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## G = C5×C12⋊C8order 480 = 25·3·5

### Direct product of C5 and C12⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C12⋊C8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C10×C3⋊C8 — C5×C12⋊C8
 Lower central C3 — C6 — C5×C12⋊C8
 Upper central C1 — C2×C20 — C4×C20

Generators and relations for C5×C12⋊C8
G = < a,b,c | a5=b12=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 116 in 76 conjugacy classes, 58 normal (46 characteristic)
C1, C2, C3, C4, C4, C4, C22, C5, C6, C8, C2×C4, C10, C12, C12, C12, C2×C6, C15, C42, C2×C8, C20, C20, C20, C2×C10, C3⋊C8, C2×C12, C30, C4⋊C8, C40, C2×C20, C2×C3⋊C8, C4×C12, C60, C60, C60, C2×C30, C4×C20, C2×C40, C12⋊C8, C5×C3⋊C8, C2×C60, C5×C4⋊C8, C10×C3⋊C8, C4×C60, C5×C12⋊C8
Quotients:

Smallest permutation representation of C5×C12⋊C8
Regular action on 480 points
Generators in S480
(1 247 120 439 148)(2 248 109 440 149)(3 249 110 441 150)(4 250 111 442 151)(5 251 112 443 152)(6 252 113 444 153)(7 241 114 433 154)(8 242 115 434 155)(9 243 116 435 156)(10 244 117 436 145)(11 245 118 437 146)(12 246 119 438 147)(13 163 330 26 460)(14 164 331 27 461)(15 165 332 28 462)(16 166 333 29 463)(17 167 334 30 464)(18 168 335 31 465)(19 157 336 32 466)(20 158 325 33 467)(21 159 326 34 468)(22 160 327 35 457)(23 161 328 36 458)(24 162 329 25 459)(37 395 451 317 272)(38 396 452 318 273)(39 385 453 319 274)(40 386 454 320 275)(41 387 455 321 276)(42 388 456 322 265)(43 389 445 323 266)(44 390 446 324 267)(45 391 447 313 268)(46 392 448 314 269)(47 393 449 315 270)(48 394 450 316 271)(49 405 364 131 136)(50 406 365 132 137)(51 407 366 121 138)(52 408 367 122 139)(53 397 368 123 140)(54 398 369 124 141)(55 399 370 125 142)(56 400 371 126 143)(57 401 372 127 144)(58 402 361 128 133)(59 403 362 129 134)(60 404 363 130 135)(61 208 262 171 183)(62 209 263 172 184)(63 210 264 173 185)(64 211 253 174 186)(65 212 254 175 187)(66 213 255 176 188)(67 214 256 177 189)(68 215 257 178 190)(69 216 258 179 191)(70 205 259 180 192)(71 206 260 169 181)(72 207 261 170 182)(73 101 240 300 427)(74 102 229 289 428)(75 103 230 290 429)(76 104 231 291 430)(77 105 232 292 431)(78 106 233 293 432)(79 107 234 294 421)(80 108 235 295 422)(81 97 236 296 423)(82 98 237 297 424)(83 99 238 298 425)(84 100 239 299 426)(85 414 375 348 304)(86 415 376 337 305)(87 416 377 338 306)(88 417 378 339 307)(89 418 379 340 308)(90 419 380 341 309)(91 420 381 342 310)(92 409 382 343 311)(93 410 383 344 312)(94 411 384 345 301)(95 412 373 346 302)(96 413 374 347 303)(193 477 358 223 282)(194 478 359 224 283)(195 479 360 225 284)(196 480 349 226 285)(197 469 350 227 286)(198 470 351 228 287)(199 471 352 217 288)(200 472 353 218 277)(201 473 354 219 278)(202 474 355 220 279)(203 475 356 221 280)(204 476 357 222 281)
(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)
(1 379 258 81 284 159 139 322)(2 378 259 80 285 158 140 321)(3 377 260 79 286 157 141 320)(4 376 261 78 287 168 142 319)(5 375 262 77 288 167 143 318)(6 374 263 76 277 166 144 317)(7 373 264 75 278 165 133 316)(8 384 253 74 279 164 134 315)(9 383 254 73 280 163 135 314)(10 382 255 84 281 162 136 313)(11 381 256 83 282 161 137 324)(12 380 257 82 283 160 138 323)(13 130 448 156 410 212 427 221)(14 129 449 155 411 211 428 220)(15 128 450 154 412 210 429 219)(16 127 451 153 413 209 430 218)(17 126 452 152 414 208 431 217)(18 125 453 151 415 207 432 228)(19 124 454 150 416 206 421 227)(20 123 455 149 417 205 422 226)(21 122 456 148 418 216 423 225)(22 121 445 147 419 215 424 224)(23 132 446 146 420 214 425 223)(24 131 447 145 409 213 426 222)(25 405 45 117 311 188 239 476)(26 404 46 116 312 187 240 475)(27 403 47 115 301 186 229 474)(28 402 48 114 302 185 230 473)(29 401 37 113 303 184 231 472)(30 400 38 112 304 183 232 471)(31 399 39 111 305 182 233 470)(32 398 40 110 306 181 234 469)(33 397 41 109 307 192 235 480)(34 408 42 120 308 191 236 479)(35 407 43 119 309 190 237 478)(36 406 44 118 310 189 238 477)(49 268 244 343 176 100 204 329)(50 267 245 342 177 99 193 328)(51 266 246 341 178 98 194 327)(52 265 247 340 179 97 195 326)(53 276 248 339 180 108 196 325)(54 275 249 338 169 107 197 336)(55 274 250 337 170 106 198 335)(56 273 251 348 171 105 199 334)(57 272 252 347 172 104 200 333)(58 271 241 346 173 103 201 332)(59 270 242 345 174 102 202 331)(60 269 243 344 175 101 203 330)(61 292 352 464 371 396 443 85)(62 291 353 463 372 395 444 96)(63 290 354 462 361 394 433 95)(64 289 355 461 362 393 434 94)(65 300 356 460 363 392 435 93)(66 299 357 459 364 391 436 92)(67 298 358 458 365 390 437 91)(68 297 359 457 366 389 438 90)(69 296 360 468 367 388 439 89)(70 295 349 467 368 387 440 88)(71 294 350 466 369 386 441 87)(72 293 351 465 370 385 442 86)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 8A ··· 8H 10A ··· 10L 12A ··· 12L 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 30A ··· 30L 40A ··· 40AF 60A ··· 60AV order 1 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2

180 irreducible representations

 dim 1 1 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 2 2 type + + + + + - - + - + image C1 C2 C2 C4 C5 C8 C10 C10 C20 C40 S3 D4 Q8 Dic3 D6 M4(2) C3⋊C8 Dic6 D12 C5×S3 C5×D4 C5×Q8 C4.Dic3 C5×Dic3 S3×C10 C5×M4(2) C5×C3⋊C8 C5×Dic6 C5×D12 C5×C4.Dic3 kernel C5×C12⋊C8 C10×C3⋊C8 C4×C60 C2×C60 C12⋊C8 C60 C2×C3⋊C8 C4×C12 C2×C12 C12 C4×C20 C60 C60 C2×C20 C2×C20 C30 C20 C20 C20 C42 C12 C12 C10 C2×C4 C2×C4 C6 C4 C4 C4 C2 # reps 1 2 1 4 4 8 8 4 16 32 1 1 1 2 1 2 4 2 2 4 4 4 4 8 4 8 16 8 8 16

Matrix representation of C5×C12⋊C8 in GL3(𝔽241) generated by

 87 0 0 0 205 0 0 0 205
,
 1 0 0 0 43 99 0 142 142
,
 233 0 0 0 172 172 0 103 69
G:=sub<GL(3,GF(241))| [87,0,0,0,205,0,0,0,205],[1,0,0,0,43,142,0,99,142],[233,0,0,0,172,103,0,172,69] >;

C5×C12⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_{12}\rtimes C_8
% in TeX

G:=Group("C5xC12:C8");
// GroupNames label

G:=SmallGroup(480,123);
// by ID

G=gap.SmallGroup(480,123);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,288,136,15686]);
// Polycyclic

G:=Group<a,b,c|a^5=b^12=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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