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G = C2×C6×C5⋊C8  order 480 = 25·3·5

Direct product of C2×C6 and C5⋊C8

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C6×C5⋊C8, (C2×C30)⋊4C8, C30⋊9(C2×C8), (C2×C10)⋊4C24, C10⋊2(C2×C24), C5⋊2(C22×C24), C15⋊11(C22×C8), C23.5(C3×F5), (C22×C6).8F5, C22.19(C6×F5), C6.55(C22×F5), (C22×C10).7C12, (C22×C30).10C4, C30.93(C22×C4), (C6×Dic5).29C4, C10.11(C22×C12), Dic5.18(C2×C12), (C2×Dic5).13C12, (C3×Dic5).74C23, Dic5.14(C22×C6), (C22×Dic5).10C6, (C6×Dic5).279C22, C2.3(C2×C6×F5), (C2×C6).62(C2×F5), (C2×C30).62(C2×C4), (C2×C6×Dic5).19C2, (C2×C10).19(C2×C12), (C2×Dic5).56(C2×C6), (C3×Dic5).74(C2×C4), SmallGroup(480,1057)

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

Subgroups: 328 in 152 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C5, C6, C6 [×6], C8 [×4], C2×C4 [×6], C23, C10, C10 [×6], C12 [×4], C2×C6 [×7], C15, C2×C8 [×6], C22×C4, Dic5, Dic5 [×3], C2×C10 [×7], C24 [×4], C2×C12 [×6], C22×C6, C30, C30 [×6], C22×C8, C5⋊C8 [×4], C2×Dic5 [×6], C22×C10, C2×C24 [×6], C22×C12, C3×Dic5, C3×Dic5 [×3], C2×C30 [×7], C2×C5⋊C8 [×6], C22×Dic5, C22×C24, C3×C5⋊C8 [×4], C6×Dic5 [×6], C22×C30, C22×C5⋊C8, C6×C5⋊C8 [×6], C2×C6×Dic5, C2×C6×C5⋊C8

Quotients:
C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C8 [×4], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C2×C8 [×6], C22×C4, F5, C24 [×4], C2×C12 [×6], C22×C6, C22×C8, C5⋊C8 [×4], C2×F5 [×3], C2×C24 [×6], C22×C12, C3×F5, C2×C5⋊C8 [×6], C22×F5, C22×C24, C3×C5⋊C8 [×4], C6×F5 [×3], C22×C5⋊C8, C6×C5⋊C8 [×6], C2×C6×F5, C2×C6×C5⋊C8

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

Smallest permutation representation
►Regular action on 480 points
Generators in S480
(1 149)(2 150)(3 151)(4 152)(5 145)(6 146)(7 147)(8 148)(9 120)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 383)(18 384)(19 377)(20 378)(21 379)(22 380)(23 381)(24 382)(25 201)(26 202)(27 203)(28 204)(29 205)(30 206)(31 207)(32 208)(33 392)(34 385)(35 386)(36 387)(37 388)(38 389)(39 390)(40 391)(41 227)(42 228)(43 229)(44 230)(45 231)(46 232)(47 225)(48 226)(49 402)(50 403)(51 404)(52 405)(53 406)(54 407)(55 408)(56 401)(57 410)(58 411)(59 412)(60 413)(61 414)(62 415)(63 416)(64 409)(65 143)(66 144)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(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)(105 297)(106 298)(107 299)(108 300)(109 301)(110 302)(111 303)(112 304)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 368)(130 361)(131 362)(132 363)(133 364)(134 365)(135 366)(136 367)(177 280)(178 273)(179 274)(180 275)(181 276)(182 277)(183 278)(184 279)(193 432)(194 425)(195 426)(196 427)(197 428)(198 429)(199 430)(200 431)(209 328)(210 321)(211 322)(212 323)(213 324)(214 325)(215 326)(216 327)(217 341)(218 342)(219 343)(220 344)(221 337)(222 338)(223 339)(224 340)(233 350)(234 351)(235 352)(236 345)(237 346)(238 347)(239 348)(240 349)(241 446)(242 447)(243 448)(244 441)(245 442)(246 443)(247 444)(248 445)(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)(281 306)(282 307)(283 308)(284 309)(285 310)(286 311)(287 312)(288 305)(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)

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

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

Matrix representation ►G ⊆ GL6(𝔽241)

24000000
010000
00240000
00024000
00002400
00000240
,
24000000
02400000
00225000
00022500
00002250
00000225
,
100000
010000
00000240
00100240
00010240
00001240
,
24000000
010000
0064154672
0070226192136
001549105142
001695517787

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] >;
 

120 conjugacy classes

class 1 2A···2G3A3B4A···4H 5 6A···6N8A···8P10A···10G12A···12P15A15B24A···24AF30A···30N
order12···2334···456···68···810···1012···12151524···2430···30
size11···1115···541···15···54···45···5445···54···4

120 irreducible representations

dim111111111111444444
type++++-+
imageC1C2C2C3C4C4C6C6C8C12C12C24F5C5⋊C8C2×F5C3×F5C3×C5⋊C8C6×F5
kernelC2×C6×C5⋊C8C6×C5⋊C8C2×C6×Dic5C22×C5⋊C8C6×Dic5C22×C30C2×C5⋊C8C22×Dic5C2×C30C2×Dic5C22×C10C2×C10C22×C6C2×C6C2×C6C23C22C22
# reps1612621221612432143286

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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