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G = C5×C12⋊C8order 480 = 25·3·5

Direct product of C5 and C12⋊C8

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C12⋊C8, C608C8, C121C40, C60.35Q8, C20.68D12, C60.189D4, C20.27Dic6, C30.41M4(2), C205(C3⋊C8), C1512(C4⋊C8), C6.7(C2×C40), (C4×C20).8S3, C12.7(C5×Q8), (C2×C60).30C4, (C4×C12).4C10, C30.69(C2×C8), (C4×C60).14C2, (C2×C12).4C20, C4.16(C5×D12), C12.32(C5×D4), C30.53(C4⋊C4), C42.2(C5×S3), C4.7(C5×Dic6), (C2×C20).444D6, C6.5(C5×M4(2)), (C2×C20).24Dic3, (C2×C60).556C22, C10.18(C4⋊Dic3), C22.8(C10×Dic3), C10.13(C4.Dic3), C4⋊(C5×C3⋊C8), C31(C5×C4⋊C8), C6.1(C5×C4⋊C4), C2.3(C10×C3⋊C8), (C2×C3⋊C8).8C10, C10.21(C2×C3⋊C8), (C10×C3⋊C8).20C2, C2.1(C5×C4⋊Dic3), (C2×C4).91(S3×C10), (C2×C6).26(C2×C20), (C2×C4).4(C5×Dic3), (C2×C30).194(C2×C4), C2.2(C5×C4.Dic3), (C2×C12).108(C2×C10), (C2×C10).60(C2×Dic3), SmallGroup(480,123)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C12⋊C8
C1C3C6C2×C6C2×C12C2×C60C10×C3⋊C8 — C5×C12⋊C8
C3C6 — C5×C12⋊C8
C1C2×C20C4×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 [×3], C3, C4 [×2], C4 [×2], C4, C22, C5, C6 [×3], C8 [×2], C2×C4 [×3], C10 [×3], C12 [×2], C12 [×2], C12, C2×C6, C15, C42, C2×C8 [×2], C20 [×2], C20 [×2], C20, C2×C10, C3⋊C8 [×2], C2×C12 [×3], C30 [×3], C4⋊C8, C40 [×2], C2×C20 [×3], C2×C3⋊C8 [×2], C4×C12, C60 [×2], C60 [×2], C60, C2×C30, C4×C20, C2×C40 [×2], C12⋊C8, C5×C3⋊C8 [×2], C2×C60 [×3], C5×C4⋊C8, C10×C3⋊C8 [×2], C4×C60, C5×C12⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C8 [×2], C2×C4, D4, Q8, C10 [×3], Dic3 [×2], D6, C4⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, C3⋊C8 [×2], Dic6, D12, C2×Dic3, C5×S3, C4⋊C8, C40 [×2], C2×C20, C5×D4, C5×Q8, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, C5×Dic3 [×2], S3×C10, C5×C4⋊C4, C2×C40, C5×M4(2), C12⋊C8, C5×C3⋊C8 [×2], C5×Dic6, C5×D12, C10×Dic3, C5×C4⋊C8, C10×C3⋊C8, C5×C4.Dic3, C5×C4⋊Dic3, C5×C12⋊C8

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

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

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

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

180 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B5C5D6A6B6C8A···8H10A···10L12A···12L15A15B15C15D20A···20P20Q···20AF30A···30L40A···40AF60A···60AV
order122234444444455556668···810···1012···121515151520···2020···2030···3040···4060···60
size111121111222211112226···61···12···222221···12···22···26···62···2

180 irreducible representations

dim111111111122222222222222222222
type+++++--+-+
imageC1C2C2C4C5C8C10C10C20C40S3D4Q8Dic3D6M4(2)C3⋊C8Dic6D12C5×S3C5×D4C5×Q8C4.Dic3C5×Dic3S3×C10C5×M4(2)C5×C3⋊C8C5×Dic6C5×D12C5×C4.Dic3
kernelC5×C12⋊C8C10×C3⋊C8C4×C60C2×C60C12⋊C8C60C2×C3⋊C8C4×C12C2×C12C12C4×C20C60C60C2×C20C2×C20C30C20C20C20C42C12C12C10C2×C4C2×C4C6C4C4C4C2
# reps1214488416321112124224444848168816

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

8700
02050
00205
,
100
04399
0142142
,
23300
0172172
010369
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

׿
×
𝔽