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G = C2×C10×C3⋊C8order 480 = 25·3·5

Direct product of C2×C10 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C10×C3⋊C8
G = < a,b,c,d | a2=b10=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 196 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, C3⋊C8, C2×C12, C22×C6, C30, C30, C22×C8, C40, C2×C20, C22×C10, C2×C3⋊C8, C22×C12, C60, C60, C2×C30, C2×C40, C22×C20, C22×C3⋊C8, C5×C3⋊C8, C2×C60, C22×C30, C22×C40, C10×C3⋊C8, C22×C60, C2×C10×C3⋊C8
Quotients: C1, C2, C4, C22, C5, S3, C8, C2×C4, C23, C10, Dic3, D6, C2×C8, C22×C4, C20, C2×C10, C3⋊C8, C2×Dic3, C22×S3, C5×S3, C22×C8, C40, C2×C20, C22×C10, C2×C3⋊C8, C22×Dic3, C5×Dic3, S3×C10, C2×C40, C22×C20, C22×C3⋊C8, C5×C3⋊C8, C10×Dic3, S3×C2×C10, C22×C40, C10×C3⋊C8, Dic3×C2×C10, C2×C10×C3⋊C8

Smallest permutation representation of C2×C10×C3⋊C8
Regular action on 480 points
Generators in S480
(1 293)(2 294)(3 295)(4 296)(5 297)(6 298)(7 299)(8 300)(9 291)(10 292)(11 380)(12 371)(13 372)(14 373)(15 374)(16 375)(17 376)(18 377)(19 378)(20 379)(21 433)(22 434)(23 435)(24 436)(25 437)(26 438)(27 439)(28 440)(29 431)(30 432)(31 443)(32 444)(33 445)(34 446)(35 447)(36 448)(37 449)(38 450)(39 441)(40 442)(41 453)(42 454)(43 455)(44 456)(45 457)(46 458)(47 459)(48 460)(49 451)(50 452)(51 150)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 407)(62 408)(63 409)(64 410)(65 401)(66 402)(67 403)(68 404)(69 405)(70 406)(71 417)(72 418)(73 419)(74 420)(75 411)(76 412)(77 413)(78 414)(79 415)(80 416)(81 367)(82 368)(83 369)(84 370)(85 361)(86 362)(87 363)(88 364)(89 365)(90 366)(91 428)(92 429)(93 430)(94 421)(95 422)(96 423)(97 424)(98 425)(99 426)(100 427)(101 238)(102 239)(103 240)(104 231)(105 232)(106 233)(107 234)(108 235)(109 236)(110 237)(111 397)(112 398)(113 399)(114 400)(115 391)(116 392)(117 393)(118 394)(119 395)(120 396)(121 246)(122 247)(123 248)(124 249)(125 250)(126 241)(127 242)(128 243)(129 244)(130 245)(131 256)(132 257)(133 258)(134 259)(135 260)(136 251)(137 252)(138 253)(139 254)(140 255)(151 302)(152 303)(153 304)(154 305)(155 306)(156 307)(157 308)(158 309)(159 310)(160 301)(161 286)(162 287)(163 288)(164 289)(165 290)(166 281)(167 282)(168 283)(169 284)(170 285)(171 277)(172 278)(173 279)(174 280)(175 271)(176 272)(177 273)(178 274)(179 275)(180 276)(181 224)(182 225)(183 226)(184 227)(185 228)(186 229)(187 230)(188 221)(189 222)(190 223)(191 316)(192 317)(193 318)(194 319)(195 320)(196 311)(197 312)(198 313)(199 314)(200 315)(201 326)(202 327)(203 328)(204 329)(205 330)(206 321)(207 322)(208 323)(209 324)(210 325)(211 336)(212 337)(213 338)(214 339)(215 340)(216 331)(217 332)(218 333)(219 334)(220 335)(261 386)(262 387)(263 388)(264 389)(265 390)(266 381)(267 382)(268 383)(269 384)(270 385)(341 466)(342 467)(343 468)(344 469)(345 470)(346 461)(347 462)(348 463)(349 464)(350 465)(351 476)(352 477)(353 478)(354 479)(355 480)(356 471)(357 472)(358 473)(359 474)(360 475)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190)(191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230)(231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250)(251 252 253 254 255 256 257 258 259 260)(261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290)(291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310)(311 312 313 314 315 316 317 318 319 320)(321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340)(341 342 343 344 345 346 347 348 349 350)(351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370)(371 372 373 374 375 376 377 378 379 380)(381 382 383 384 385 386 387 388 389 390)(391 392 393 394 395 396 397 398 399 400)(401 402 403 404 405 406 407 408 409 410)(411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430)(431 432 433 434 435 436 437 438 439 440)(441 442 443 444 445 446 447 448 449 450)(451 452 453 454 455 456 457 458 459 460)(461 462 463 464 465 466 467 468 469 470)(471 472 473 474 475 476 477 478 479 480)
(1 135 203)(2 136 204)(3 137 205)(4 138 206)(5 139 207)(6 140 208)(7 131 209)(8 132 210)(9 133 201)(10 134 202)(11 77 36)(12 78 37)(13 79 38)(14 80 39)(15 71 40)(16 72 31)(17 73 32)(18 74 33)(19 75 34)(20 76 35)(21 82 384)(22 83 385)(23 84 386)(24 85 387)(25 86 388)(26 87 389)(27 88 390)(28 89 381)(29 90 382)(30 81 383)(41 95 473)(42 96 474)(43 97 475)(44 98 476)(45 99 477)(46 100 478)(47 91 479)(48 92 480)(49 93 471)(50 94 472)(51 124 199)(52 125 200)(53 126 191)(54 127 192)(55 128 193)(56 129 194)(57 130 195)(58 121 196)(59 122 197)(60 123 198)(61 468 120)(62 469 111)(63 470 112)(64 461 113)(65 462 114)(66 463 115)(67 464 116)(68 465 117)(69 466 118)(70 467 119)(101 151 218)(102 152 219)(103 153 220)(104 154 211)(105 155 212)(106 156 213)(107 157 214)(108 158 215)(109 159 216)(110 160 217)(141 250 315)(142 241 316)(143 242 317)(144 243 318)(145 244 319)(146 245 320)(147 246 311)(148 247 312)(149 248 313)(150 249 314)(161 179 189)(162 180 190)(163 171 181)(164 172 182)(165 173 183)(166 174 184)(167 175 185)(168 176 186)(169 177 187)(170 178 188)(221 285 274)(222 286 275)(223 287 276)(224 288 277)(225 289 278)(226 290 279)(227 281 280)(228 282 271)(229 283 272)(230 284 273)(231 305 336)(232 306 337)(233 307 338)(234 308 339)(235 309 340)(236 310 331)(237 301 332)(238 302 333)(239 303 334)(240 304 335)(251 329 294)(252 330 295)(253 321 296)(254 322 297)(255 323 298)(256 324 299)(257 325 300)(258 326 291)(259 327 292)(260 328 293)(261 435 370)(262 436 361)(263 437 362)(264 438 363)(265 439 364)(266 440 365)(267 431 366)(268 432 367)(269 433 368)(270 434 369)(341 394 405)(342 395 406)(343 396 407)(344 397 408)(345 398 409)(346 399 410)(347 400 401)(348 391 402)(349 392 403)(350 393 404)(351 456 425)(352 457 426)(353 458 427)(354 459 428)(355 460 429)(356 451 430)(357 452 421)(358 453 422)(359 454 423)(360 455 424)(371 414 449)(372 415 450)(373 416 441)(374 417 442)(375 418 443)(376 419 444)(377 420 445)(378 411 446)(379 412 447)(380 413 448)
(1 25 190 72 152 463 197 91)(2 26 181 73 153 464 198 92)(3 27 182 74 154 465 199 93)(4 28 183 75 155 466 200 94)(5 29 184 76 156 467 191 95)(6 30 185 77 157 468 192 96)(7 21 186 78 158 469 193 97)(8 22 187 79 159 470 194 98)(9 23 188 80 160 461 195 99)(10 24 189 71 151 462 196 100)(11 214 61 54 42 140 383 167)(12 215 62 55 43 131 384 168)(13 216 63 56 44 132 385 169)(14 217 64 57 45 133 386 170)(15 218 65 58 46 134 387 161)(16 219 66 59 47 135 388 162)(17 220 67 60 48 136 389 163)(18 211 68 51 49 137 390 164)(19 212 69 52 50 138 381 165)(20 213 70 53 41 139 382 166)(31 102 115 122 479 203 86 180)(32 103 116 123 480 204 87 171)(33 104 117 124 471 205 88 172)(34 105 118 125 472 206 89 173)(35 106 119 126 473 207 90 174)(36 107 120 127 474 208 81 175)(37 108 111 128 475 209 82 176)(38 109 112 129 476 210 83 177)(39 110 113 130 477 201 84 178)(40 101 114 121 478 202 85 179)(141 452 253 266 290 378 337 405)(142 453 254 267 281 379 338 406)(143 454 255 268 282 380 339 407)(144 455 256 269 283 371 340 408)(145 456 257 270 284 372 331 409)(146 457 258 261 285 373 332 410)(147 458 259 262 286 374 333 401)(148 459 260 263 287 375 334 402)(149 460 251 264 288 376 335 403)(150 451 252 265 289 377 336 404)(221 416 301 346 320 426 291 435)(222 417 302 347 311 427 292 436)(223 418 303 348 312 428 293 437)(224 419 304 349 313 429 294 438)(225 420 305 350 314 430 295 439)(226 411 306 341 315 421 296 440)(227 412 307 342 316 422 297 431)(228 413 308 343 317 423 298 432)(229 414 309 344 318 424 299 433)(230 415 310 345 319 425 300 434)(231 393 249 356 330 364 278 445)(232 394 250 357 321 365 279 446)(233 395 241 358 322 366 280 447)(234 396 242 359 323 367 271 448)(235 397 243 360 324 368 272 449)(236 398 244 351 325 369 273 450)(237 399 245 352 326 370 274 441)(238 400 246 353 327 361 275 442)(239 391 247 354 328 362 276 443)(240 392 248 355 329 363 277 444)

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

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

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

240 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 5A 5B 5C 5D 6A ··· 6G 8A ··· 8P 10A ··· 10AB 12A ··· 12H 15A 15B 15C 15D 20A ··· 20AF 30A ··· 30AB 40A ··· 40BL 60A ··· 60AF order 1 2 ··· 2 3 4 ··· 4 5 5 5 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 ··· 1 2 1 ··· 1 1 1 1 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

240 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 C4 C4 C5 C8 C10 C10 C20 C20 C40 S3 Dic3 D6 Dic3 C3⋊C8 C5×S3 C5×Dic3 S3×C10 C5×Dic3 C5×C3⋊C8 kernel C2×C10×C3⋊C8 C10×C3⋊C8 C22×C60 C2×C60 C22×C30 C22×C3⋊C8 C2×C30 C2×C3⋊C8 C22×C12 C2×C12 C22×C6 C2×C6 C22×C20 C2×C20 C2×C20 C22×C10 C2×C10 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 4 16 24 4 24 8 64 1 3 3 1 8 4 12 12 4 32

Matrix representation of C2×C10×C3⋊C8 in GL4(𝔽241) generated by

 1 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 154 0 0 0 0 154
,
 1 0 0 0 0 1 0 0 0 0 240 240 0 0 1 0
,
 240 0 0 0 0 240 0 0 0 0 34 159 0 0 125 207
G:=sub<GL(4,GF(241))| [1,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,154,0,0,0,0,154],[1,0,0,0,0,1,0,0,0,0,240,1,0,0,240,0],[240,0,0,0,0,240,0,0,0,0,34,125,0,0,159,207] >;

C2×C10×C3⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_{10}\times C_3\rtimes C_8
% in TeX

G:=Group("C2xC10xC3:C8");
// GroupNames label

G:=SmallGroup(480,799);
// by ID

G=gap.SmallGroup(480,799);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^3=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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