223)(179,399,382,185,224)(180,400,383,186,225), (1,409,376,455,248,330,154,284)(2,410,377,456,249,331,155,285)(3,411,378,445,250,332,156,286)(4,412,379,446,251,333,145,287)(5,413,380,447,252,334,146,288)(6,414,381,448,241,335,147,277)(7,415,382,449,242,336,148,278)(8,416,383,450,243,325,149,279)(9,417,384,451,244,326,150,280)(10,418,373,452,245,327,151,281)(11,419,374,453,246,328,152,282)(12,420,375,454,247,329,153,283)(13,94,274,103,219,111,134,342)(14,95,275,104,220,112,135,343)(15,96,276,105,221,113,136,344)(16,85,265,106,222,114,137,345)(17,86,266,107,223,115,138,346)(18,87,267,108,224,116,139,347)(19,88,268,97,225,117,140,348)(20,89,269,98,226,118,141,337)(21,90,270,99,227,119,142,338)(22,91,271,100,228,120,143,339)(23,92,272,101,217,109,144,340)(24,93,273,102,218,110,133,341)(25,463,230,290,129,257,169,368)(26,464,231,291,130,258,170,369)(27,465,232,292,131,259,171,370)(28,466,233,293,132,260,172,371)(29,467,234,294,121,261,173,372)(30,468,235,295,122,262,174,361)(31,457,236,296,123,263,175,362)(32,458,237,297,124,264,176,363)(33,459,238,298,125,253,177,364)(34,460,239,299,126,254,178,365)(35,461,240,300,127,255,179,366)(36,462,229,289,128,256,180,367)(37,184,391,211,62,351,305,164)(38,185,392,212,63,352,306,165)(39,186,393,213,64,353,307,166)(40,187,394,214,65,354,308,167)(41,188,395,215,66,355,309,168)(42,189,396,216,67,356,310,157)(43,190,385,205,68,357,311,158)(44,191,386,206,69,358,312,159)(45,192,387,207,70,359,301,160)(46,181,388,208,71,360,302,161)(47,182,389,209,72,349,303,162)(48,183,390,210,61,350,304,163)(49,319,402,198,74,426,435,469)(50,320,403,199,75,427,436,470)(51,321,404,200,76,428,437,471)(52,322,405,201,77,429,438,472)(53,323,406,202,78,430,439,473)(54,324,407,203,79,431,440,474)(55,313,408,204,80,432,441,475)(56,314,397,193,81,421,442,476)(57,315,398,194,82,422,443,477)(58,316,399,195,83,423,444,478)(59,317,400,196,84,424,433,479)(60,318,401,197,73,425,434,480)>;
G:=Group( (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), (1,52,121,273,159)(2,53,122,274,160)(3,54,123,275,161)(4,55,124,276,162)(5,56,125,265,163)(6,57,126,266,164)(7,58,127,267,165)(8,59,128,268,166)(9,60,129,269,167)(10,49,130,270,168)(11,50,131,271,157)(12,51,132,272,158)(13,235,439,155,359)(14,236,440,156,360)(15,237,441,145,349)(16,238,442,146,350)(17,239,443,147,351)(18,240,444,148,352)(19,229,433,149,353)(20,230,434,150,354)(21,231,435,151,355)(22,232,436,152,356)(23,233,437,153,357)(24,234,438,154,358)(25,141,214,244,73)(26,142,215,245,74)(27,143,216,246,75)(28,144,205,247,76)(29,133,206,248,77)(30,134,207,249,78)(31,135,208,250,79)(32,136,209,251,80)(33,137,210,252,81)(34,138,211,241,82)(35,139,212,242,83)(36,140,213,243,84)(37,254,414,107,315)(38,255,415,108,316)(39,256,416,97,317)(40,257,417,98,318)(41,258,418,99,319)(42,259,419,100,320)(43,260,420,101,321)(44,261,409,102,322)(45,262,410,103,323)(46,263,411,104,324)(47,264,412,105,313)(48,253,413,106,314)(61,459,334,345,421)(62,460,335,346,422)(63,461,336,347,423)(64,462,325,348,424)(65,463,326,337,425)(66,464,327,338,426)(67,465,328,339,427)(68,466,329,340,428)(69,467,330,341,429)(70,468,331,342,430)(71,457,332,343,431)(72,458,333,344,432)(85,288,298,304,476)(86,277,299,305,477)(87,278,300,306,478)(88,279,289,307,479)(89,280,290,308,480)(90,281,291,309,469)(91,282,292,310,470)(92,283,293,311,471)(93,284,294,312,472)(94,285,295,301,473)(95,286,296,302,474)(96,287,297,303,475)(109,454,371,385,200)(110,455,372,386,201)(111,456,361,387,202)(112,445,362,388,203)(113,446,363,389,204)(114,447,364,390,193)(115,448,365,391,194)(116,449,366,392,195)(117,450,367,393,196)(118,451,368,394,197)(119,452,369,395,198)(120,453,370,396,199)(169,401,384,187,226)(170,402,373,188,227)(171,403,374,189,228)(172,404,375,190,217)(173,405,376,191,218)(174,406,377,192,219)(175,407,378,181,220)(176,408,379,182,221)(177,397,380,183,222)(178,398,381,184,223)(179,399,382,185,224)(180,400,383,186,225), (1,409,376,455,248,330,154,284)(2,410,377,456,249,331,155,285)(3,411,378,445,250,332,156,286)(4,412,379,446,251,333,145,287)(5,413,380,447,252,334,146,288)(6,414,381,448,241,335,147,277)(7,415,382,449,242,336,148,278)(8,416,383,450,243,325,149,279)(9,417,384,451,244,326,150,280)(10,418,373,452,245,327,151,281)(11,419,374,453,246,328,152,282)(12,420,375,454,247,329,153,283)(13,94,274,103,219,111,134,342)(14,95,275,104,220,112,135,343)(15,96,276,105,221,113,136,344)(16,85,265,106,222,114,137,345)(17,86,266,107,223,115,138,346)(18,87,267,108,224,116,139,347)(19,88,268,97,225,117,140,348)(20,89,269,98,226,118,141,337)(21,90,270,99,227,119,142,338)(22,91,271,100,228,120,143,339)(23,92,272,101,217,109,144,340)(24,93,273,102,218,110,133,341)(25,463,230,290,129,257,169,368)(26,464,231,291,130,258,170,369)(27,465,232,292,131,259,171,370)(28,466,233,293,132,260,172,371)(29,467,234,294,121,261,173,372)(30,468,235,295,122,262,174,361)(31,457,236,296,123,263,175,362)(32,458,237,297,124,264,176,363)(33,459,238,298,125,253,177,364)(34,460,239,299,126,254,178,365)(35,461,240,300,127,255,179,366)(36,462,229,289,128,256,180,367)(37,184,391,211,62,351,305,164)(38,185,392,212,63,352,306,165)(39,186,393,213,64,353,307,166)(40,187,394,214,65,354,308,167)(41,188,395,215,66,355,309,168)(42,189,396,216,67,356,310,157)(43,190,385,205,68,357,311,158)(44,191,386,206,69,358,312,159)(45,192,387,207,70,359,301,160)(46,181,388,208,71,360,302,161)(47,182,389,209,72,349,303,162)(48,183,390,210,61,350,304,163)(49,319,402,198,74,426,435,469)(50,320,403,199,75,427,436,470)(51,321,404,200,76,428,437,471)(52,322,405,201,77,429,438,472)(53,323,406,202,78,430,439,473)(54,324,407,203,79,431,440,474)(55,313,408,204,80,432,441,475)(56,314,397,193,81,421,442,476)(57,315,398,194,82,422,443,477)(58,316,399,195,83,423,444,478)(59,317,400,196,84,424,433,479)(60,318,401,197,73,425,434,480) );
G=PermutationGroup([[(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)], [(1,52,121,273,159),(2,53,122,274,160),(3,54,123,275,161),(4,55,124,276,162),(5,56,125,265,163),(6,57,126,266,164),(7,58,127,267,165),(8,59,128,268,166),(9,60,129,269,167),(10,49,130,270,168),(11,50,131,271,157),(12,51,132,272,158),(13,235,439,155,359),(14,236,440,156,360),(15,237,441,145,349),(16,238,442,146,350),(17,239,443,147,351),(18,240,444,148,352),(19,229,433,149,353),(20,230,434,150,354),(21,231,435,151,355),(22,232,436,152,356),(23,233,437,153,357),(24,234,438,154,358),(25,141,214,244,73),(26,142,215,245,74),(27,143,216,246,75),(28,144,205,247,76),(29,133,206,248,77),(30,134,207,249,78),(31,135,208,250,79),(32,136,209,251,80),(33,137,210,252,81),(34,138,211,241,82),(35,139,212,242,83),(36,140,213,243,84),(37,254,414,107,315),(38,255,415,108,316),(39,256,416,97,317),(40,257,417,98,318),(41,258,418,99,319),(42,259,419,100,320),(43,260,420,101,321),(44,261,409,102,322),(45,262,410,103,323),(46,263,411,104,324),(47,264,412,105,313),(48,253,413,106,314),(61,459,334,345,421),(62,460,335,346,422),(63,461,336,347,423),(64,462,325,348,424),(65,463,326,337,425),(66,464,327,338,426),(67,465,328,339,427),(68,466,329,340,428),(69,467,330,341,429),(70,468,331,342,430),(71,457,332,343,431),(72,458,333,344,432),(85,288,298,304,476),(86,277,299,305,477),(87,278,300,306,478),(88,279,289,307,479),(89,280,290,308,480),(90,281,291,309,469),(91,282,292,310,470),(92,283,293,311,471),(93,284,294,312,472),(94,285,295,301,473),(95,286,296,302,474),(96,287,297,303,475),(109,454,371,385,200),(110,455,372,386,201),(111,456,361,387,202),(112,445,362,388,203),(113,446,363,389,204),(114,447,364,390,193),(115,448,365,391,194),(116,449,366,392,195),(117,450,367,393,196),(118,451,368,394,197),(119,452,369,395,198),(120,453,370,396,199),(169,401,384,187,226),(170,402,373,188,227),(171,403,374,189,228),(172,404,375,190,217),(173,405,376,191,218),(174,406,377,192,219),(175,407,378,181,220),(176,408,379,182,221),(177,397,380,183,222),(178,398,381,184,223),(179,399,382,185,224),(180,400,383,186,225)], [(1,409,376,455,248,330,154,284),(2,410,377,456,249,331,155,285),(3,411,378,445,250,332,156,286),(4,412,379,446,251,333,145,287),(5,413,380,447,252,334,146,288),(6,414,381,448,241,335,147,277),(7,415,382,449,242,336,148,278),(8,416,383,450,243,325,149,279),(9,417,384,451,244,326,150,280),(10,418,373,452,245,327,151,281),(11,419,374,453,246,328,152,282),(12,420,375,454,247,329,153,283),(13,94,274,103,219,111,134,342),(14,95,275,104,220,112,135,343),(15,96,276,105,221,113,136,344),(16,85,265,106,222,114,137,345),(17,86,266,107,223,115,138,346),(18,87,267,108,224,116,139,347),(19,88,268,97,225,117,140,348),(20,89,269,98,226,118,141,337),(21,90,270,99,227,119,142,338),(22,91,271,100,228,120,143,339),(23,92,272,101,217,109,144,340),(24,93,273,102,218,110,133,341),(25,463,230,290,129,257,169,368),(26,464,231,291,130,258,170,369),(27,465,232,292,131,259,171,370),(28,466,233,293,132,260,172,371),(29,467,234,294,121,261,173,372),(30,468,235,295,122,262,174,361),(31,457,236,296,123,263,175,362),(32,458,237,297,124,264,176,363),(33,459,238,298,125,253,177,364),(34,460,239,299,126,254,178,365),(35,461,240,300,127,255,179,366),(36,462,229,289,128,256,180,367),(37,184,391,211,62,351,305,164),(38,185,392,212,63,352,306,165),(39,186,393,213,64,353,307,166),(40,187,394,214,65,354,308,167),(41,188,395,215,66,355,309,168),(42,189,396,216,67,356,310,157),(43,190,385,205,68,357,311,158),(44,191,386,206,69,358,312,159),(45,192,387,207,70,359,301,160),(46,181,388,208,71,360,302,161),(47,182,389,209,72,349,303,162),(48,183,390,210,61,350,304,163),(49,319,402,198,74,426,435,469),(50,320,403,199,75,427,436,470),(51,321,404,200,76,428,437,471),(52,322,405,201,77,429,438,472),(53,323,406,202,78,430,439,473),(54,324,407,203,79,431,440,474),(55,313,408,204,80,432,441,475),(56,314,397,193,81,421,442,476),(57,315,398,194,82,422,443,477),(58,316,399,195,83,423,444,478),(59,317,400,196,84,424,433,479),(60,318,401,197,73,425,434,480)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 6A | ··· | 6F | 8A | ··· | 8P | 10A | 10B | 10C | 12A | ··· | 12H | 12I | ··· | 12X | 15A | 15B | 20A | 20B | 20C | 20D | 24A | ··· | 24AF | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C12 | C24 | C24 | F5 | C5⋊C8 | C2×F5 | C3×F5 | D5⋊C8 | C4×F5 | C3×C5⋊C8 | C6×F5 | C3×D5⋊C8 | C12×F5 |
| kernel | C12×C5⋊C8 | C12×Dic5 | C6×C5⋊C8 | C4×C5⋊C8 | C3×C5⋊C8 | C6×Dic5 | C2×C60 | C4×Dic5 | C2×C5⋊C8 | C3×Dic5 | C60 | C5⋊C8 | C2×Dic5 | C2×C20 | Dic5 | C20 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 4 | 4 | 16 | 16 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C12×C5⋊C8 ►in GL5(𝔽241)
| 177 | 0 | 0 | 0 | 0 |
| 0 | 181 | 0 | 0 | 0 |
| 0 | 0 | 181 | 0 | 0 |
| 0 | 0 | 0 | 181 | 0 |
| 0 | 0 | 0 | 0 | 181 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 1 | 240 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 187 | 189 | 140 | 47 |
| 0 | 86 | 236 | 59 | 234 |
| 0 | 5 | 182 | 7 | 133 |
| 0 | 194 | 81 | 54 | 52 |
G:=sub<GL(5,GF(241))| [177,0,0,0,0,0,181,0,0,0,0,0,181,0,0,0,0,0,181,0,0,0,0,0,181],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[240,0,0,0,0,0,187,86,5,194,0,189,236,182,81,0,140,59,7,54,0,47,234,133,52] >;
C12×C5⋊C8 in GAP, Magma, Sage, TeX
C_{12}\times C_5\rtimes C_8 % in TeX
G:=Group("C12xC5:C8"); // GroupNames label
G:=SmallGroup(480,280);
// by ID
G=gap.SmallGroup(480,280);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,176,136,9414,1595]);
// Polycyclic
G:=Group<a,b,c|a^12=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations