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## G = C3×Dic5⋊C8order 480 = 25·3·5

### Direct product of C3 and Dic5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×Dic5⋊C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C6×Dic5 — C6×C5⋊C8 — C3×Dic5⋊C8
 Lower central C5 — C10 — C3×Dic5⋊C8
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×Dic5⋊C8
G = < a,b,c,d | a3=b10=d8=1, c2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 200 in 76 conjugacy classes, 44 normal (36 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C30, C4⋊C8, C5⋊C8, C2×Dic5, C2×C20, C4×C12, C2×C24, C3×Dic5, C3×Dic5, C60, C2×C30, C4×Dic5, C2×C5⋊C8, C3×C4⋊C8, C3×C5⋊C8, C6×Dic5, C2×C60, Dic5⋊C8, C12×Dic5, C6×C5⋊C8, C3×Dic5⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, C2×C8, M4(2), F5, C24, C2×C12, C3×D4, C3×Q8, C4⋊C8, C2×F5, C3×C4⋊C4, C2×C24, C3×M4(2), C3×F5, D5⋊C8, C4⋊F5, C22.F5, C3×C4⋊C8, C6×F5, Dic5⋊C8, C3×D5⋊C8, C3×C4⋊F5, C3×C22.F5, C3×Dic5⋊C8

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5 6A ··· 6F 8A ··· 8H 10A 10B 10C 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P 15A 15B 20A 20B 20C 20D 24A ··· 24P 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 6 ··· 6 8 ··· 8 10 10 10 12 12 12 12 12 ··· 12 12 12 12 12 15 15 20 20 20 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 2 5 5 5 5 10 10 4 1 ··· 1 10 ··· 10 4 4 4 2 2 2 2 5 ··· 5 10 10 10 10 4 4 4 4 4 4 10 ··· 10 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + - + + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 Q8 M4(2) C3×D4 C3×Q8 C3×M4(2) F5 C2×F5 C3×F5 D5⋊C8 C4⋊F5 C22.F5 C6×F5 C3×D5⋊C8 C3×C4⋊F5 C3×C22.F5 kernel C3×Dic5⋊C8 C12×Dic5 C6×C5⋊C8 Dic5⋊C8 C6×Dic5 C2×C60 C4×Dic5 C2×C5⋊C8 C3×Dic5 C2×Dic5 C2×C20 Dic5 C3×Dic5 C3×Dic5 C30 Dic5 Dic5 C10 C2×C12 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 2 2 2 2 2 4 8 4 4 16 1 1 2 2 2 4 1 1 2 2 2 2 2 4 4 4

Matrix representation of C3×Dic5⋊C8 in GL6(𝔽241)

 15 0 0 0 0 0 0 15 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 1 0 0 240 0 0 1 0 0 0 240 0 1 0 0 0 0 240 1
,
 32 231 0 0 0 0 223 209 0 0 0 0 0 0 91 14 238 202 0 0 105 11 199 139 0 0 102 213 136 230 0 0 63 150 227 3
,
 175 215 0 0 0 0 10 66 0 0 0 0 0 0 224 11 76 205 0 0 59 216 42 188 0 0 25 199 53 23 0 0 36 34 17 230

G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,1,1,1],[32,223,0,0,0,0,231,209,0,0,0,0,0,0,91,105,102,63,0,0,14,11,213,150,0,0,238,199,136,227,0,0,202,139,230,3],[175,10,0,0,0,0,215,66,0,0,0,0,0,0,224,59,25,36,0,0,11,216,199,34,0,0,76,42,53,17,0,0,205,188,23,230] >;

C3×Dic5⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic5:C8");
// GroupNames label

G:=SmallGroup(480,284);
// by ID

G=gap.SmallGroup(480,284);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,512,136,9414,1595]);
// Polycyclic

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

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