direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic3⋊C8, Dic3⋊C40, C60.36Q8, C60.236D4, C20.28Dic6, C30.31M4(2), C15⋊15(C4⋊C8), C2.4(S3×C40), (C2×C40).1S3, C6.4(C2×C40), C12.8(C5×Q8), C10.28(S3×C8), (C2×C120).3C2, (C2×C24).1C10, C30.56(C2×C8), (C5×Dic3)⋊5C8, C12.51(C5×D4), C30.56(C4⋊C4), C4.8(C5×Dic6), (C2×C20).446D6, C22.9(S3×C20), C6.1(C5×M4(2)), (C4×Dic3).6C10, (C2×Dic3).2C20, C10.15(C8⋊S3), C20.119(C3⋊D4), (C2×C60).558C22, (Dic3×C20).15C2, (C10×Dic3).19C4, C10.24(Dic3⋊C4), C3⋊2(C5×C4⋊C8), C6.4(C5×C4⋊C4), (C2×C3⋊C8).9C10, (C2×C8).1(C5×S3), (C10×C3⋊C8).21C2, C2.1(C5×C8⋊S3), C4.26(C5×C3⋊D4), (C2×C10).81(C4×S3), (C2×C4).93(S3×C10), (C2×C6).10(C2×C20), C2.1(C5×Dic3⋊C4), (C2×C30).155(C2×C4), (C2×C12).110(C2×C10), SmallGroup(480,133)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic3⋊C8
G = < a,b,c,d | a5=b6=d8=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >
Subgroups: 132 in 76 conjugacy classes, 50 normal (46 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C2×C8, C2×C8, C20, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C4⋊C8, C40, C2×C20, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C5×Dic3, C60, C2×C30, C4×C20, C2×C40, C2×C40, Dic3⋊C8, C5×C3⋊C8, C120, C10×Dic3, C2×C60, C5×C4⋊C8, C10×C3⋊C8, Dic3×C20, C2×C120, C5×Dic3⋊C8
Quotients: C1, C2, C4, C22, C5, S3, C8, C2×C4, D4, Q8, C10, D6, C4⋊C4, C2×C8, M4(2), C20, C2×C10, Dic6, C4×S3, C3⋊D4, C5×S3, C4⋊C8, C40, C2×C20, C5×D4, C5×Q8, S3×C8, C8⋊S3, Dic3⋊C4, S3×C10, C5×C4⋊C4, C2×C40, C5×M4(2), Dic3⋊C8, C5×Dic6, S3×C20, C5×C3⋊D4, C5×C4⋊C8, S3×C40, C5×C8⋊S3, C5×Dic3⋊C4, C5×Dic3⋊C8
►Regular action on 480 points
Generators in S480
(1 245 119 219 95)(2 246 120 220 96)(3 247 113 221 89)(4 248 114 222 90)(5 241 115 223 91)(6 242 116 224 92)(7 243 117 217 93)(8 244 118 218 94)(9 355 343 331 319)(10 356 344 332 320)(11 357 337 333 313)(12 358 338 334 314)(13 359 339 335 315)(14 360 340 336 316)(15 353 341 329 317)(16 354 342 330 318)(17 65 173 41 149)(18 66 174 42 150)(19 67 175 43 151)(20 68 176 44 152)(21 69 169 45 145)(22 70 170 46 146)(23 71 171 47 147)(24 72 172 48 148)(25 199 183 55 159)(26 200 184 56 160)(27 193 177 49 153)(28 194 178 50 154)(29 195 179 51 155)(30 196 180 52 156)(31 197 181 53 157)(32 198 182 54 158)(33 141 189 57 165)(34 142 190 58 166)(35 143 191 59 167)(36 144 192 60 168)(37 137 185 61 161)(38 138 186 62 162)(39 139 187 63 163)(40 140 188 64 164)(73 361 225 97 201)(74 362 226 98 202)(75 363 227 99 203)(76 364 228 100 204)(77 365 229 101 205)(78 366 230 102 206)(79 367 231 103 207)(80 368 232 104 208)(81 252 237 105 213)(82 253 238 106 214)(83 254 239 107 215)(84 255 240 108 216)(85 256 233 109 209)(86 249 234 110 210)(87 250 235 111 211)(88 251 236 112 212)(121 465 457 441 433)(122 466 458 442 434)(123 467 459 443 435)(124 468 460 444 436)(125 469 461 445 437)(126 470 462 446 438)(127 471 463 447 439)(128 472 464 448 440)(129 473 352 449 328)(130 474 345 450 321)(131 475 346 451 322)(132 476 347 452 323)(133 477 348 453 324)(134 478 349 454 325)(135 479 350 455 326)(136 480 351 456 327)(257 305 293 281 269)(258 306 294 282 270)(259 307 295 283 271)(260 308 296 284 272)(261 309 289 285 265)(262 310 290 286 266)(263 311 291 287 267)(264 312 292 288 268)(273 378 426 297 402)(274 379 427 298 403)(275 380 428 299 404)(276 381 429 300 405)(277 382 430 301 406)(278 383 431 302 407)(279 384 432 303 408)(280 377 425 304 401)(369 417 409 393 385)(370 418 410 394 386)(371 419 411 395 387)(372 420 412 396 388)(373 421 413 397 389)(374 422 414 398 390)(375 423 415 399 391)(376 424 416 400 392)
(1 77 139 17 159 83)(2 78 140 18 160 84)(3 79 141 19 153 85)(4 80 142 20 154 86)(5 73 143 21 155 87)(6 74 144 22 156 88)(7 75 137 23 157 81)(8 76 138 24 158 82)(9 129 422 306 432 127)(10 130 423 307 425 128)(11 131 424 308 426 121)(12 132 417 309 427 122)(13 133 418 310 428 123)(14 134 419 311 429 124)(15 135 420 312 430 125)(16 136 421 305 431 126)(25 254 245 365 187 65)(26 255 246 366 188 66)(27 256 247 367 189 67)(28 249 248 368 190 68)(29 250 241 361 191 69)(30 251 242 362 192 70)(31 252 243 363 185 71)(32 253 244 364 186 72)(33 151 49 209 89 207)(34 152 50 210 90 208)(35 145 51 211 91 201)(36 146 52 212 92 202)(37 147 53 213 93 203)(38 148 54 214 94 204)(39 149 55 215 95 205)(40 150 56 216 96 206)(41 183 107 219 101 163)(42 184 108 220 102 164)(43 177 109 221 103 165)(44 178 110 222 104 166)(45 179 111 223 97 167)(46 180 112 224 98 168)(47 181 105 217 99 161)(48 182 106 218 100 162)(57 175 193 233 113 231)(58 176 194 234 114 232)(59 169 195 235 115 225)(60 170 196 236 116 226)(61 171 197 237 117 227)(62 172 198 238 118 228)(63 173 199 239 119 229)(64 174 200 240 120 230)(257 383 438 318 327 373)(258 384 439 319 328 374)(259 377 440 320 321 375)(260 378 433 313 322 376)(261 379 434 314 323 369)(262 380 435 315 324 370)(263 381 436 316 325 371)(264 382 437 317 326 372)(265 274 442 334 452 385)(266 275 443 335 453 386)(267 276 444 336 454 387)(268 277 445 329 455 388)(269 278 446 330 456 389)(270 279 447 331 449 390)(271 280 448 332 450 391)(272 273 441 333 451 392)(281 407 462 342 351 397)(282 408 463 343 352 398)(283 401 464 344 345 399)(284 402 457 337 346 400)(285 403 458 338 347 393)(286 404 459 339 348 394)(287 405 460 340 349 395)(288 406 461 341 350 396)(289 298 466 358 476 409)(290 299 467 359 477 410)(291 300 468 360 478 411)(292 301 469 353 479 412)(293 302 470 354 480 413)(294 303 471 355 473 414)(295 304 472 356 474 415)(296 297 465 357 475 416)
(1 313 17 260)(2 261 18 314)(3 315 19 262)(4 263 20 316)(5 317 21 264)(6 257 22 318)(7 319 23 258)(8 259 24 320)(9 71 306 243)(10 244 307 72)(11 65 308 245)(12 246 309 66)(13 67 310 247)(14 248 311 68)(15 69 312 241)(16 242 305 70)(25 424 365 121)(26 122 366 417)(27 418 367 123)(28 124 368 419)(29 420 361 125)(30 126 362 421)(31 422 363 127)(32 128 364 423)(33 275 209 453)(34 454 210 276)(35 277 211 455)(36 456 212 278)(37 279 213 449)(38 450 214 280)(39 273 215 451)(40 452 216 274)(41 284 219 337)(42 338 220 285)(43 286 221 339)(44 340 222 287)(45 288 223 341)(46 342 224 281)(47 282 217 343)(48 344 218 283)(49 386 207 443)(50 444 208 387)(51 388 201 445)(52 446 202 389)(53 390 203 447)(54 448 204 391)(55 392 205 441)(56 442 206 385)(57 299 233 477)(58 478 234 300)(59 301 235 479)(60 480 236 302)(61 303 237 473)(62 474 238 304)(63 297 239 475)(64 476 240 298)(73 437 155 372)(74 373 156 438)(75 439 157 374)(76 375 158 440)(77 433 159 376)(78 369 160 434)(79 435 153 370)(80 371 154 436)(81 328 137 384)(82 377 138 321)(83 322 139 378)(84 379 140 323)(85 324 141 380)(86 381 142 325)(87 326 143 382)(88 383 144 327)(89 335 151 266)(90 267 152 336)(91 329 145 268)(92 269 146 330)(93 331 147 270)(94 271 148 332)(95 333 149 272)(96 265 150 334)(97 461 179 396)(98 397 180 462)(99 463 181 398)(100 399 182 464)(101 457 183 400)(102 393 184 458)(103 459 177 394)(104 395 178 460)(105 352 161 408)(106 401 162 345)(107 346 163 402)(108 403 164 347)(109 348 165 404)(110 405 166 349)(111 350 167 406)(112 407 168 351)(113 359 175 290)(114 291 176 360)(115 353 169 292)(116 293 170 354)(117 355 171 294)(118 295 172 356)(119 357 173 296)(120 289 174 358)(129 185 432 252)(130 253 425 186)(131 187 426 254)(132 255 427 188)(133 189 428 256)(134 249 429 190)(135 191 430 250)(136 251 431 192)(193 410 231 467)(194 468 232 411)(195 412 225 469)(196 470 226 413)(197 414 227 471)(198 472 228 415)(199 416 229 465)(200 466 230 409)
(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)
G:=sub<Sym(480)| (1,245,119,219,95)(2,246,120,220,96)(3,247,113,221,89)(4,248,114,222,90)(5,241,115,223,91)(6,242,116,224,92)(7,243,117,217,93)(8,244,118,218,94)(9,355,343,331,319)(10,356,344,332,320)(11,357,337,333,313)(12,358,338,334,314)(13,359,339,335,315)(14,360,340,336,316)(15,353,341,329,317)(16,354,342,330,318)(17,65,173,41,149)(18,66,174,42,150)(19,67,175,43,151)(20,68,176,44,152)(21,69,169,45,145)(22,70,170,46,146)(23,71,171,47,147)(24,72,172,48,148)(25,199,183,55,159)(26,200,184,56,160)(27,193,177,49,153)(28,194,178,50,154)(29,195,179,51,155)(30,196,180,52,156)(31,197,181,53,157)(32,198,182,54,158)(33,141,189,57,165)(34,142,190,58,166)(35,143,191,59,167)(36,144,192,60,168)(37,137,185,61,161)(38,138,186,62,162)(39,139,187,63,163)(40,140,188,64,164)(73,361,225,97,201)(74,362,226,98,202)(75,363,227,99,203)(76,364,228,100,204)(77,365,229,101,205)(78,366,230,102,206)(79,367,231,103,207)(80,368,232,104,208)(81,252,237,105,213)(82,253,238,106,214)(83,254,239,107,215)(84,255,240,108,216)(85,256,233,109,209)(86,249,234,110,210)(87,250,235,111,211)(88,251,236,112,212)(121,465,457,441,433)(122,466,458,442,434)(123,467,459,443,435)(124,468,460,444,436)(125,469,461,445,437)(126,470,462,446,438)(127,471,463,447,439)(128,472,464,448,440)(129,473,352,449,328)(130,474,345,450,321)(131,475,346,451,322)(132,476,347,452,323)(133,477,348,453,324)(134,478,349,454,325)(135,479,350,455,326)(136,480,351,456,327)(257,305,293,281,269)(258,306,294,282,270)(259,307,295,283,271)(260,308,296,284,272)(261,309,289,285,265)(262,310,290,286,266)(263,311,291,287,267)(264,312,292,288,268)(273,378,426,297,402)(274,379,427,298,403)(275,380,428,299,404)(276,381,429,300,405)(277,382,430,301,406)(278,383,431,302,407)(279,384,432,303,408)(280,377,425,304,401)(369,417,409,393,385)(370,418,410,394,386)(371,419,411,395,387)(372,420,412,396,388)(373,421,413,397,389)(374,422,414,398,390)(375,423,415,399,391)(376,424,416,400,392), (1,77,139,17,159,83)(2,78,140,18,160,84)(3,79,141,19,153,85)(4,80,142,20,154,86)(5,73,143,21,155,87)(6,74,144,22,156,88)(7,75,137,23,157,81)(8,76,138,24,158,82)(9,129,422,306,432,127)(10,130,423,307,425,128)(11,131,424,308,426,121)(12,132,417,309,427,122)(13,133,418,310,428,123)(14,134,419,311,429,124)(15,135,420,312,430,125)(16,136,421,305,431,126)(25,254,245,365,187,65)(26,255,246,366,188,66)(27,256,247,367,189,67)(28,249,248,368,190,68)(29,250,241,361,191,69)(30,251,242,362,192,70)(31,252,243,363,185,71)(32,253,244,364,186,72)(33,151,49,209,89,207)(34,152,50,210,90,208)(35,145,51,211,91,201)(36,146,52,212,92,202)(37,147,53,213,93,203)(38,148,54,214,94,204)(39,149,55,215,95,205)(40,150,56,216,96,206)(41,183,107,219,101,163)(42,184,108,220,102,164)(43,177,109,221,103,165)(44,178,110,222,104,166)(45,179,111,223,97,167)(46,180,112,224,98,168)(47,181,105,217,99,161)(48,182,106,218,100,162)(57,175,193,233,113,231)(58,176,194,234,114,232)(59,169,195,235,115,225)(60,170,196,236,116,226)(61,171,197,237,117,227)(62,172,198,238,118,228)(63,173,199,239,119,229)(64,174,200,240,120,230)(257,383,438,318,327,373)(258,384,439,319,328,374)(259,377,440,320,321,375)(260,378,433,313,322,376)(261,379,434,314,323,369)(262,380,435,315,324,370)(263,381,436,316,325,371)(264,382,437,317,326,372)(265,274,442,334,452,385)(266,275,443,335,453,386)(267,276,444,336,454,387)(268,277,445,329,455,388)(269,278,446,330,456,389)(270,279,447,331,449,390)(271,280,448,332,450,391)(272,273,441,333,451,392)(281,407,462,342,351,397)(282,408,463,343,352,398)(283,401,464,344,345,399)(284,402,457,337,346,400)(285,403,458,338,347,393)(286,404,459,339,348,394)(287,405,460,340,349,395)(288,406,461,341,350,396)(289,298,466,358,476,409)(290,299,467,359,477,410)(291,300,468,360,478,411)(292,301,469,353,479,412)(293,302,470,354,480,413)(294,303,471,355,473,414)(295,304,472,356,474,415)(296,297,465,357,475,416), (1,313,17,260)(2,261,18,314)(3,315,19,262)(4,263,20,316)(5,317,21,264)(6,257,22,318)(7,319,23,258)(8,259,24,320)(9,71,306,243)(10,244,307,72)(11,65,308,245)(12,246,309,66)(13,67,310,247)(14,248,311,68)(15,69,312,241)(16,242,305,70)(25,424,365,121)(26,122,366,417)(27,418,367,123)(28,124,368,419)(29,420,361,125)(30,126,362,421)(31,422,363,127)(32,128,364,423)(33,275,209,453)(34,454,210,276)(35,277,211,455)(36,456,212,278)(37,279,213,449)(38,450,214,280)(39,273,215,451)(40,452,216,274)(41,284,219,337)(42,338,220,285)(43,286,221,339)(44,340,222,287)(45,288,223,341)(46,342,224,281)(47,282,217,343)(48,344,218,283)(49,386,207,443)(50,444,208,387)(51,388,201,445)(52,446,202,389)(53,390,203,447)(54,448,204,391)(55,392,205,441)(56,442,206,385)(57,299,233,477)(58,478,234,300)(59,301,235,479)(60,480,236,302)(61,303,237,473)(62,474,238,304)(63,297,239,475)(64,476,240,298)(73,437,155,372)(74,373,156,438)(75,439,157,374)(76,375,158,440)(77,433,159,376)(78,369,160,434)(79,435,153,370)(80,371,154,436)(81,328,137,384)(82,377,138,321)(83,322,139,378)(84,379,140,323)(85,324,141,380)(86,381,142,325)(87,326,143,382)(88,383,144,327)(89,335,151,266)(90,267,152,336)(91,329,145,268)(92,269,146,330)(93,331,147,270)(94,271,148,332)(95,333,149,272)(96,265,150,334)(97,461,179,396)(98,397,180,462)(99,463,181,398)(100,399,182,464)(101,457,183,400)(102,393,184,458)(103,459,177,394)(104,395,178,460)(105,352,161,408)(106,401,162,345)(107,346,163,402)(108,403,164,347)(109,348,165,404)(110,405,166,349)(111,350,167,406)(112,407,168,351)(113,359,175,290)(114,291,176,360)(115,353,169,292)(116,293,170,354)(117,355,171,294)(118,295,172,356)(119,357,173,296)(120,289,174,358)(129,185,432,252)(130,253,425,186)(131,187,426,254)(132,255,427,188)(133,189,428,256)(134,249,429,190)(135,191,430,250)(136,251,431,192)(193,410,231,467)(194,468,232,411)(195,412,225,469)(196,470,226,413)(197,414,227,471)(198,472,228,415)(199,416,229,465)(200,466,230,409), (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)>;
G:=Group( (1,245,119,219,95)(2,246,120,220,96)(3,247,113,221,89)(4,248,114,222,90)(5,241,115,223,91)(6,242,116,224,92)(7,243,117,217,93)(8,244,118,218,94)(9,355,343,331,319)(10,356,344,332,320)(11,357,337,333,313)(12,358,338,334,314)(13,359,339,335,315)(14,360,340,336,316)(15,353,341,329,317)(16,354,342,330,318)(17,65,173,41,149)(18,66,174,42,150)(19,67,175,43,151)(20,68,176,44,152)(21,69,169,45,145)(22,70,170,46,146)(23,71,171,47,147)(24,72,172,48,148)(25,199,183,55,159)(26,200,184,56,160)(27,193,177,49,153)(28,194,178,50,154)(29,195,179,51,155)(30,196,180,52,156)(31,197,181,53,157)(32,198,182,54,158)(33,141,189,57,165)(34,142,190,58,166)(35,143,191,59,167)(36,144,192,60,168)(37,137,185,61,161)(38,138,186,62,162)(39,139,187,63,163)(40,140,188,64,164)(73,361,225,97,201)(74,362,226,98,202)(75,363,227,99,203)(76,364,228,100,204)(77,365,229,101,205)(78,366,230,102,206)(79,367,231,103,207)(80,368,232,104,208)(81,252,237,105,213)(82,253,238,106,214)(83,254,239,107,215)(84,255,240,108,216)(85,256,233,109,209)(86,249,234,110,210)(87,250,235,111,211)(88,251,236,112,212)(121,465,457,441,433)(122,466,458,442,434)(123,467,459,443,435)(124,468,460,444,436)(125,469,461,445,437)(126,470,462,446,438)(127,471,463,447,439)(128,472,464,448,440)(129,473,352,449,328)(130,474,345,450,321)(131,475,346,451,322)(132,476,347,452,323)(133,477,348,453,324)(134,478,349,454,325)(135,479,350,455,326)(136,480,351,456,327)(257,305,293,281,269)(258,306,294,282,270)(259,307,295,283,271)(260,308,296,284,272)(261,309,289,285,265)(262,310,290,286,266)(263,311,291,287,267)(264,312,292,288,268)(273,378,426,297,402)(274,379,427,298,403)(275,380,428,299,404)(276,381,429,300,405)(277,382,430,301,406)(278,383,431,302,407)(279,384,432,303,408)(280,377,425,304,401)(369,417,409,393,385)(370,418,410,394,386)(371,419,411,395,387)(372,420,412,396,388)(373,421,413,397,389)(374,422,414,398,390)(375,423,415,399,391)(376,424,416,400,392), (1,77,139,17,159,83)(2,78,140,18,160,84)(3,79,141,19,153,85)(4,80,142,20,154,86)(5,73,143,21,155,87)(6,74,144,22,156,88)(7,75,137,23,157,81)(8,76,138,24,158,82)(9,129,422,306,432,127)(10,130,423,307,425,128)(11,131,424,308,426,121)(12,132,417,309,427,122)(13,133,418,310,428,123)(14,134,419,311,429,124)(15,135,420,312,430,125)(16,136,421,305,431,126)(25,254,245,365,187,65)(26,255,246,366,188,66)(27,256,247,367,189,67)(28,249,248,368,190,68)(29,250,241,361,191,69)(30,251,242,362,192,70)(31,252,243,363,185,71)(32,253,244,364,186,72)(33,151,49,209,89,207)(34,152,50,210,90,208)(35,145,51,211,91,201)(36,146,52,212,92,202)(37,147,53,213,93,203)(38,148,54,214,94,204)(39,149,55,215,95,205)(40,150,56,216,96,206)(41,183,107,219,101,163)(42,184,108,220,102,164)(43,177,109,221,103,165)(44,178,110,222,104,166)(45,179,111,223,97,167)(46,180,112,224,98,168)(47,181,105,217,99,161)(48,182,106,218,100,162)(57,175,193,233,113,231)(58,176,194,234,114,232)(59,169,195,235,115,225)(60,170,196,236,116,226)(61,171,197,237,117,227)(62,172,198,238,118,228)(63,173,199,239,119,229)(64,174,200,240,120,230)(257,383,438,318,327,373)(258,384,439,319,328,374)(259,377,440,320,321,375)(260,378,433,313,322,376)(261,379,434,314,323,369)(262,380,435,315,324,370)(263,381,436,316,325,371)(264,382,437,317,326,372)(265,274,442,334,452,385)(266,275,443,335,453,386)(267,276,444,336,454,387)(268,277,445,329,455,388)(269,278,446,330,456,389)(270,279,447,331,449,390)(271,280,448,332,450,391)(272,273,441,333,451,392)(281,407,462,342,351,397)(282,408,463,343,352,398)(283,401,464,344,345,399)(284,402,457,337,346,400)(285,403,458,338,347,393)(286,404,459,339,348,394)(287,405,460,340,349,395)(288,406,461,341,350,396)(289,298,466,358,476,409)(290,299,467,359,477,410)(291,300,468,360,478,411)(292,301,469,353,479,412)(293,302,470,354,480,413)(294,303,471,355,473,414)(295,304,472,356,474,415)(296,297,465,357,475,416), (1,313,17,260)(2,261,18,314)(3,315,19,262)(4,263,20,316)(5,317,21,264)(6,257,22,318)(7,319,23,258)(8,259,24,320)(9,71,306,243)(10,244,307,72)(11,65,308,245)(12,246,309,66)(13,67,310,247)(14,248,311,68)(15,69,312,241)(16,242,305,70)(25,424,365,121)(26,122,366,417)(27,418,367,123)(28,124,368,419)(29,420,361,125)(30,126,362,421)(31,422,363,127)(32,128,364,423)(33,275,209,453)(34,454,210,276)(35,277,211,455)(36,456,212,278)(37,279,213,449)(38,450,214,280)(39,273,215,451)(40,452,216,274)(41,284,219,337)(42,338,220,285)(43,286,221,339)(44,340,222,287)(45,288,223,341)(46,342,224,281)(47,282,217,343)(48,344,218,283)(49,386,207,443)(50,444,208,387)(51,388,201,445)(52,446,202,389)(53,390,203,447)(54,448,204,391)(55,392,205,441)(56,442,206,385)(57,299,233,477)(58,478,234,300)(59,301,235,479)(60,480,236,302)(61,303,237,473)(62,474,238,304)(63,297,239,475)(64,476,240,298)(73,437,155,372)(74,373,156,438)(75,439,157,374)(76,375,158,440)(77,433,159,376)(78,369,160,434)(79,435,153,370)(80,371,154,436)(81,328,137,384)(82,377,138,321)(83,322,139,378)(84,379,140,323)(85,324,141,380)(86,381,142,325)(87,326,143,382)(88,383,144,327)(89,335,151,266)(90,267,152,336)(91,329,145,268)(92,269,146,330)(93,331,147,270)(94,271,148,332)(95,333,149,272)(96,265,150,334)(97,461,179,396)(98,397,180,462)(99,463,181,398)(100,399,182,464)(101,457,183,400)(102,393,184,458)(103,459,177,394)(104,395,178,460)(105,352,161,408)(106,401,162,345)(107,346,163,402)(108,403,164,347)(109,348,165,404)(110,405,166,349)(111,350,167,406)(112,407,168,351)(113,359,175,290)(114,291,176,360)(115,353,169,292)(116,293,170,354)(117,355,171,294)(118,295,172,356)(119,357,173,296)(120,289,174,358)(129,185,432,252)(130,253,425,186)(131,187,426,254)(132,255,427,188)(133,189,428,256)(134,249,429,190)(135,191,430,250)(136,251,431,192)(193,410,231,467)(194,468,232,411)(195,412,225,469)(196,470,226,413)(197,414,227,471)(198,472,228,415)(199,416,229,465)(200,466,230,409), (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) );
G=PermutationGroup([[(1,245,119,219,95),(2,246,120,220,96),(3,247,113,221,89),(4,248,114,222,90),(5,241,115,223,91),(6,242,116,224,92),(7,243,117,217,93),(8,244,118,218,94),(9,355,343,331,319),(10,356,344,332,320),(11,357,337,333,313),(12,358,338,334,314),(13,359,339,335,315),(14,360,340,336,316),(15,353,341,329,317),(16,354,342,330,318),(17,65,173,41,149),(18,66,174,42,150),(19,67,175,43,151),(20,68,176,44,152),(21,69,169,45,145),(22,70,170,46,146),(23,71,171,47,147),(24,72,172,48,148),(25,199,183,55,159),(26,200,184,56,160),(27,193,177,49,153),(28,194,178,50,154),(29,195,179,51,155),(30,196,180,52,156),(31,197,181,53,157),(32,198,182,54,158),(33,141,189,57,165),(34,142,190,58,166),(35,143,191,59,167),(36,144,192,60,168),(37,137,185,61,161),(38,138,186,62,162),(39,139,187,63,163),(40,140,188,64,164),(73,361,225,97,201),(74,362,226,98,202),(75,363,227,99,203),(76,364,228,100,204),(77,365,229,101,205),(78,366,230,102,206),(79,367,231,103,207),(80,368,232,104,208),(81,252,237,105,213),(82,253,238,106,214),(83,254,239,107,215),(84,255,240,108,216),(85,256,233,109,209),(86,249,234,110,210),(87,250,235,111,211),(88,251,236,112,212),(121,465,457,441,433),(122,466,458,442,434),(123,467,459,443,435),(124,468,460,444,436),(125,469,461,445,437),(126,470,462,446,438),(127,471,463,447,439),(128,472,464,448,440),(129,473,352,449,328),(130,474,345,450,321),(131,475,346,451,322),(132,476,347,452,323),(133,477,348,453,324),(134,478,349,454,325),(135,479,350,455,326),(136,480,351,456,327),(257,305,293,281,269),(258,306,294,282,270),(259,307,295,283,271),(260,308,296,284,272),(261,309,289,285,265),(262,310,290,286,266),(263,311,291,287,267),(264,312,292,288,268),(273,378,426,297,402),(274,379,427,298,403),(275,380,428,299,404),(276,381,429,300,405),(277,382,430,301,406),(278,383,431,302,407),(279,384,432,303,408),(280,377,425,304,401),(369,417,409,393,385),(370,418,410,394,386),(371,419,411,395,387),(372,420,412,396,388),(373,421,413,397,389),(374,422,414,398,390),(375,423,415,399,391),(376,424,416,400,392)], [(1,77,139,17,159,83),(2,78,140,18,160,84),(3,79,141,19,153,85),(4,80,142,20,154,86),(5,73,143,21,155,87),(6,74,144,22,156,88),(7,75,137,23,157,81),(8,76,138,24,158,82),(9,129,422,306,432,127),(10,130,423,307,425,128),(11,131,424,308,426,121),(12,132,417,309,427,122),(13,133,418,310,428,123),(14,134,419,311,429,124),(15,135,420,312,430,125),(16,136,421,305,431,126),(25,254,245,365,187,65),(26,255,246,366,188,66),(27,256,247,367,189,67),(28,249,248,368,190,68),(29,250,241,361,191,69),(30,251,242,362,192,70),(31,252,243,363,185,71),(32,253,244,364,186,72),(33,151,49,209,89,207),(34,152,50,210,90,208),(35,145,51,211,91,201),(36,146,52,212,92,202),(37,147,53,213,93,203),(38,148,54,214,94,204),(39,149,55,215,95,205),(40,150,56,216,96,206),(41,183,107,219,101,163),(42,184,108,220,102,164),(43,177,109,221,103,165),(44,178,110,222,104,166),(45,179,111,223,97,167),(46,180,112,224,98,168),(47,181,105,217,99,161),(48,182,106,218,100,162),(57,175,193,233,113,231),(58,176,194,234,114,232),(59,169,195,235,115,225),(60,170,196,236,116,226),(61,171,197,237,117,227),(62,172,198,238,118,228),(63,173,199,239,119,229),(64,174,200,240,120,230),(257,383,438,318,327,373),(258,384,439,319,328,374),(259,377,440,320,321,375),(260,378,433,313,322,376),(261,379,434,314,323,369),(262,380,435,315,324,370),(263,381,436,316,325,371),(264,382,437,317,326,372),(265,274,442,334,452,385),(266,275,443,335,453,386),(267,276,444,336,454,387),(268,277,445,329,455,388),(269,278,446,330,456,389),(270,279,447,331,449,390),(271,280,448,332,450,391),(272,273,441,333,451,392),(281,407,462,342,351,397),(282,408,463,343,352,398),(283,401,464,344,345,399),(284,402,457,337,346,400),(285,403,458,338,347,393),(286,404,459,339,348,394),(287,405,460,340,349,395),(288,406,461,341,350,396),(289,298,466,358,476,409),(290,299,467,359,477,410),(291,300,468,360,478,411),(292,301,469,353,479,412),(293,302,470,354,480,413),(294,303,471,355,473,414),(295,304,472,356,474,415),(296,297,465,357,475,416)], [(1,313,17,260),(2,261,18,314),(3,315,19,262),(4,263,20,316),(5,317,21,264),(6,257,22,318),(7,319,23,258),(8,259,24,320),(9,71,306,243),(10,244,307,72),(11,65,308,245),(12,246,309,66),(13,67,310,247),(14,248,311,68),(15,69,312,241),(16,242,305,70),(25,424,365,121),(26,122,366,417),(27,418,367,123),(28,124,368,419),(29,420,361,125),(30,126,362,421),(31,422,363,127),(32,128,364,423),(33,275,209,453),(34,454,210,276),(35,277,211,455),(36,456,212,278),(37,279,213,449),(38,450,214,280),(39,273,215,451),(40,452,216,274),(41,284,219,337),(42,338,220,285),(43,286,221,339),(44,340,222,287),(45,288,223,341),(46,342,224,281),(47,282,217,343),(48,344,218,283),(49,386,207,443),(50,444,208,387),(51,388,201,445),(52,446,202,389),(53,390,203,447),(54,448,204,391),(55,392,205,441),(56,442,206,385),(57,299,233,477),(58,478,234,300),(59,301,235,479),(60,480,236,302),(61,303,237,473),(62,474,238,304),(63,297,239,475),(64,476,240,298),(73,437,155,372),(74,373,156,438),(75,439,157,374),(76,375,158,440),(77,433,159,376),(78,369,160,434),(79,435,153,370),(80,371,154,436),(81,328,137,384),(82,377,138,321),(83,322,139,378),(84,379,140,323),(85,324,141,380),(86,381,142,325),(87,326,143,382),(88,383,144,327),(89,335,151,266),(90,267,152,336),(91,329,145,268),(92,269,146,330),(93,331,147,270),(94,271,148,332),(95,333,149,272),(96,265,150,334),(97,461,179,396),(98,397,180,462),(99,463,181,398),(100,399,182,464),(101,457,183,400),(102,393,184,458),(103,459,177,394),(104,395,178,460),(105,352,161,408),(106,401,162,345),(107,346,163,402),(108,403,164,347),(109,348,165,404),(110,405,166,349),(111,350,167,406),(112,407,168,351),(113,359,175,290),(114,291,176,360),(115,353,169,292),(116,293,170,354),(117,355,171,294),(118,295,172,356),(119,357,173,296),(120,289,174,358),(129,185,432,252),(130,253,425,186),(131,187,426,254),(132,255,427,188),(133,189,428,256),(134,249,429,190),(135,191,430,250),(136,251,431,192),(193,410,231,467),(194,468,232,411),(195,412,225,469),(196,470,226,413),(197,414,227,471),(198,472,228,415),(199,416,229,465),(200,466,230,409)], [(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)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10L | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AF | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 40Q | ··· | 40AF | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C40 | S3 | D4 | Q8 | D6 | M4(2) | Dic6 | C3⋊D4 | C4×S3 | C5×S3 | C5×D4 | C5×Q8 | S3×C8 | C8⋊S3 | S3×C10 | C5×M4(2) | C5×Dic6 | C5×C3⋊D4 | S3×C20 | S3×C40 | C5×C8⋊S3 |
| kernel | C5×Dic3⋊C8 | C10×C3⋊C8 | Dic3×C20 | C2×C120 | C10×Dic3 | Dic3⋊C8 | C5×Dic3 | C2×C3⋊C8 | C4×Dic3 | C2×C24 | C2×Dic3 | Dic3 | C2×C40 | C60 | C60 | C2×C20 | C30 | C20 | C20 | C2×C10 | C2×C8 | C12 | C12 | C10 | C10 | C2×C4 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 4 | 4 | 4 | 16 | 32 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C5×Dic3⋊C8 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 146 | 148 |
| 0 | 0 | 53 | 95 |
| 30 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 171 | 140 |
| 0 | 0 | 101 | 70 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,205,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,1],[1,0,0,0,0,240,0,0,0,0,146,53,0,0,148,95],[30,0,0,0,0,1,0,0,0,0,171,101,0,0,140,70] >;
C5×Dic3⋊C8 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_3\rtimes C_8 % in TeX
G:=Group("C5xDic3:C8"); // GroupNames label
G:=SmallGroup(480,133);
// by ID
G=gap.SmallGroup(480,133);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,280,589,148,136,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^6=d^8=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations