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G = C5×C8⋊Dic3order 480 = 25·3·5

Direct product of C5 and C8⋊Dic3

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C8⋊Dic3, C242C20, C12016C4, C60.30Q8, C4010Dic3, C30.23SD16, C20.24Dic6, C82(C5×Dic3), C12.4(C5×Q8), (C2×C40).16S3, (C2×C24).8C10, C1512(C4.Q8), C30.57(C4⋊C4), C6.2(C5×SD16), C4.4(C5×Dic6), (C2×C120).24C2, C12.34(C2×C20), C60.244(C2×C4), (C2×C30).119D4, (C2×C20).421D6, (C2×C10).49D12, C4⋊Dic3.2C10, C4.6(C10×Dic3), C22.8(C5×D12), C20.66(C2×Dic3), C10.10(C24⋊C2), (C2×C60).522C22, C10.19(C4⋊Dic3), C32(C5×C4.Q8), C6.5(C5×C4⋊C4), (C2×C8).6(C5×S3), C2.2(C5×C24⋊C2), (C2×C6).13(C5×D4), C2.3(C5×C4⋊Dic3), (C2×C4).71(S3×C10), (C2×C12).86(C2×C10), (C5×C4⋊Dic3).14C2, SmallGroup(480,136)

Series: Derived Chief Lower central Upper central

C1C12 — C5×C8⋊Dic3
C1C3C6C12C2×C12C2×C60C5×C4⋊Dic3 — C5×C8⋊Dic3
C3C6C12 — C5×C8⋊Dic3
C1C2×C10C2×C20C2×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 [×2], C3, C4 [×2], C4 [×2], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, C20 [×2], C20 [×2], C2×C10, C24 [×2], C2×Dic3 [×2], C2×C12, C30, C30 [×2], C4.Q8, C40 [×2], C2×C20, C2×C20 [×2], C4⋊Dic3 [×2], C2×C24, C5×Dic3 [×2], C60 [×2], C2×C30, C5×C4⋊C4 [×2], C2×C40, C8⋊Dic3, C120 [×2], C10×Dic3 [×2], C2×C60, C5×C4.Q8, C5×C4⋊Dic3 [×2], C2×C120, C5×C8⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4, Q8, C10 [×3], Dic3 [×2], D6, C4⋊C4, SD16 [×2], C20 [×2], C2×C10, Dic6, D12, C2×Dic3, C5×S3, C4.Q8, C2×C20, C5×D4, C5×Q8, C24⋊C2 [×2], C4⋊Dic3, C5×Dic3 [×2], S3×C10, C5×C4⋊C4, C5×SD16 [×2], C8⋊Dic3, C5×Dic6, C5×D12, C10×Dic3, C5×C4.Q8, C5×C24⋊C2 [×2], 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)(9 341 463 317 439)(10 342 464 318 440)(11 343 457 319 433)(12 344 458 320 434)(13 337 459 313 435)(14 338 460 314 436)(15 339 461 315 437)(16 340 462 316 438)(17 197 72 26 48)(18 198 65 27 41)(19 199 66 28 42)(20 200 67 29 43)(21 193 68 30 44)(22 194 69 31 45)(23 195 70 32 46)(24 196 71 25 47)(33 149 50 57 173)(34 150 51 58 174)(35 151 52 59 175)(36 152 53 60 176)(37 145 54 61 169)(38 146 55 62 170)(39 147 56 63 171)(40 148 49 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)(121 185 184 161 160)(122 186 177 162 153)(123 187 178 163 154)(124 188 179 164 155)(125 189 180 165 156)(126 190 181 166 157)(127 191 182 167 158)(128 192 183 168 159)(129 472 357 448 333)(130 465 358 441 334)(131 466 359 442 335)(132 467 360 443 336)(133 468 353 444 329)(134 469 354 445 330)(135 470 355 446 331)(136 471 356 447 332)(137 475 351 451 327)(138 476 352 452 328)(139 477 345 453 321)(140 478 346 454 322)(141 479 347 455 323)(142 480 348 456 324)(143 473 349 449 325)(144 474 350 450 326)(201 254 234 225 210)(202 255 235 226 211)(203 256 236 227 212)(204 249 237 228 213)(205 250 238 229 214)(206 251 239 230 215)(207 252 240 231 216)(208 253 233 232 209)(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)(273 381 429 297 405)(274 382 430 298 406)(275 383 431 299 407)(276 384 432 300 408)(277 377 425 301 401)(278 378 426 302 402)(279 379 427 303 403)(280 380 428 304 404)(281 372 420 305 396)(282 373 421 306 397)(283 374 422 307 398)(284 375 423 308 399)(285 376 424 309 400)(286 369 417 310 393)(287 370 418 311 394)(288 371 419 312 395)
(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 49 23 126)(2 96 207 50 24 127)(3 89 208 51 17 128)(4 90 201 52 18 121)(5 91 202 53 19 122)(6 92 203 54 20 123)(7 93 204 55 21 124)(8 94 205 56 22 125)(9 426 421 290 139 129)(10 427 422 291 140 130)(11 428 423 292 141 131)(12 429 424 293 142 132)(13 430 417 294 143 133)(14 431 418 295 144 134)(15 432 419 296 137 135)(16 425 420 289 138 136)(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)(57 196 191 366 264 252)(58 197 192 367 257 253)(59 198 185 368 258 254)(60 199 186 361 259 255)(61 200 187 362 260 256)(62 193 188 363 261 249)(63 194 189 364 262 250)(64 195 190 365 263 251)(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)(241 328 332 438 377 372)(242 321 333 439 378 373)(243 322 334 440 379 374)(244 323 335 433 380 375)(245 324 336 434 381 376)(246 325 329 435 382 369)(247 326 330 436 383 370)(248 327 331 437 384 371)(265 352 356 462 401 396)(266 345 357 463 402 397)(267 346 358 464 403 398)(268 347 359 457 404 399)(269 348 360 458 405 400)(270 349 353 459 406 393)(271 350 354 460 407 394)(272 351 355 461 408 395)(273 285 391 456 443 320)(274 286 392 449 444 313)(275 287 385 450 445 314)(276 288 386 451 446 315)(277 281 387 452 447 316)(278 282 388 453 448 317)(279 283 389 454 441 318)(280 284 390 455 442 319)(297 309 415 480 467 344)(298 310 416 473 468 337)(299 311 409 474 469 338)(300 312 410 475 470 339)(301 305 411 476 471 340)(302 306 412 477 472 341)(303 307 413 478 465 342)(304 308 414 479 466 343)
(1 242 49 439)(2 245 50 434)(3 248 51 437)(4 243 52 440)(5 246 53 435)(6 241 54 438)(7 244 55 433)(8 247 56 436)(9 365 290 64)(10 368 291 59)(11 363 292 62)(12 366 293 57)(13 361 294 60)(14 364 295 63)(15 367 296 58)(16 362 289 61)(17 331 89 371)(18 334 90 374)(19 329 91 369)(20 332 92 372)(21 335 93 375)(22 330 94 370)(23 333 95 373)(24 336 96 376)(25 360 120 400)(26 355 113 395)(27 358 114 398)(28 353 115 393)(29 356 116 396)(30 359 117 399)(31 354 118 394)(32 357 119 397)(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)(41 441 76 283)(42 444 77 286)(43 447 78 281)(44 442 79 284)(45 445 80 287)(46 448 73 282)(47 443 74 285)(48 446 75 288)(65 465 100 307)(66 468 101 310)(67 471 102 305)(68 466 103 308)(69 469 104 311)(70 472 97 306)(71 467 98 309)(72 470 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)(121 322 201 379)(122 325 202 382)(123 328 203 377)(124 323 204 380)(125 326 205 383)(126 321 206 378)(127 324 207 381)(128 327 208 384)(129 263 421 195)(130 258 422 198)(131 261 423 193)(132 264 424 196)(133 259 417 199)(134 262 418 194)(135 257 419 197)(136 260 420 200)(137 253 432 192)(138 256 425 187)(139 251 426 190)(140 254 427 185)(141 249 428 188)(142 252 429 191)(143 255 430 186)(144 250 431 189)(153 449 211 274)(154 452 212 277)(155 455 213 280)(156 450 214 275)(157 453 215 278)(158 456 216 273)(159 451 209 276)(160 454 210 279)(161 346 225 403)(162 349 226 406)(163 352 227 401)(164 347 228 404)(165 350 229 407)(166 345 230 402)(167 348 231 405)(168 351 232 408)(177 473 235 298)(178 476 236 301)(179 479 237 304)(180 474 238 299)(181 477 239 302)(182 480 240 297)(183 475 233 300)(184 478 234 303)

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

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

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

150 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B5C5D6A6B6C8A8B8C8D10A···10L12A12B12C12D15A15B15C15D20A···20H20I···20X24A···24H30A···30L40A···40P60A···60P120A···120AF
order122234444445555666888810···10121212121515151520···2020···2024···2430···3040···4060···60120···120
size111122212121212111122222221···1222222222···212···122···22···22···22···22···2

150 irreducible representations

dim11111111222222222222222222
type++++-+-+-+
imageC1C2C2C4C5C10C10C20S3Q8D4Dic3D6SD16Dic6D12C5×S3C5×Q8C5×D4C24⋊C2C5×Dic3S3×C10C5×SD16C5×Dic6C5×D12C5×C24⋊C2
kernelC5×C8⋊Dic3C5×C4⋊Dic3C2×C120C120C8⋊Dic3C4⋊Dic3C2×C24C24C2×C40C60C2×C30C40C2×C20C30C20C2×C10C2×C8C12C2×C6C10C8C2×C4C6C4C22C2
# reps12144841611121422444884168832

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

100
0910
0091
,
24000
0175147
09428
,
24000
0240240
010
,
17700
0168223
05573
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

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