Copied to
clipboard

## G = Dic5×C3⋊C8order 480 = 25·3·5

### Direct product of Dic5 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — Dic5×C3⋊C8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C12×Dic5 — Dic5×C3⋊C8
 Lower central C15 — Dic5×C3⋊C8
 Upper central C1 — C2×C4

Generators and relations for Dic5×C3⋊C8
G = < a,b,c,d | a10=c3=d8=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 236 in 88 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C30, C30, C4×C8, C52C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C3×Dic5, C60, C2×C30, C2×C52C8, C4×Dic5, C2×C40, C4×C3⋊C8, C5×C3⋊C8, C153C8, C6×Dic5, C2×C60, C8×Dic5, C12×Dic5, C10×C3⋊C8, C2×C153C8, Dic5×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D5, Dic3, D6, C42, C2×C8, Dic5, D10, C3⋊C8, C4×S3, C2×Dic3, C4×C8, C4×D5, C2×Dic5, C2×C3⋊C8, C4×Dic3, S3×D5, C8×D5, C4×Dic5, C4×C3⋊C8, D5×Dic3, S3×Dic5, D30.C2, C8×Dic5, D5×C3⋊C8, Dic3×Dic5, Dic5×C3⋊C8

Smallest permutation representation of Dic5×C3⋊C8
Regular action on 480 points
Generators in S480
(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)
(1 282 6 287)(2 281 7 286)(3 290 8 285)(4 289 9 284)(5 288 10 283)(11 353 16 358)(12 352 17 357)(13 351 18 356)(14 360 19 355)(15 359 20 354)(21 361 26 366)(22 370 27 365)(23 369 28 364)(24 368 29 363)(25 367 30 362)(31 335 36 340)(32 334 37 339)(33 333 38 338)(34 332 39 337)(35 331 40 336)(41 311 46 316)(42 320 47 315)(43 319 48 314)(44 318 49 313)(45 317 50 312)(51 252 56 257)(52 251 57 256)(53 260 58 255)(54 259 59 254)(55 258 60 253)(61 133 66 138)(62 132 67 137)(63 131 68 136)(64 140 69 135)(65 139 70 134)(71 297 76 292)(72 296 77 291)(73 295 78 300)(74 294 79 299)(75 293 80 298)(81 346 86 341)(82 345 87 350)(83 344 88 349)(84 343 89 348)(85 342 90 347)(91 322 96 327)(92 321 97 326)(93 330 98 325)(94 329 99 324)(95 328 100 323)(101 268 106 263)(102 267 107 262)(103 266 108 261)(104 265 109 270)(105 264 110 269)(111 278 116 273)(112 277 117 272)(113 276 118 271)(114 275 119 280)(115 274 120 279)(121 303 126 308)(122 302 127 307)(123 301 128 306)(124 310 129 305)(125 309 130 304)(141 401 146 406)(142 410 147 405)(143 409 148 404)(144 408 149 403)(145 407 150 402)(151 455 156 460)(152 454 157 459)(153 453 158 458)(154 452 159 457)(155 451 160 456)(161 431 166 436)(162 440 167 435)(163 439 168 434)(164 438 169 433)(165 437 170 432)(171 372 176 377)(172 371 177 376)(173 380 178 375)(174 379 179 374)(175 378 180 373)(181 480 186 475)(182 479 187 474)(183 478 188 473)(184 477 189 472)(185 476 190 471)(191 417 196 412)(192 416 197 411)(193 415 198 420)(194 414 199 419)(195 413 200 418)(201 466 206 461)(202 465 207 470)(203 464 208 469)(204 463 209 468)(205 462 210 467)(211 442 216 447)(212 441 217 446)(213 450 218 445)(214 449 219 444)(215 448 220 443)(221 388 226 383)(222 387 227 382)(223 386 228 381)(224 385 229 390)(225 384 230 389)(231 398 236 393)(232 397 237 392)(233 396 238 391)(234 395 239 400)(235 394 240 399)(241 430 246 425)(242 429 247 424)(243 428 248 423)(244 427 249 422)(245 426 250 421)
(1 99 81)(2 100 82)(3 91 83)(4 92 84)(5 93 85)(6 94 86)(7 95 87)(8 96 88)(9 97 89)(10 98 90)(11 51 23)(12 52 24)(13 53 25)(14 54 26)(15 55 27)(16 56 28)(17 57 29)(18 58 30)(19 59 21)(20 60 22)(31 110 49)(32 101 50)(33 102 41)(34 103 42)(35 104 43)(36 105 44)(37 106 45)(38 107 46)(39 108 47)(40 109 48)(61 375 476)(62 376 477)(63 377 478)(64 378 479)(65 379 480)(66 380 471)(67 371 472)(68 372 473)(69 373 474)(70 374 475)(71 123 114)(72 124 115)(73 125 116)(74 126 117)(75 127 118)(76 128 119)(77 129 120)(78 130 111)(79 121 112)(80 122 113)(131 171 188)(132 172 189)(133 173 190)(134 174 181)(135 175 182)(136 176 183)(137 177 184)(138 178 185)(139 179 186)(140 180 187)(141 202 220)(142 203 211)(143 204 212)(144 205 213)(145 206 214)(146 207 215)(147 208 216)(148 209 217)(149 210 218)(150 201 219)(151 169 230)(152 170 221)(153 161 222)(154 162 223)(155 163 224)(156 164 225)(157 165 226)(158 166 227)(159 167 228)(160 168 229)(191 234 250)(192 235 241)(193 236 242)(194 237 243)(195 238 244)(196 239 245)(197 240 246)(198 231 247)(199 232 248)(200 233 249)(251 368 352)(252 369 353)(253 370 354)(254 361 355)(255 362 356)(256 363 357)(257 364 358)(258 365 359)(259 366 360)(260 367 351)(261 315 337)(262 316 338)(263 317 339)(264 318 340)(265 319 331)(266 320 332)(267 311 333)(268 312 334)(269 313 335)(270 314 336)(271 293 307)(272 294 308)(273 295 309)(274 296 310)(275 297 301)(276 298 302)(277 299 303)(278 300 304)(279 291 305)(280 292 306)(281 323 345)(282 324 346)(283 325 347)(284 326 348)(285 327 349)(286 328 350)(287 329 341)(288 330 342)(289 321 343)(290 322 344)(381 457 435)(382 458 436)(383 459 437)(384 460 438)(385 451 439)(386 452 440)(387 453 431)(388 454 432)(389 455 433)(390 456 434)(391 427 413)(392 428 414)(393 429 415)(394 430 416)(395 421 417)(396 422 418)(397 423 419)(398 424 420)(399 425 411)(400 426 412)(401 465 443)(402 466 444)(403 467 445)(404 468 446)(405 469 447)(406 470 448)(407 461 449)(408 462 450)(409 463 441)(410 464 442)
(1 234 119 163 48 190 18 145)(2 235 120 164 49 181 19 146)(3 236 111 165 50 182 20 147)(4 237 112 166 41 183 11 148)(5 238 113 167 42 184 12 149)(6 239 114 168 43 185 13 150)(7 240 115 169 44 186 14 141)(8 231 116 170 45 187 15 142)(9 232 117 161 46 188 16 143)(10 233 118 162 47 189 17 144)(21 207 82 241 129 225 110 134)(22 208 83 242 130 226 101 135)(23 209 84 243 121 227 102 136)(24 210 85 244 122 228 103 137)(25 201 86 245 123 229 104 138)(26 202 87 246 124 230 105 139)(27 203 88 247 125 221 106 140)(28 204 89 248 126 222 107 131)(29 205 90 249 127 223 108 132)(30 206 81 250 128 224 109 133)(31 174 59 215 100 192 77 156)(32 175 60 216 91 193 78 157)(33 176 51 217 92 194 79 158)(34 177 52 218 93 195 80 159)(35 178 53 219 94 196 71 160)(36 179 54 220 95 197 72 151)(37 180 55 211 96 198 73 152)(38 171 56 212 97 199 74 153)(39 172 57 213 98 200 75 154)(40 173 58 214 99 191 76 155)(61 367 466 341 426 301 390 265)(62 368 467 342 427 302 381 266)(63 369 468 343 428 303 382 267)(64 370 469 344 429 304 383 268)(65 361 470 345 430 305 384 269)(66 362 461 346 421 306 385 270)(67 363 462 347 422 307 386 261)(68 364 463 348 423 308 387 262)(69 365 464 349 424 309 388 263)(70 366 465 350 425 310 389 264)(251 445 330 413 298 457 332 376)(252 446 321 414 299 458 333 377)(253 447 322 415 300 459 334 378)(254 448 323 416 291 460 335 379)(255 449 324 417 292 451 336 380)(256 450 325 418 293 452 337 371)(257 441 326 419 294 453 338 372)(258 442 327 420 295 454 339 373)(259 443 328 411 296 455 340 374)(260 444 329 412 297 456 331 375)(271 440 315 472 357 408 283 396)(272 431 316 473 358 409 284 397)(273 432 317 474 359 410 285 398)(274 433 318 475 360 401 286 399)(275 434 319 476 351 402 287 400)(276 435 320 477 352 403 288 391)(277 436 311 478 353 404 289 392)(278 437 312 479 354 405 290 393)(279 438 313 480 355 406 281 394)(280 439 314 471 356 407 282 395)

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

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

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

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + - + - + + - + - image C1 C2 C2 C2 C4 C4 C4 C8 S3 D5 Dic3 D6 Dic5 D10 C3⋊C8 C4×S3 C4×D5 C4×D5 C8×D5 S3×D5 S3×Dic5 D30.C2 D5×Dic3 D5×C3⋊C8 kernel Dic5×C3⋊C8 C12×Dic5 C10×C3⋊C8 C2×C15⋊3C8 C5×C3⋊C8 C15⋊3C8 C6×Dic5 C3×Dic5 C4×Dic5 C2×C3⋊C8 C2×Dic5 C2×C20 C3⋊C8 C2×C12 Dic5 C20 C12 C2×C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 1 1 1 4 4 4 16 1 2 2 1 4 2 8 4 4 4 16 2 2 2 2 8

Matrix representation of Dic5×C3⋊C8 in GL4(𝔽241) generated by

 52 240 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 239 35 0 0 172 2 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 49 0 0 177 239
,
 177 0 0 0 0 177 0 0 0 0 109 2 0 0 173 132
G:=sub<GL(4,GF(241))| [52,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[239,172,0,0,35,2,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,177,0,0,49,239],[177,0,0,0,0,177,0,0,0,0,109,173,0,0,2,132] >;

Dic5×C3⋊C8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_3\rtimes C_8
% in TeX

G:=Group("Dic5xC3:C8");
// GroupNames label

G:=SmallGroup(480,25);
// by ID

G=gap.SmallGroup(480,25);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^3=d^8=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽