Copied to
clipboard

## G = C5×C8⋊Dic3order 480 = 25·3·5

### Direct product of C5 and C8⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C8⋊Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C5×C4⋊Dic3 — C5×C8⋊Dic3
 Lower central C3 — C6 — C12 — C5×C8⋊Dic3
 Upper central C1 — C2×C10 — C2×C20 — C2×C40

Generators and relations for C5×C8⋊Dic3
G = < a,b,c,d | a5=b8=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >

Subgroups: 164 in 72 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, C20, C20, C2×C10, C24, C2×Dic3, C2×C12, C30, C30, C4.Q8, C40, C2×C20, C2×C20, C4⋊Dic3, C2×C24, C5×Dic3, C60, C2×C30, C5×C4⋊C4, C2×C40, C8⋊Dic3, C120, C10×Dic3, C2×C60, C5×C4.Q8, C5×C4⋊Dic3, C2×C120, C5×C8⋊Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C10, Dic3, D6, C4⋊C4, SD16, C20, C2×C10, Dic6, D12, C2×Dic3, C5×S3, C4.Q8, C2×C20, C5×D4, C5×Q8, C24⋊C2, C4⋊Dic3, C5×Dic3, S3×C10, C5×C4⋊C4, C5×SD16, C8⋊Dic3, C5×Dic6, C5×D12, C10×Dic3, C5×C4.Q8, C5×C24⋊C2, C5×C4⋊Dic3, C5×C8⋊Dic3

Smallest permutation representation of C5×C8⋊Dic3
Regular action on 480 points
Generators in S480
(1 365 105 221 81)(2 366 106 222 82)(3 367 107 223 83)(4 368 108 224 84)(5 361 109 217 85)(6 362 110 218 86)(7 363 111 219 87)(8 364 112 220 88)(9 347 355 323 331)(10 348 356 324 332)(11 349 357 325 333)(12 350 358 326 334)(13 351 359 327 335)(14 352 360 328 336)(15 345 353 321 329)(16 346 354 322 330)(17 56 70 32 46)(18 49 71 25 47)(19 50 72 26 48)(20 51 65 27 41)(21 52 66 28 42)(22 53 67 29 43)(23 54 68 30 44)(24 55 69 31 45)(33 149 188 57 173)(34 150 189 58 174)(35 151 190 59 175)(36 152 191 60 176)(37 145 192 61 169)(38 146 185 62 170)(39 147 186 63 171)(40 148 187 64 172)(73 95 263 97 119)(74 96 264 98 120)(75 89 257 99 113)(76 90 258 100 114)(77 91 259 101 115)(78 92 260 102 116)(79 93 261 103 117)(80 94 262 104 118)(121 194 184 161 160)(122 195 177 162 153)(123 196 178 163 154)(124 197 179 164 155)(125 198 180 165 156)(126 199 181 166 157)(127 200 182 167 158)(128 193 183 168 159)(129 343 457 319 433)(130 344 458 320 434)(131 337 459 313 435)(132 338 460 314 436)(133 339 461 315 437)(134 340 462 316 438)(135 341 463 317 439)(136 342 464 318 440)(137 475 466 451 442)(138 476 467 452 443)(139 477 468 453 444)(140 478 469 454 445)(141 479 470 455 446)(142 480 471 456 447)(143 473 472 449 448)(144 474 465 450 441)(201 369 234 225 210)(202 370 235 226 211)(203 371 236 227 212)(204 372 237 228 213)(205 373 238 229 214)(206 374 239 230 215)(207 375 240 231 216)(208 376 233 232 209)(241 289 411 265 387)(242 290 412 266 388)(243 291 413 267 389)(244 292 414 268 390)(245 293 415 269 391)(246 294 416 270 392)(247 295 409 271 385)(248 296 410 272 386)(249 297 305 273 281)(250 298 306 274 282)(251 299 307 275 283)(252 300 308 276 284)(253 301 309 277 285)(254 302 310 278 286)(255 303 311 279 287)(256 304 312 280 288)(377 425 424 401 400)(378 426 417 402 393)(379 427 418 403 394)(380 428 419 404 395)(381 429 420 405 396)(382 430 421 406 397)(383 431 422 407 398)(384 432 423 408 399)
(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 95 206 187 17 126)(2 96 207 188 18 127)(3 89 208 189 19 128)(4 90 201 190 20 121)(5 91 202 191 21 122)(6 92 203 192 22 123)(7 93 204 185 23 124)(8 94 205 186 24 125)(9 133 432 304 296 137)(10 134 425 297 289 138)(11 135 426 298 290 139)(12 136 427 299 291 140)(13 129 428 300 292 141)(14 130 429 301 293 142)(15 131 430 302 294 143)(16 132 431 303 295 144)(25 167 222 120 231 33)(26 168 223 113 232 34)(27 161 224 114 225 35)(28 162 217 115 226 36)(29 163 218 116 227 37)(30 164 219 117 228 38)(31 165 220 118 229 39)(32 166 221 119 230 40)(41 160 84 76 210 151)(42 153 85 77 211 152)(43 154 86 78 212 145)(44 155 87 79 213 146)(45 156 88 80 214 147)(46 157 81 73 215 148)(47 158 82 74 216 149)(48 159 83 75 209 150)(49 200 366 264 375 57)(50 193 367 257 376 58)(51 194 368 258 369 59)(52 195 361 259 370 60)(53 196 362 260 371 61)(54 197 363 261 372 62)(55 198 364 262 373 63)(56 199 365 263 374 64)(65 184 108 100 234 175)(66 177 109 101 235 176)(67 178 110 102 236 169)(68 179 111 103 237 170)(69 180 112 104 238 171)(70 181 105 97 239 172)(71 182 106 98 240 173)(72 183 107 99 233 174)(241 443 332 438 377 249)(242 444 333 439 378 250)(243 445 334 440 379 251)(244 446 335 433 380 252)(245 447 336 434 381 253)(246 448 329 435 382 254)(247 441 330 436 383 255)(248 442 331 437 384 256)(265 467 356 462 401 273)(266 468 357 463 402 274)(267 469 358 464 403 275)(268 470 359 457 404 276)(269 471 360 458 405 277)(270 472 353 459 406 278)(271 465 354 460 407 279)(272 466 355 461 408 280)(281 387 452 324 316 400)(282 388 453 325 317 393)(283 389 454 326 318 394)(284 390 455 327 319 395)(285 391 456 328 320 396)(286 392 449 321 313 397)(287 385 450 322 314 398)(288 386 451 323 315 399)(305 411 476 348 340 424)(306 412 477 349 341 417)(307 413 478 350 342 418)(308 414 479 351 343 419)(309 415 480 352 344 420)(310 416 473 345 337 421)(311 409 474 346 338 422)(312 410 475 347 339 423)
(1 242 187 439)(2 245 188 434)(3 248 189 437)(4 243 190 440)(5 246 191 435)(6 241 192 438)(7 244 185 433)(8 247 186 436)(9 257 304 50)(10 260 297 53)(11 263 298 56)(12 258 299 51)(13 261 300 54)(14 264 301 49)(15 259 302 52)(16 262 303 55)(17 333 95 250)(18 336 96 253)(19 331 89 256)(20 334 90 251)(21 329 91 254)(22 332 92 249)(23 335 93 252)(24 330 94 255)(25 360 120 277)(26 355 113 280)(27 358 114 275)(28 353 115 278)(29 356 116 273)(30 359 117 276)(31 354 118 279)(32 357 119 274)(33 458 222 269)(34 461 223 272)(35 464 224 267)(36 459 217 270)(37 462 218 265)(38 457 219 268)(39 460 220 271)(40 463 221 266)(41 326 76 283)(42 321 77 286)(43 324 78 281)(44 327 79 284)(45 322 80 287)(46 325 73 282)(47 328 74 285)(48 323 75 288)(57 130 366 293)(58 133 367 296)(59 136 368 291)(60 131 361 294)(61 134 362 289)(62 129 363 292)(63 132 364 295)(64 135 365 290)(65 350 100 307)(66 345 101 310)(67 348 102 305)(68 351 103 308)(69 346 104 311)(70 349 97 306)(71 352 98 309)(72 347 99 312)(81 388 148 317)(82 391 149 320)(83 386 150 315)(84 389 151 318)(85 392 152 313)(86 387 145 316)(87 390 146 319)(88 385 147 314)(105 412 172 341)(106 415 173 344)(107 410 174 339)(108 413 175 342)(109 416 176 337)(110 411 169 340)(111 414 170 343)(112 409 171 338)(121 445 201 379)(122 448 202 382)(123 443 203 377)(124 446 204 380)(125 441 205 383)(126 444 206 378)(127 447 207 381)(128 442 208 384)(137 376 432 193)(138 371 425 196)(139 374 426 199)(140 369 427 194)(141 372 428 197)(142 375 429 200)(143 370 430 195)(144 373 431 198)(153 449 211 397)(154 452 212 400)(155 455 213 395)(156 450 214 398)(157 453 215 393)(158 456 216 396)(159 451 209 399)(160 454 210 394)(161 469 225 403)(162 472 226 406)(163 467 227 401)(164 470 228 404)(165 465 229 407)(166 468 230 402)(167 471 231 405)(168 466 232 408)(177 473 235 421)(178 476 236 424)(179 479 237 419)(180 474 238 422)(181 477 239 417)(182 480 240 420)(183 475 233 423)(184 478 234 418)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I ··· 20X 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 2 2 2 12 12 12 12 1 1 1 1 2 2 2 2 2 2 2 1 ··· 1 2 2 2 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C4 C5 C10 C10 C20 S3 Q8 D4 Dic3 D6 SD16 Dic6 D12 C5×S3 C5×Q8 C5×D4 C24⋊C2 C5×Dic3 S3×C10 C5×SD16 C5×Dic6 C5×D12 C5×C24⋊C2 kernel C5×C8⋊Dic3 C5×C4⋊Dic3 C2×C120 C120 C8⋊Dic3 C4⋊Dic3 C2×C24 C24 C2×C40 C60 C2×C30 C40 C2×C20 C30 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 2 1 4 4 8 4 16 1 1 1 2 1 4 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C5×C8⋊Dic3 in GL3(𝔽241) generated by

 1 0 0 0 91 0 0 0 91
,
 240 0 0 0 175 147 0 94 28
,
 240 0 0 0 240 240 0 1 0
,
 177 0 0 0 168 223 0 55 73
G:=sub<GL(3,GF(241))| [1,0,0,0,91,0,0,0,91],[240,0,0,0,175,94,0,147,28],[240,0,0,0,240,1,0,240,0],[177,0,0,0,168,55,0,223,73] >;

C5×C8⋊Dic3 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes {\rm Dic}_3
% in TeX

G:=Group("C5xC8:Dic3");
// GroupNames label

G:=SmallGroup(480,136);
// by ID

G=gap.SmallGroup(480,136);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,288,4204,102,15686]);
// Polycyclic

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

׿
×
𝔽