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## G = C12×C5⋊2C8order 480 = 25·3·5

### Direct product of C12 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C12×C5⋊2C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C6×C5⋊2C8 — C12×C5⋊2C8
 Lower central C5 — C12×C5⋊2C8
 Upper central C1 — C4×C12

Generators and relations for C12×C52C8
G = < a,b,c | a12=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 144 in 88 conjugacy classes, 74 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C2×C6, C15, C42, C2×C8, C20, C2×C10, C24, C2×C12, C2×C12, C30, C30, C4×C8, C52C8, C2×C20, C2×C20, C4×C12, C2×C24, C60, C2×C30, C2×C52C8, C4×C20, C4×C24, C3×C52C8, C2×C60, C2×C60, C4×C52C8, C6×C52C8, C4×C60, C12×C52C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D5, C12, C2×C6, C42, C2×C8, Dic5, D10, C24, C2×C12, C3×D5, C4×C8, C52C8, C4×D5, C2×Dic5, C4×C12, C2×C24, C3×Dic5, C6×D5, C2×C52C8, C4×Dic5, C4×C24, C3×C52C8, D5×C12, C6×Dic5, C4×C52C8, C6×C52C8, C12×Dic5, C12×C52C8

Smallest permutation representation of C12×C52C8
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 88 259 23 352)(2 89 260 24 353)(3 90 261 13 354)(4 91 262 14 355)(5 92 263 15 356)(6 93 264 16 357)(7 94 253 17 358)(8 95 254 18 359)(9 96 255 19 360)(10 85 256 20 349)(11 86 257 21 350)(12 87 258 22 351)(25 432 383 216 59)(26 421 384 205 60)(27 422 373 206 49)(28 423 374 207 50)(29 424 375 208 51)(30 425 376 209 52)(31 426 377 210 53)(32 427 378 211 54)(33 428 379 212 55)(34 429 380 213 56)(35 430 381 214 57)(36 431 382 215 58)(37 226 361 337 289)(38 227 362 338 290)(39 228 363 339 291)(40 217 364 340 292)(41 218 365 341 293)(42 219 366 342 294)(43 220 367 343 295)(44 221 368 344 296)(45 222 369 345 297)(46 223 370 346 298)(47 224 371 347 299)(48 225 372 348 300)(61 401 230 310 452)(62 402 231 311 453)(63 403 232 312 454)(64 404 233 301 455)(65 405 234 302 456)(66 406 235 303 445)(67 407 236 304 446)(68 408 237 305 447)(69 397 238 306 448)(70 398 239 307 449)(71 399 240 308 450)(72 400 229 309 451)(73 323 336 416 143)(74 324 325 417 144)(75 313 326 418 133)(76 314 327 419 134)(77 315 328 420 135)(78 316 329 409 136)(79 317 330 410 137)(80 318 331 411 138)(81 319 332 412 139)(82 320 333 413 140)(83 321 334 414 141)(84 322 335 415 142)(97 122 277 249 162)(98 123 278 250 163)(99 124 279 251 164)(100 125 280 252 165)(101 126 281 241 166)(102 127 282 242 167)(103 128 283 243 168)(104 129 284 244 157)(105 130 285 245 158)(106 131 286 246 159)(107 132 287 247 160)(108 121 288 248 161)(109 202 272 181 395)(110 203 273 182 396)(111 204 274 183 385)(112 193 275 184 386)(113 194 276 185 387)(114 195 265 186 388)(115 196 266 187 389)(116 197 267 188 390)(117 198 268 189 391)(118 199 269 190 392)(119 200 270 191 393)(120 201 271 192 394)(145 470 468 438 173)(146 471 457 439 174)(147 472 458 440 175)(148 473 459 441 176)(149 474 460 442 177)(150 475 461 443 178)(151 476 462 444 179)(152 477 463 433 180)(153 478 464 434 169)(154 479 465 435 170)(155 480 466 436 171)(156 469 467 437 172)
(1 376 123 174 228 456 196 313)(2 377 124 175 217 445 197 314)(3 378 125 176 218 446 198 315)(4 379 126 177 219 447 199 316)(5 380 127 178 220 448 200 317)(6 381 128 179 221 449 201 318)(7 382 129 180 222 450 202 319)(8 383 130 169 223 451 203 320)(9 384 131 170 224 452 204 321)(10 373 132 171 225 453 193 322)(11 374 121 172 226 454 194 323)(12 375 122 173 227 455 195 324)(13 54 165 473 293 407 391 420)(14 55 166 474 294 408 392 409)(15 56 167 475 295 397 393 410)(16 57 168 476 296 398 394 411)(17 58 157 477 297 399 395 412)(18 59 158 478 298 400 396 413)(19 60 159 479 299 401 385 414)(20 49 160 480 300 402 386 415)(21 50 161 469 289 403 387 416)(22 51 162 470 290 404 388 417)(23 52 163 471 291 405 389 418)(24 53 164 472 292 406 390 419)(25 245 464 346 229 182 140 254)(26 246 465 347 230 183 141 255)(27 247 466 348 231 184 142 256)(28 248 467 337 232 185 143 257)(29 249 468 338 233 186 144 258)(30 250 457 339 234 187 133 259)(31 251 458 340 235 188 134 260)(32 252 459 341 236 189 135 261)(33 241 460 342 237 190 136 262)(34 242 461 343 238 191 137 263)(35 243 462 344 239 192 138 264)(36 244 463 345 240 181 139 253)(37 63 113 336 350 207 108 156)(38 64 114 325 351 208 97 145)(39 65 115 326 352 209 98 146)(40 66 116 327 353 210 99 147)(41 67 117 328 354 211 100 148)(42 68 118 329 355 212 101 149)(43 69 119 330 356 213 102 150)(44 70 120 331 357 214 103 151)(45 71 109 332 358 215 104 152)(46 72 110 333 359 216 105 153)(47 61 111 334 360 205 106 154)(48 62 112 335 349 206 107 155)(73 86 423 288 437 361 312 276)(74 87 424 277 438 362 301 265)(75 88 425 278 439 363 302 266)(76 89 426 279 440 364 303 267)(77 90 427 280 441 365 304 268)(78 91 428 281 442 366 305 269)(79 92 429 282 443 367 306 270)(80 93 430 283 444 368 307 271)(81 94 431 284 433 369 308 272)(82 95 432 285 434 370 309 273)(83 96 421 286 435 371 310 274)(84 85 422 287 436 372 311 275)

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

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

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

192 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4L 5A 5B 6A ··· 6F 8A ··· 8P 10A ··· 10F 12A ··· 12X 15A 15B 15C 15D 20A ··· 20X 24A ··· 24AF 30A ··· 30L 60A ··· 60AV order 1 2 2 2 3 3 4 ··· 4 5 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 1 ··· 1 2 2 1 ··· 1 5 ··· 5 2 ··· 2 1 ··· 1 2 2 2 2 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

192 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D5 Dic5 D10 C3×D5 C5⋊2C8 C4×D5 C3×Dic5 C6×D5 C3×C5⋊2C8 D5×C12 kernel C12×C5⋊2C8 C6×C5⋊2C8 C4×C60 C4×C5⋊2C8 C3×C5⋊2C8 C2×C60 C2×C5⋊2C8 C4×C20 C60 C5⋊2C8 C2×C20 C20 C4×C12 C2×C12 C2×C12 C42 C12 C12 C2×C4 C2×C4 C4 C4 # reps 1 2 1 2 8 4 4 2 16 16 8 32 2 4 2 4 16 8 8 4 32 16

Matrix representation of C12×C52C8 in GL4(𝔽241) generated by

 16 0 0 0 0 64 0 0 0 0 226 0 0 0 0 226
,
 1 0 0 0 0 1 0 0 0 0 51 240 0 0 1 0
,
 211 0 0 0 0 1 0 0 0 0 1 34 0 0 85 240
G:=sub<GL(4,GF(241))| [16,0,0,0,0,64,0,0,0,0,226,0,0,0,0,226],[1,0,0,0,0,1,0,0,0,0,51,1,0,0,240,0],[211,0,0,0,0,1,0,0,0,0,1,85,0,0,34,240] >;

C12×C52C8 in GAP, Magma, Sage, TeX

C_{12}\times C_5\rtimes_2C_8
% in TeX

G:=Group("C12xC5:2C8");
// GroupNames label

G:=SmallGroup(480,80);
// by ID

G=gap.SmallGroup(480,80);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,176,136,18822]);
// 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^-1>;
// generators/relations

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