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## G = C5×Dic3⋊C8order 480 = 25·3·5

### Direct product of C5 and Dic3⋊C8

Series: Derived Chief Lower central Upper central

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

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 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×2], Dic3, C12 [×2], C2×C6, C15, C42, C2×C8, C2×C8, C20 [×2], C20 [×3], C2×C10, C3⋊C8, C24, C2×Dic3 [×2], C2×C12, C30 [×3], C4⋊C8, C40 [×2], C2×C20, C2×C20 [×2], C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3 [×2], C5×Dic3, C60 [×2], C2×C30, C4×C20, C2×C40, C2×C40, Dic3⋊C8, C5×C3⋊C8, C120, C10×Dic3 [×2], C2×C60, C5×C4⋊C8, C10×C3⋊C8, Dic3×C20, C2×C120, C5×Dic3⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C8 [×2], C2×C4, D4, Q8, C10 [×3], D6, C4⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, Dic6, C4×S3, C3⋊D4, C5×S3, C4⋊C8, C40 [×2], 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

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

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