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## G = C2×C6×C5⋊2C8order 480 = 25·3·5

### Direct product of C2×C6 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C6×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C60 — C3×C5⋊2C8 — C6×C5⋊2C8 — C2×C6×C5⋊2C8
 Lower central C5 — C2×C6×C5⋊2C8
 Upper central C1 — C22×C12

Generators and relations for C2×C6×C52C8
G = < a,b,c,d | a2=b6=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 240 in 152 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C12, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C24, C2×C12, C22×C6, C30, C30, C22×C8, C52C8, C2×C20, C22×C10, C2×C24, C22×C12, C60, C60, C2×C30, C2×C52C8, C22×C20, C22×C24, C3×C52C8, C2×C60, C22×C30, C22×C52C8, C6×C52C8, C22×C60, C2×C6×C52C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, D5, C12, C2×C6, C2×C8, C22×C4, Dic5, D10, C24, C2×C12, C22×C6, C3×D5, C22×C8, C52C8, C2×Dic5, C22×D5, C2×C24, C22×C12, C3×Dic5, C6×D5, C2×C52C8, C22×Dic5, C22×C24, C3×C52C8, C6×Dic5, D5×C2×C6, C22×C52C8, C6×C52C8, C2×C6×Dic5, C2×C6×C52C8

Smallest permutation representation of C2×C6×C52C8
Regular action on 480 points
Generators in S480
(1 69)(2 70)(3 71)(4 72)(5 65)(6 66)(7 67)(8 68)(9 346)(10 347)(11 348)(12 349)(13 350)(14 351)(15 352)(16 345)(17 301)(18 302)(19 303)(20 304)(21 297)(22 298)(23 299)(24 300)(25 127)(26 128)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 310)(34 311)(35 312)(36 305)(37 306)(38 307)(39 308)(40 309)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 328)(50 321)(51 322)(52 323)(53 324)(54 325)(55 326)(56 327)(57 336)(58 329)(59 330)(60 331)(61 332)(62 333)(63 334)(64 335)(73 196)(74 197)(75 198)(76 199)(77 200)(78 193)(79 194)(80 195)(81 204)(82 205)(83 206)(84 207)(85 208)(86 201)(87 202)(88 203)(89 212)(90 213)(91 214)(92 215)(93 216)(94 209)(95 210)(96 211)(97 220)(98 221)(99 222)(100 223)(101 224)(102 217)(103 218)(104 219)(105 228)(106 229)(107 230)(108 231)(109 232)(110 225)(111 226)(112 227)(113 236)(114 237)(115 238)(116 239)(117 240)(118 233)(119 234)(120 235)(129 442)(130 443)(131 444)(132 445)(133 446)(134 447)(135 448)(136 441)(137 463)(138 464)(139 457)(140 458)(141 459)(142 460)(143 461)(144 462)(153 472)(154 465)(155 466)(156 467)(157 468)(158 469)(159 470)(160 471)(161 422)(162 423)(163 424)(164 417)(165 418)(166 419)(167 420)(168 421)(169 430)(170 431)(171 432)(172 425)(173 426)(174 427)(175 428)(176 429)(177 438)(178 439)(179 440)(180 433)(181 434)(182 435)(183 436)(184 437)(185 358)(186 359)(187 360)(188 353)(189 354)(190 355)(191 356)(192 357)(241 366)(242 367)(243 368)(244 361)(245 362)(246 363)(247 364)(248 365)(249 374)(250 375)(251 376)(252 369)(253 370)(254 371)(255 372)(256 373)(257 382)(258 383)(259 384)(260 377)(261 378)(262 379)(263 380)(264 381)(265 390)(266 391)(267 392)(268 385)(269 386)(270 387)(271 388)(272 389)(273 398)(274 399)(275 400)(276 393)(277 394)(278 395)(279 396)(280 397)(281 406)(282 407)(283 408)(284 401)(285 402)(286 403)(287 404)(288 405)(289 414)(290 415)(291 416)(292 409)(293 410)(294 411)(295 412)(296 413)(313 454)(314 455)(315 456)(316 449)(317 450)(318 451)(319 452)(320 453)(337 480)(338 473)(339 474)(340 475)(341 476)(342 477)(343 478)(344 479)
(1 228 203 29 420 48)(2 229 204 30 421 41)(3 230 205 31 422 42)(4 231 206 32 423 43)(5 232 207 25 424 44)(6 225 208 26 417 45)(7 226 201 27 418 46)(8 227 202 28 419 47)(9 266 196 240 471 33)(10 267 197 233 472 34)(11 268 198 234 465 35)(12 269 199 235 466 36)(13 270 200 236 467 37)(14 271 193 237 468 38)(15 272 194 238 469 39)(16 265 195 239 470 40)(17 480 458 191 224 261)(18 473 459 192 217 262)(19 474 460 185 218 263)(20 475 461 186 219 264)(21 476 462 187 220 257)(22 477 463 188 221 258)(23 478 464 189 222 259)(24 479 457 190 223 260)(49 441 425 212 244 288)(50 442 426 213 245 281)(51 443 427 214 246 282)(52 444 428 215 247 283)(53 445 429 216 248 284)(54 446 430 209 241 285)(55 447 431 210 242 286)(56 448 432 211 243 287)(57 449 433 280 252 296)(58 450 434 273 253 289)(59 451 435 274 254 290)(60 452 436 275 255 291)(61 453 437 276 256 292)(62 454 438 277 249 293)(63 455 439 278 250 294)(64 456 440 279 251 295)(65 109 84 127 163 148)(66 110 85 128 164 149)(67 111 86 121 165 150)(68 112 87 122 166 151)(69 105 88 123 167 152)(70 106 81 124 168 145)(71 107 82 125 161 146)(72 108 83 126 162 147)(73 117 160 310 346 391)(74 118 153 311 347 392)(75 119 154 312 348 385)(76 120 155 305 349 386)(77 113 156 306 350 387)(78 114 157 307 351 388)(79 115 158 308 352 389)(80 116 159 309 345 390)(89 361 405 328 136 172)(90 362 406 321 129 173)(91 363 407 322 130 174)(92 364 408 323 131 175)(93 365 401 324 132 176)(94 366 402 325 133 169)(95 367 403 326 134 170)(96 368 404 327 135 171)(97 382 297 341 144 360)(98 383 298 342 137 353)(99 384 299 343 138 354)(100 377 300 344 139 355)(101 378 301 337 140 356)(102 379 302 338 141 357)(103 380 303 339 142 358)(104 381 304 340 143 359)(177 394 374 410 333 313)(178 395 375 411 334 314)(179 396 376 412 335 315)(180 397 369 413 336 316)(181 398 370 414 329 317)(182 399 371 415 330 318)(183 400 372 416 331 319)(184 393 373 409 332 320)
(1 465 188 51 59)(2 60 52 189 466)(3 467 190 53 61)(4 62 54 191 468)(5 469 192 55 63)(6 64 56 185 470)(7 471 186 49 57)(8 58 50 187 472)(9 264 425 433 201)(10 202 434 426 257)(11 258 427 435 203)(12 204 436 428 259)(13 260 429 437 205)(14 206 438 430 261)(15 262 431 439 207)(16 208 440 432 263)(17 271 32 277 209)(18 210 278 25 272)(19 265 26 279 211)(20 212 280 27 266)(21 267 28 273 213)(22 214 274 29 268)(23 269 30 275 215)(24 216 276 31 270)(33 219 441 449 226)(34 227 450 442 220)(35 221 443 451 228)(36 229 452 444 222)(37 223 445 453 230)(38 231 454 446 224)(39 217 447 455 232)(40 225 456 448 218)(41 291 283 464 235)(42 236 457 284 292)(43 293 285 458 237)(44 238 459 286 294)(45 295 287 460 239)(46 240 461 288 296)(47 289 281 462 233)(48 234 463 282 290)(65 158 357 326 334)(66 335 327 358 159)(67 160 359 328 336)(68 329 321 360 153)(69 154 353 322 330)(70 331 323 354 155)(71 156 355 324 332)(72 333 325 356 157)(73 340 361 369 165)(74 166 370 362 341)(75 342 363 371 167)(76 168 372 364 343)(77 344 365 373 161)(78 162 374 366 337)(79 338 367 375 163)(80 164 376 368 339)(81 183 175 384 349)(82 350 377 176 184)(83 177 169 378 351)(84 352 379 170 178)(85 179 171 380 345)(86 346 381 172 180)(87 181 173 382 347)(88 348 383 174 182)(89 397 121 391 304)(90 297 392 122 398)(91 399 123 385 298)(92 299 386 124 400)(93 393 125 387 300)(94 301 388 126 394)(95 395 127 389 302)(96 303 390 128 396)(97 311 112 317 129)(98 130 318 105 312)(99 305 106 319 131)(100 132 320 107 306)(101 307 108 313 133)(102 134 314 109 308)(103 309 110 315 135)(104 136 316 111 310)(113 139 401 409 146)(114 147 410 402 140)(115 141 403 411 148)(116 149 412 404 142)(117 143 405 413 150)(118 151 414 406 144)(119 137 407 415 152)(120 145 416 408 138)(193 423 249 241 480)(194 473 242 250 424)(195 417 251 243 474)(196 475 244 252 418)(197 419 253 245 476)(198 477 246 254 420)(199 421 255 247 478)(200 479 248 256 422)
(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)| (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,346)(10,347)(11,348)(12,349)(13,350)(14,351)(15,352)(16,345)(17,301)(18,302)(19,303)(20,304)(21,297)(22,298)(23,299)(24,300)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,310)(34,311)(35,312)(36,305)(37,306)(38,307)(39,308)(40,309)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,328)(50,321)(51,322)(52,323)(53,324)(54,325)(55,326)(56,327)(57,336)(58,329)(59,330)(60,331)(61,332)(62,333)(63,334)(64,335)(73,196)(74,197)(75,198)(76,199)(77,200)(78,193)(79,194)(80,195)(81,204)(82,205)(83,206)(84,207)(85,208)(86,201)(87,202)(88,203)(89,212)(90,213)(91,214)(92,215)(93,216)(94,209)(95,210)(96,211)(97,220)(98,221)(99,222)(100,223)(101,224)(102,217)(103,218)(104,219)(105,228)(106,229)(107,230)(108,231)(109,232)(110,225)(111,226)(112,227)(113,236)(114,237)(115,238)(116,239)(117,240)(118,233)(119,234)(120,235)(129,442)(130,443)(131,444)(132,445)(133,446)(134,447)(135,448)(136,441)(137,463)(138,464)(139,457)(140,458)(141,459)(142,460)(143,461)(144,462)(153,472)(154,465)(155,466)(156,467)(157,468)(158,469)(159,470)(160,471)(161,422)(162,423)(163,424)(164,417)(165,418)(166,419)(167,420)(168,421)(169,430)(170,431)(171,432)(172,425)(173,426)(174,427)(175,428)(176,429)(177,438)(178,439)(179,440)(180,433)(181,434)(182,435)(183,436)(184,437)(185,358)(186,359)(187,360)(188,353)(189,354)(190,355)(191,356)(192,357)(241,366)(242,367)(243,368)(244,361)(245,362)(246,363)(247,364)(248,365)(249,374)(250,375)(251,376)(252,369)(253,370)(254,371)(255,372)(256,373)(257,382)(258,383)(259,384)(260,377)(261,378)(262,379)(263,380)(264,381)(265,390)(266,391)(267,392)(268,385)(269,386)(270,387)(271,388)(272,389)(273,398)(274,399)(275,400)(276,393)(277,394)(278,395)(279,396)(280,397)(281,406)(282,407)(283,408)(284,401)(285,402)(286,403)(287,404)(288,405)(289,414)(290,415)(291,416)(292,409)(293,410)(294,411)(295,412)(296,413)(313,454)(314,455)(315,456)(316,449)(317,450)(318,451)(319,452)(320,453)(337,480)(338,473)(339,474)(340,475)(341,476)(342,477)(343,478)(344,479), (1,228,203,29,420,48)(2,229,204,30,421,41)(3,230,205,31,422,42)(4,231,206,32,423,43)(5,232,207,25,424,44)(6,225,208,26,417,45)(7,226,201,27,418,46)(8,227,202,28,419,47)(9,266,196,240,471,33)(10,267,197,233,472,34)(11,268,198,234,465,35)(12,269,199,235,466,36)(13,270,200,236,467,37)(14,271,193,237,468,38)(15,272,194,238,469,39)(16,265,195,239,470,40)(17,480,458,191,224,261)(18,473,459,192,217,262)(19,474,460,185,218,263)(20,475,461,186,219,264)(21,476,462,187,220,257)(22,477,463,188,221,258)(23,478,464,189,222,259)(24,479,457,190,223,260)(49,441,425,212,244,288)(50,442,426,213,245,281)(51,443,427,214,246,282)(52,444,428,215,247,283)(53,445,429,216,248,284)(54,446,430,209,241,285)(55,447,431,210,242,286)(56,448,432,211,243,287)(57,449,433,280,252,296)(58,450,434,273,253,289)(59,451,435,274,254,290)(60,452,436,275,255,291)(61,453,437,276,256,292)(62,454,438,277,249,293)(63,455,439,278,250,294)(64,456,440,279,251,295)(65,109,84,127,163,148)(66,110,85,128,164,149)(67,111,86,121,165,150)(68,112,87,122,166,151)(69,105,88,123,167,152)(70,106,81,124,168,145)(71,107,82,125,161,146)(72,108,83,126,162,147)(73,117,160,310,346,391)(74,118,153,311,347,392)(75,119,154,312,348,385)(76,120,155,305,349,386)(77,113,156,306,350,387)(78,114,157,307,351,388)(79,115,158,308,352,389)(80,116,159,309,345,390)(89,361,405,328,136,172)(90,362,406,321,129,173)(91,363,407,322,130,174)(92,364,408,323,131,175)(93,365,401,324,132,176)(94,366,402,325,133,169)(95,367,403,326,134,170)(96,368,404,327,135,171)(97,382,297,341,144,360)(98,383,298,342,137,353)(99,384,299,343,138,354)(100,377,300,344,139,355)(101,378,301,337,140,356)(102,379,302,338,141,357)(103,380,303,339,142,358)(104,381,304,340,143,359)(177,394,374,410,333,313)(178,395,375,411,334,314)(179,396,376,412,335,315)(180,397,369,413,336,316)(181,398,370,414,329,317)(182,399,371,415,330,318)(183,400,372,416,331,319)(184,393,373,409,332,320), (1,465,188,51,59)(2,60,52,189,466)(3,467,190,53,61)(4,62,54,191,468)(5,469,192,55,63)(6,64,56,185,470)(7,471,186,49,57)(8,58,50,187,472)(9,264,425,433,201)(10,202,434,426,257)(11,258,427,435,203)(12,204,436,428,259)(13,260,429,437,205)(14,206,438,430,261)(15,262,431,439,207)(16,208,440,432,263)(17,271,32,277,209)(18,210,278,25,272)(19,265,26,279,211)(20,212,280,27,266)(21,267,28,273,213)(22,214,274,29,268)(23,269,30,275,215)(24,216,276,31,270)(33,219,441,449,226)(34,227,450,442,220)(35,221,443,451,228)(36,229,452,444,222)(37,223,445,453,230)(38,231,454,446,224)(39,217,447,455,232)(40,225,456,448,218)(41,291,283,464,235)(42,236,457,284,292)(43,293,285,458,237)(44,238,459,286,294)(45,295,287,460,239)(46,240,461,288,296)(47,289,281,462,233)(48,234,463,282,290)(65,158,357,326,334)(66,335,327,358,159)(67,160,359,328,336)(68,329,321,360,153)(69,154,353,322,330)(70,331,323,354,155)(71,156,355,324,332)(72,333,325,356,157)(73,340,361,369,165)(74,166,370,362,341)(75,342,363,371,167)(76,168,372,364,343)(77,344,365,373,161)(78,162,374,366,337)(79,338,367,375,163)(80,164,376,368,339)(81,183,175,384,349)(82,350,377,176,184)(83,177,169,378,351)(84,352,379,170,178)(85,179,171,380,345)(86,346,381,172,180)(87,181,173,382,347)(88,348,383,174,182)(89,397,121,391,304)(90,297,392,122,398)(91,399,123,385,298)(92,299,386,124,400)(93,393,125,387,300)(94,301,388,126,394)(95,395,127,389,302)(96,303,390,128,396)(97,311,112,317,129)(98,130,318,105,312)(99,305,106,319,131)(100,132,320,107,306)(101,307,108,313,133)(102,134,314,109,308)(103,309,110,315,135)(104,136,316,111,310)(113,139,401,409,146)(114,147,410,402,140)(115,141,403,411,148)(116,149,412,404,142)(117,143,405,413,150)(118,151,414,406,144)(119,137,407,415,152)(120,145,416,408,138)(193,423,249,241,480)(194,473,242,250,424)(195,417,251,243,474)(196,475,244,252,418)(197,419,253,245,476)(198,477,246,254,420)(199,421,255,247,478)(200,479,248,256,422), (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( (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,346)(10,347)(11,348)(12,349)(13,350)(14,351)(15,352)(16,345)(17,301)(18,302)(19,303)(20,304)(21,297)(22,298)(23,299)(24,300)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,310)(34,311)(35,312)(36,305)(37,306)(38,307)(39,308)(40,309)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,328)(50,321)(51,322)(52,323)(53,324)(54,325)(55,326)(56,327)(57,336)(58,329)(59,330)(60,331)(61,332)(62,333)(63,334)(64,335)(73,196)(74,197)(75,198)(76,199)(77,200)(78,193)(79,194)(80,195)(81,204)(82,205)(83,206)(84,207)(85,208)(86,201)(87,202)(88,203)(89,212)(90,213)(91,214)(92,215)(93,216)(94,209)(95,210)(96,211)(97,220)(98,221)(99,222)(100,223)(101,224)(102,217)(103,218)(104,219)(105,228)(106,229)(107,230)(108,231)(109,232)(110,225)(111,226)(112,227)(113,236)(114,237)(115,238)(116,239)(117,240)(118,233)(119,234)(120,235)(129,442)(130,443)(131,444)(132,445)(133,446)(134,447)(135,448)(136,441)(137,463)(138,464)(139,457)(140,458)(141,459)(142,460)(143,461)(144,462)(153,472)(154,465)(155,466)(156,467)(157,468)(158,469)(159,470)(160,471)(161,422)(162,423)(163,424)(164,417)(165,418)(166,419)(167,420)(168,421)(169,430)(170,431)(171,432)(172,425)(173,426)(174,427)(175,428)(176,429)(177,438)(178,439)(179,440)(180,433)(181,434)(182,435)(183,436)(184,437)(185,358)(186,359)(187,360)(188,353)(189,354)(190,355)(191,356)(192,357)(241,366)(242,367)(243,368)(244,361)(245,362)(246,363)(247,364)(248,365)(249,374)(250,375)(251,376)(252,369)(253,370)(254,371)(255,372)(256,373)(257,382)(258,383)(259,384)(260,377)(261,378)(262,379)(263,380)(264,381)(265,390)(266,391)(267,392)(268,385)(269,386)(270,387)(271,388)(272,389)(273,398)(274,399)(275,400)(276,393)(277,394)(278,395)(279,396)(280,397)(281,406)(282,407)(283,408)(284,401)(285,402)(286,403)(287,404)(288,405)(289,414)(290,415)(291,416)(292,409)(293,410)(294,411)(295,412)(296,413)(313,454)(314,455)(315,456)(316,449)(317,450)(318,451)(319,452)(320,453)(337,480)(338,473)(339,474)(340,475)(341,476)(342,477)(343,478)(344,479), (1,228,203,29,420,48)(2,229,204,30,421,41)(3,230,205,31,422,42)(4,231,206,32,423,43)(5,232,207,25,424,44)(6,225,208,26,417,45)(7,226,201,27,418,46)(8,227,202,28,419,47)(9,266,196,240,471,33)(10,267,197,233,472,34)(11,268,198,234,465,35)(12,269,199,235,466,36)(13,270,200,236,467,37)(14,271,193,237,468,38)(15,272,194,238,469,39)(16,265,195,239,470,40)(17,480,458,191,224,261)(18,473,459,192,217,262)(19,474,460,185,218,263)(20,475,461,186,219,264)(21,476,462,187,220,257)(22,477,463,188,221,258)(23,478,464,189,222,259)(24,479,457,190,223,260)(49,441,425,212,244,288)(50,442,426,213,245,281)(51,443,427,214,246,282)(52,444,428,215,247,283)(53,445,429,216,248,284)(54,446,430,209,241,285)(55,447,431,210,242,286)(56,448,432,211,243,287)(57,449,433,280,252,296)(58,450,434,273,253,289)(59,451,435,274,254,290)(60,452,436,275,255,291)(61,453,437,276,256,292)(62,454,438,277,249,293)(63,455,439,278,250,294)(64,456,440,279,251,295)(65,109,84,127,163,148)(66,110,85,128,164,149)(67,111,86,121,165,150)(68,112,87,122,166,151)(69,105,88,123,167,152)(70,106,81,124,168,145)(71,107,82,125,161,146)(72,108,83,126,162,147)(73,117,160,310,346,391)(74,118,153,311,347,392)(75,119,154,312,348,385)(76,120,155,305,349,386)(77,113,156,306,350,387)(78,114,157,307,351,388)(79,115,158,308,352,389)(80,116,159,309,345,390)(89,361,405,328,136,172)(90,362,406,321,129,173)(91,363,407,322,130,174)(92,364,408,323,131,175)(93,365,401,324,132,176)(94,366,402,325,133,169)(95,367,403,326,134,170)(96,368,404,327,135,171)(97,382,297,341,144,360)(98,383,298,342,137,353)(99,384,299,343,138,354)(100,377,300,344,139,355)(101,378,301,337,140,356)(102,379,302,338,141,357)(103,380,303,339,142,358)(104,381,304,340,143,359)(177,394,374,410,333,313)(178,395,375,411,334,314)(179,396,376,412,335,315)(180,397,369,413,336,316)(181,398,370,414,329,317)(182,399,371,415,330,318)(183,400,372,416,331,319)(184,393,373,409,332,320), (1,465,188,51,59)(2,60,52,189,466)(3,467,190,53,61)(4,62,54,191,468)(5,469,192,55,63)(6,64,56,185,470)(7,471,186,49,57)(8,58,50,187,472)(9,264,425,433,201)(10,202,434,426,257)(11,258,427,435,203)(12,204,436,428,259)(13,260,429,437,205)(14,206,438,430,261)(15,262,431,439,207)(16,208,440,432,263)(17,271,32,277,209)(18,210,278,25,272)(19,265,26,279,211)(20,212,280,27,266)(21,267,28,273,213)(22,214,274,29,268)(23,269,30,275,215)(24,216,276,31,270)(33,219,441,449,226)(34,227,450,442,220)(35,221,443,451,228)(36,229,452,444,222)(37,223,445,453,230)(38,231,454,446,224)(39,217,447,455,232)(40,225,456,448,218)(41,291,283,464,235)(42,236,457,284,292)(43,293,285,458,237)(44,238,459,286,294)(45,295,287,460,239)(46,240,461,288,296)(47,289,281,462,233)(48,234,463,282,290)(65,158,357,326,334)(66,335,327,358,159)(67,160,359,328,336)(68,329,321,360,153)(69,154,353,322,330)(70,331,323,354,155)(71,156,355,324,332)(72,333,325,356,157)(73,340,361,369,165)(74,166,370,362,341)(75,342,363,371,167)(76,168,372,364,343)(77,344,365,373,161)(78,162,374,366,337)(79,338,367,375,163)(80,164,376,368,339)(81,183,175,384,349)(82,350,377,176,184)(83,177,169,378,351)(84,352,379,170,178)(85,179,171,380,345)(86,346,381,172,180)(87,181,173,382,347)(88,348,383,174,182)(89,397,121,391,304)(90,297,392,122,398)(91,399,123,385,298)(92,299,386,124,400)(93,393,125,387,300)(94,301,388,126,394)(95,395,127,389,302)(96,303,390,128,396)(97,311,112,317,129)(98,130,318,105,312)(99,305,106,319,131)(100,132,320,107,306)(101,307,108,313,133)(102,134,314,109,308)(103,309,110,315,135)(104,136,316,111,310)(113,139,401,409,146)(114,147,410,402,140)(115,141,403,411,148)(116,149,412,404,142)(117,143,405,413,150)(118,151,414,406,144)(119,137,407,415,152)(120,145,416,408,138)(193,423,249,241,480)(194,473,242,250,424)(195,417,251,243,474)(196,475,244,252,418)(197,419,253,245,476)(198,477,246,254,420)(199,421,255,247,478)(200,479,248,256,422), (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([[(1,69),(2,70),(3,71),(4,72),(5,65),(6,66),(7,67),(8,68),(9,346),(10,347),(11,348),(12,349),(13,350),(14,351),(15,352),(16,345),(17,301),(18,302),(19,303),(20,304),(21,297),(22,298),(23,299),(24,300),(25,127),(26,128),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,310),(34,311),(35,312),(36,305),(37,306),(38,307),(39,308),(40,309),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,328),(50,321),(51,322),(52,323),(53,324),(54,325),(55,326),(56,327),(57,336),(58,329),(59,330),(60,331),(61,332),(62,333),(63,334),(64,335),(73,196),(74,197),(75,198),(76,199),(77,200),(78,193),(79,194),(80,195),(81,204),(82,205),(83,206),(84,207),(85,208),(86,201),(87,202),(88,203),(89,212),(90,213),(91,214),(92,215),(93,216),(94,209),(95,210),(96,211),(97,220),(98,221),(99,222),(100,223),(101,224),(102,217),(103,218),(104,219),(105,228),(106,229),(107,230),(108,231),(109,232),(110,225),(111,226),(112,227),(113,236),(114,237),(115,238),(116,239),(117,240),(118,233),(119,234),(120,235),(129,442),(130,443),(131,444),(132,445),(133,446),(134,447),(135,448),(136,441),(137,463),(138,464),(139,457),(140,458),(141,459),(142,460),(143,461),(144,462),(153,472),(154,465),(155,466),(156,467),(157,468),(158,469),(159,470),(160,471),(161,422),(162,423),(163,424),(164,417),(165,418),(166,419),(167,420),(168,421),(169,430),(170,431),(171,432),(172,425),(173,426),(174,427),(175,428),(176,429),(177,438),(178,439),(179,440),(180,433),(181,434),(182,435),(183,436),(184,437),(185,358),(186,359),(187,360),(188,353),(189,354),(190,355),(191,356),(192,357),(241,366),(242,367),(243,368),(244,361),(245,362),(246,363),(247,364),(248,365),(249,374),(250,375),(251,376),(252,369),(253,370),(254,371),(255,372),(256,373),(257,382),(258,383),(259,384),(260,377),(261,378),(262,379),(263,380),(264,381),(265,390),(266,391),(267,392),(268,385),(269,386),(270,387),(271,388),(272,389),(273,398),(274,399),(275,400),(276,393),(277,394),(278,395),(279,396),(280,397),(281,406),(282,407),(283,408),(284,401),(285,402),(286,403),(287,404),(288,405),(289,414),(290,415),(291,416),(292,409),(293,410),(294,411),(295,412),(296,413),(313,454),(314,455),(315,456),(316,449),(317,450),(318,451),(319,452),(320,453),(337,480),(338,473),(339,474),(340,475),(341,476),(342,477),(343,478),(344,479)], [(1,228,203,29,420,48),(2,229,204,30,421,41),(3,230,205,31,422,42),(4,231,206,32,423,43),(5,232,207,25,424,44),(6,225,208,26,417,45),(7,226,201,27,418,46),(8,227,202,28,419,47),(9,266,196,240,471,33),(10,267,197,233,472,34),(11,268,198,234,465,35),(12,269,199,235,466,36),(13,270,200,236,467,37),(14,271,193,237,468,38),(15,272,194,238,469,39),(16,265,195,239,470,40),(17,480,458,191,224,261),(18,473,459,192,217,262),(19,474,460,185,218,263),(20,475,461,186,219,264),(21,476,462,187,220,257),(22,477,463,188,221,258),(23,478,464,189,222,259),(24,479,457,190,223,260),(49,441,425,212,244,288),(50,442,426,213,245,281),(51,443,427,214,246,282),(52,444,428,215,247,283),(53,445,429,216,248,284),(54,446,430,209,241,285),(55,447,431,210,242,286),(56,448,432,211,243,287),(57,449,433,280,252,296),(58,450,434,273,253,289),(59,451,435,274,254,290),(60,452,436,275,255,291),(61,453,437,276,256,292),(62,454,438,277,249,293),(63,455,439,278,250,294),(64,456,440,279,251,295),(65,109,84,127,163,148),(66,110,85,128,164,149),(67,111,86,121,165,150),(68,112,87,122,166,151),(69,105,88,123,167,152),(70,106,81,124,168,145),(71,107,82,125,161,146),(72,108,83,126,162,147),(73,117,160,310,346,391),(74,118,153,311,347,392),(75,119,154,312,348,385),(76,120,155,305,349,386),(77,113,156,306,350,387),(78,114,157,307,351,388),(79,115,158,308,352,389),(80,116,159,309,345,390),(89,361,405,328,136,172),(90,362,406,321,129,173),(91,363,407,322,130,174),(92,364,408,323,131,175),(93,365,401,324,132,176),(94,366,402,325,133,169),(95,367,403,326,134,170),(96,368,404,327,135,171),(97,382,297,341,144,360),(98,383,298,342,137,353),(99,384,299,343,138,354),(100,377,300,344,139,355),(101,378,301,337,140,356),(102,379,302,338,141,357),(103,380,303,339,142,358),(104,381,304,340,143,359),(177,394,374,410,333,313),(178,395,375,411,334,314),(179,396,376,412,335,315),(180,397,369,413,336,316),(181,398,370,414,329,317),(182,399,371,415,330,318),(183,400,372,416,331,319),(184,393,373,409,332,320)], [(1,465,188,51,59),(2,60,52,189,466),(3,467,190,53,61),(4,62,54,191,468),(5,469,192,55,63),(6,64,56,185,470),(7,471,186,49,57),(8,58,50,187,472),(9,264,425,433,201),(10,202,434,426,257),(11,258,427,435,203),(12,204,436,428,259),(13,260,429,437,205),(14,206,438,430,261),(15,262,431,439,207),(16,208,440,432,263),(17,271,32,277,209),(18,210,278,25,272),(19,265,26,279,211),(20,212,280,27,266),(21,267,28,273,213),(22,214,274,29,268),(23,269,30,275,215),(24,216,276,31,270),(33,219,441,449,226),(34,227,450,442,220),(35,221,443,451,228),(36,229,452,444,222),(37,223,445,453,230),(38,231,454,446,224),(39,217,447,455,232),(40,225,456,448,218),(41,291,283,464,235),(42,236,457,284,292),(43,293,285,458,237),(44,238,459,286,294),(45,295,287,460,239),(46,240,461,288,296),(47,289,281,462,233),(48,234,463,282,290),(65,158,357,326,334),(66,335,327,358,159),(67,160,359,328,336),(68,329,321,360,153),(69,154,353,322,330),(70,331,323,354,155),(71,156,355,324,332),(72,333,325,356,157),(73,340,361,369,165),(74,166,370,362,341),(75,342,363,371,167),(76,168,372,364,343),(77,344,365,373,161),(78,162,374,366,337),(79,338,367,375,163),(80,164,376,368,339),(81,183,175,384,349),(82,350,377,176,184),(83,177,169,378,351),(84,352,379,170,178),(85,179,171,380,345),(86,346,381,172,180),(87,181,173,382,347),(88,348,383,174,182),(89,397,121,391,304),(90,297,392,122,398),(91,399,123,385,298),(92,299,386,124,400),(93,393,125,387,300),(94,301,388,126,394),(95,395,127,389,302),(96,303,390,128,396),(97,311,112,317,129),(98,130,318,105,312),(99,305,106,319,131),(100,132,320,107,306),(101,307,108,313,133),(102,134,314,109,308),(103,309,110,315,135),(104,136,316,111,310),(113,139,401,409,146),(114,147,410,402,140),(115,141,403,411,148),(116,149,412,404,142),(117,143,405,413,150),(118,151,414,406,144),(119,137,407,415,152),(120,145,416,408,138),(193,423,249,241,480),(194,473,242,250,424),(195,417,251,243,474),(196,475,244,252,418),(197,419,253,245,476),(198,477,246,254,420),(199,421,255,247,478),(200,479,248,256,422)], [(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)]])

192 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 5A 5B 6A ··· 6N 8A ··· 8P 10A ··· 10N 12A ··· 12P 15A 15B 15C 15D 20A ··· 20P 24A ··· 24AF 30A ··· 30AB 60A ··· 60AF order 1 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 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 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C6×D5 C3×Dic5 C3×C5⋊2C8 kernel C2×C6×C5⋊2C8 C6×C5⋊2C8 C22×C60 C22×C5⋊2C8 C2×C60 C22×C30 C2×C5⋊2C8 C22×C20 C2×C30 C2×C20 C22×C10 C2×C10 C22×C12 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 6 1 2 6 2 12 2 16 12 4 32 2 6 6 2 4 16 12 12 4 32

Matrix representation of C2×C6×C52C8 in GL4(𝔽241) generated by

 1 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 226 0 0 0 0 240 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 189 240 0 0 1 0
,
 64 0 0 0 0 240 0 0 0 0 70 203 0 0 178 171
G:=sub<GL(4,GF(241))| [1,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[226,0,0,0,0,240,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,189,1,0,0,240,0],[64,0,0,0,0,240,0,0,0,0,70,178,0,0,203,171] >;

C2×C6×C52C8 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_5\rtimes_2C_8
% in TeX

G:=Group("C2xC6xC5:2C8");
// GroupNames label

G:=SmallGroup(480,713);
// by ID

G=gap.SmallGroup(480,713);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,102,18822]);
// 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^-1>;
// generators/relations

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