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## G = C2×C6×C5⋊C8order 480 = 25·3·5

### Direct product of C2×C6 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C6×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C3×Dic5 — C3×C5⋊C8 — C6×C5⋊C8 — C2×C6×C5⋊C8
 Lower central C5 — C2×C6×C5⋊C8
 Upper central C1 — C22×C6

Generators and relations for C2×C6×C5⋊C8
G = < a,b,c,d | a2=b6=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 328 in 152 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C2×C10, C24, C2×C12, C22×C6, C30, C30, C22×C8, C5⋊C8, C2×Dic5, C22×C10, C2×C24, C22×C12, C3×Dic5, C3×Dic5, C2×C30, C2×C5⋊C8, C22×Dic5, C22×C24, C3×C5⋊C8, C6×Dic5, C22×C30, C22×C5⋊C8, C6×C5⋊C8, C2×C6×Dic5, C2×C6×C5⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C12, C2×C6, C2×C8, C22×C4, F5, C24, C2×C12, C22×C6, C22×C8, C5⋊C8, C2×F5, C2×C24, C22×C12, C3×F5, C2×C5⋊C8, C22×F5, C22×C24, C3×C5⋊C8, C6×F5, C22×C5⋊C8, C6×C5⋊C8, C2×C6×F5, C2×C6×C5⋊C8

Smallest permutation representation of C2×C6×C5⋊C8
Regular action on 480 points
Generators in S480
(1 441)(2 442)(3 443)(4 444)(5 445)(6 446)(7 447)(8 448)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 105)(16 106)(17 383)(18 384)(19 377)(20 378)(21 379)(22 380)(23 381)(24 382)(25 389)(26 390)(27 391)(28 392)(29 385)(30 386)(31 387)(32 388)(33 209)(34 210)(35 211)(36 212)(37 213)(38 214)(39 215)(40 216)(41 407)(42 408)(43 401)(44 402)(45 403)(46 404)(47 405)(48 406)(49 235)(50 236)(51 237)(52 238)(53 239)(54 240)(55 233)(56 234)(57 410)(58 411)(59 412)(60 413)(61 414)(62 415)(63 416)(64 409)(65 150)(66 151)(67 152)(68 145)(69 146)(70 147)(71 148)(72 149)(73 158)(74 159)(75 160)(76 153)(77 154)(78 155)(79 156)(80 157)(81 166)(82 167)(83 168)(84 161)(85 162)(86 163)(87 164)(88 165)(89 174)(90 175)(91 176)(92 169)(93 170)(94 171)(95 172)(96 173)(97 473)(98 474)(99 475)(100 476)(101 477)(102 478)(103 479)(104 480)(113 305)(114 306)(115 307)(116 308)(117 309)(118 310)(119 311)(120 312)(121 365)(122 366)(123 367)(124 368)(125 361)(126 362)(127 363)(128 364)(129 193)(130 194)(131 195)(132 196)(133 197)(134 198)(135 199)(136 200)(137 243)(138 244)(139 245)(140 246)(141 247)(142 248)(143 241)(144 242)(177 280)(178 273)(179 274)(180 275)(181 276)(182 277)(183 278)(184 279)(185 429)(186 430)(187 431)(188 432)(189 425)(190 426)(191 427)(192 428)(201 325)(202 326)(203 327)(204 328)(205 321)(206 322)(207 323)(208 324)(217 341)(218 342)(219 343)(220 344)(221 337)(222 338)(223 339)(224 340)(225 347)(226 348)(227 349)(228 350)(229 351)(230 352)(231 345)(232 346)(249 454)(250 455)(251 456)(252 449)(253 450)(254 451)(255 452)(256 453)(257 462)(258 463)(259 464)(260 457)(261 458)(262 459)(263 460)(264 461)(265 470)(266 471)(267 472)(268 465)(269 466)(270 467)(271 468)(272 469)(281 301)(282 302)(283 303)(284 304)(285 297)(286 298)(287 299)(288 300)(289 314)(290 315)(291 316)(292 317)(293 318)(294 319)(295 320)(296 313)(329 398)(330 399)(331 400)(332 393)(333 394)(334 395)(335 396)(336 397)(353 420)(354 421)(355 422)(356 423)(357 424)(358 417)(359 418)(360 419)(369 438)(370 439)(371 440)(372 433)(373 434)(374 435)(375 436)(376 437)
(1 288 83 208 124 401)(2 281 84 201 125 402)(3 282 85 202 126 403)(4 283 86 203 127 404)(5 284 87 204 128 405)(6 285 88 205 121 406)(7 286 81 206 122 407)(8 287 82 207 123 408)(9 346 31 187 471 148)(10 347 32 188 472 149)(11 348 25 189 465 150)(12 349 26 190 466 151)(13 350 27 191 467 152)(14 351 28 192 468 145)(15 352 29 185 469 146)(16 345 30 186 470 147)(17 360 338 142 477 463)(18 353 339 143 478 464)(19 354 340 144 479 457)(20 355 341 137 480 458)(21 356 342 138 473 459)(22 357 343 139 474 460)(23 358 344 140 475 461)(24 359 337 141 476 462)(33 197 173 158 309 56)(34 198 174 159 310 49)(35 199 175 160 311 50)(36 200 176 153 312 51)(37 193 169 154 305 52)(38 194 170 155 306 53)(39 195 171 156 307 54)(40 196 172 157 308 55)(41 447 298 166 322 366)(42 448 299 167 323 367)(43 441 300 168 324 368)(44 442 301 161 325 361)(45 443 302 162 326 362)(46 444 303 163 327 363)(47 445 304 164 328 364)(48 446 297 165 321 365)(57 329 373 182 454 318)(58 330 374 183 455 319)(59 331 375 184 456 320)(60 332 376 177 449 313)(61 333 369 178 450 314)(62 334 370 179 451 315)(63 335 371 180 452 316)(64 336 372 181 453 317)(65 109 226 389 425 268)(66 110 227 390 426 269)(67 111 228 391 427 270)(68 112 229 392 428 271)(69 105 230 385 429 272)(70 106 231 386 430 265)(71 107 232 387 431 266)(72 108 225 388 432 267)(73 117 234 209 133 96)(74 118 235 210 134 89)(75 119 236 211 135 90)(76 120 237 212 136 91)(77 113 238 213 129 92)(78 114 239 214 130 93)(79 115 240 215 131 94)(80 116 233 216 132 95)(97 262 379 423 218 244)(98 263 380 424 219 245)(99 264 381 417 220 246)(100 257 382 418 221 247)(101 258 383 419 222 248)(102 259 384 420 223 241)(103 260 377 421 224 242)(104 261 378 422 217 243)(249 293 410 398 434 277)(250 294 411 399 435 278)(251 295 412 400 436 279)(252 296 413 393 437 280)(253 289 414 394 438 273)(254 290 415 395 439 274)(255 291 416 396 440 275)(256 292 409 397 433 276)
(1 239 248 232 415)(2 225 240 416 241)(3 409 226 242 233)(4 243 410 234 227)(5 235 244 228 411)(6 229 236 412 245)(7 413 230 246 237)(8 247 414 238 231)(9 315 43 306 338)(10 307 316 339 44)(11 340 308 45 317)(12 46 341 318 309)(13 319 47 310 342)(14 311 320 343 48)(15 344 312 41 313)(16 42 337 314 305)(17 471 179 324 170)(18 325 472 171 180)(19 172 326 181 465)(20 182 173 466 327)(21 467 183 328 174)(22 321 468 175 184)(23 176 322 177 469)(24 178 169 470 323)(25 479 40 302 336)(26 303 480 329 33)(27 330 304 34 473)(28 35 331 474 297)(29 475 36 298 332)(30 299 476 333 37)(31 334 300 38 477)(32 39 335 478 301)(49 138 350 58 445)(50 59 139 446 351)(51 447 60 352 140)(52 345 448 141 61)(53 142 346 62 441)(54 63 143 442 347)(55 443 64 348 144)(56 349 444 137 57)(65 421 80 126 256)(66 127 422 249 73)(67 250 128 74 423)(68 75 251 424 121)(69 417 76 122 252)(70 123 418 253 77)(71 254 124 78 419)(72 79 255 420 125)(81 437 429 264 136)(82 257 438 129 430)(83 130 258 431 439)(84 432 131 440 259)(85 433 425 260 132)(86 261 434 133 426)(87 134 262 427 435)(88 428 135 436 263)(89 379 270 278 204)(90 279 380 205 271)(91 206 280 272 381)(92 265 207 382 273)(93 383 266 274 208)(94 275 384 201 267)(95 202 276 268 377)(96 269 203 378 277)(97 391 399 284 210)(98 285 392 211 400)(99 212 286 393 385)(100 394 213 386 287)(101 387 395 288 214)(102 281 388 215 396)(103 216 282 397 389)(104 398 209 390 283)(105 220 120 407 296)(106 408 221 289 113)(107 290 401 114 222)(108 115 291 223 402)(109 224 116 403 292)(110 404 217 293 117)(111 294 405 118 218)(112 119 295 219 406)(145 160 456 357 365)(146 358 153 366 449)(147 367 359 450 154)(148 451 368 155 360)(149 156 452 353 361)(150 354 157 362 453)(151 363 355 454 158)(152 455 364 159 356)(161 188 195 371 464)(162 372 189 457 196)(163 458 373 197 190)(164 198 459 191 374)(165 192 199 375 460)(166 376 185 461 200)(167 462 369 193 186)(168 194 463 187 370)
(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)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 5 6A ··· 6N 8A ··· 8P 10A ··· 10G 12A ··· 12P 15A 15B 24A ··· 24AF 30A ··· 30N order 1 2 ··· 2 3 3 4 ··· 4 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 24 ··· 24 30 ··· 30 size 1 1 ··· 1 1 1 5 ··· 5 4 1 ··· 1 5 ··· 5 4 ··· 4 5 ··· 5 4 4 5 ··· 5 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 F5 C5⋊C8 C2×F5 C3×F5 C3×C5⋊C8 C6×F5 kernel C2×C6×C5⋊C8 C6×C5⋊C8 C2×C6×Dic5 C22×C5⋊C8 C6×Dic5 C22×C30 C2×C5⋊C8 C22×Dic5 C2×C30 C2×Dic5 C22×C10 C2×C10 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 2 6 2 12 2 16 12 4 32 1 4 3 2 8 6

Matrix representation of C2×C6×C5⋊C8 in GL6(𝔽241)

 240 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 240 0 0 0 1 0 240 0 0 0 0 1 240
,
 240 0 0 0 0 0 0 1 0 0 0 0 0 0 64 154 6 72 0 0 70 226 192 136 0 0 15 49 105 142 0 0 169 55 177 87

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,64,70,15,169,0,0,154,226,49,55,0,0,6,192,105,177,0,0,72,136,142,87] >;

C2×C6×C5⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_5\rtimes C_8
% in TeX

G:=Group("C2xC6xC5:C8");
// GroupNames label

G:=SmallGroup(480,1057);
// by ID

G=gap.SmallGroup(480,1057);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,102,9414,818]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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