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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 [×6], C3, C4, C4 [×3], C22 [×7], C5, C6, C6 [×6], C8 [×4], C2×C4 [×6], C23, C10, C10 [×6], C12, C12 [×3], C2×C6 [×7], C15, C2×C8 [×6], C22×C4, C20, C20 [×3], C2×C10 [×7], C3⋊C8 [×4], C2×C12 [×6], C22×C6, C30, C30 [×6], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C2×C3⋊C8 [×6], C22×C12, C60, C60 [×3], C2×C30 [×7], C2×C40 [×6], C22×C20, C22×C3⋊C8, C5×C3⋊C8 [×4], C2×C60 [×6], C22×C30, C22×C40, C10×C3⋊C8 [×6], C22×C60, C2×C10×C3⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C8 [×4], C2×C4 [×6], C23, C10 [×7], Dic3 [×4], D6 [×3], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C3⋊C8 [×4], C2×Dic3 [×6], C22×S3, C5×S3, C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C2×C3⋊C8 [×6], C22×Dic3, C5×Dic3 [×4], S3×C10 [×3], C2×C40 [×6], C22×C20, C22×C3⋊C8, C5×C3⋊C8 [×4], C10×Dic3 [×6], S3×C2×C10, C22×C40, C10×C3⋊C8 [×6], Dic3×C2×C10, C2×C10×C3⋊C8

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

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

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

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

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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