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## G = C5×C4.Dic6order 480 = 25·3·5

### Direct product of C5 and C4.Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C4.Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C20 — C5×C4.Dic6
 Lower central C3 — C2×C6 — C5×C4.Dic6
 Upper central C1 — C2×C10 — C5×C4⋊C4

Generators and relations for C5×C4.Dic6
G = < a,b,c,d | a5=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 212 in 112 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, C20, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×C20, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C60, C60, C2×C30, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C4.Dic6, C10×Dic3, C10×Dic3, C2×C60, C2×C60, C5×C42.C2, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×C4⋊Dic3, C15×C4⋊C4, C5×C4.Dic6
Quotients: C1, C2, C22, C5, S3, Q8, C23, C10, D6, C2×Q8, C4○D4, C2×C10, Dic6, C22×S3, C5×S3, C42.C2, C5×Q8, C22×C10, C2×Dic6, D42S3, Q83S3, S3×C10, Q8×C10, C5×C4○D4, C4.Dic6, C5×Dic6, S3×C2×C10, C5×C42.C2, C10×Dic6, C5×D42S3, C5×Q83S3, C5×C4.Dic6

Smallest permutation representation of C5×C4.Dic6
Regular action on 480 points
Generators in S480
(1 59 282 181 93)(2 60 283 182 94)(3 49 284 183 95)(4 50 285 184 96)(5 51 286 185 85)(6 52 287 186 86)(7 53 288 187 87)(8 54 277 188 88)(9 55 278 189 89)(10 56 279 190 90)(11 57 280 191 91)(12 58 281 192 92)(13 69 25 302 420)(14 70 26 303 409)(15 71 27 304 410)(16 72 28 305 411)(17 61 29 306 412)(18 62 30 307 413)(19 63 31 308 414)(20 64 32 309 415)(21 65 33 310 416)(22 66 34 311 417)(23 67 35 312 418)(24 68 36 301 419)(37 386 470 111 269)(38 387 471 112 270)(39 388 472 113 271)(40 389 473 114 272)(41 390 474 115 273)(42 391 475 116 274)(43 392 476 117 275)(44 393 477 118 276)(45 394 478 119 265)(46 395 479 120 266)(47 396 480 109 267)(48 385 469 110 268)(73 465 198 256 399)(74 466 199 257 400)(75 467 200 258 401)(76 468 201 259 402)(77 457 202 260 403)(78 458 203 261 404)(79 459 204 262 405)(80 460 193 263 406)(81 461 194 264 407)(82 462 195 253 408)(83 463 196 254 397)(84 464 197 255 398)(97 361 175 220 155)(98 362 176 221 156)(99 363 177 222 145)(100 364 178 223 146)(101 365 179 224 147)(102 366 180 225 148)(103 367 169 226 149)(104 368 170 227 150)(105 369 171 228 151)(106 370 172 217 152)(107 371 173 218 153)(108 372 174 219 154)(121 239 207 354 442)(122 240 208 355 443)(123 229 209 356 444)(124 230 210 357 433)(125 231 211 358 434)(126 232 212 359 435)(127 233 213 360 436)(128 234 214 349 437)(129 235 215 350 438)(130 236 216 351 439)(131 237 205 352 440)(132 238 206 353 441)(133 338 334 244 296)(134 339 335 245 297)(135 340 336 246 298)(136 341 325 247 299)(137 342 326 248 300)(138 343 327 249 289)(139 344 328 250 290)(140 345 329 251 291)(141 346 330 252 292)(142 347 331 241 293)(143 348 332 242 294)(144 337 333 243 295)(157 456 313 429 376)(158 445 314 430 377)(159 446 315 431 378)(160 447 316 432 379)(161 448 317 421 380)(162 449 318 422 381)(163 450 319 423 382)(164 451 320 424 383)(165 452 321 425 384)(166 453 322 426 373)(167 454 323 427 374)(168 455 324 428 375)
(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)
(1 254 434 39)(2 261 435 46)(3 256 436 41)(4 263 437 48)(5 258 438 43)(6 253 439 38)(7 260 440 45)(8 255 441 40)(9 262 442 47)(10 257 443 42)(11 264 444 37)(12 259 433 44)(13 428 294 224)(14 423 295 219)(15 430 296 226)(16 425 297 221)(17 432 298 228)(18 427 299 223)(19 422 300 218)(20 429 289 225)(21 424 290 220)(22 431 291 227)(23 426 292 222)(24 421 293 217)(25 168 348 101)(26 163 337 108)(27 158 338 103)(28 165 339 98)(29 160 340 105)(30 167 341 100)(31 162 342 107)(32 157 343 102)(33 164 344 97)(34 159 345 104)(35 166 346 99)(36 161 347 106)(49 399 127 390)(50 406 128 385)(51 401 129 392)(52 408 130 387)(53 403 131 394)(54 398 132 389)(55 405 121 396)(56 400 122 391)(57 407 123 386)(58 402 124 393)(59 397 125 388)(60 404 126 395)(61 379 135 151)(62 374 136 146)(63 381 137 153)(64 376 138 148)(65 383 139 155)(66 378 140 150)(67 373 141 145)(68 380 142 152)(69 375 143 147)(70 382 144 154)(71 377 133 149)(72 384 134 156)(73 233 474 284)(74 240 475 279)(75 235 476 286)(76 230 477 281)(77 237 478 288)(78 232 479 283)(79 239 480 278)(80 234 469 285)(81 229 470 280)(82 236 471 287)(83 231 472 282)(84 238 473 277)(85 200 350 275)(86 195 351 270)(87 202 352 265)(88 197 353 272)(89 204 354 267)(90 199 355 274)(91 194 356 269)(92 201 357 276)(93 196 358 271)(94 203 359 266)(95 198 360 273)(96 193 349 268)(109 189 459 207)(110 184 460 214)(111 191 461 209)(112 186 462 216)(113 181 463 211)(114 188 464 206)(115 183 465 213)(116 190 466 208)(117 185 467 215)(118 192 468 210)(119 187 457 205)(120 182 458 212)(169 410 314 244)(170 417 315 251)(171 412 316 246)(172 419 317 241)(173 414 318 248)(174 409 319 243)(175 416 320 250)(176 411 321 245)(177 418 322 252)(178 413 323 247)(179 420 324 242)(180 415 313 249)(301 448 331 370)(302 455 332 365)(303 450 333 372)(304 445 334 367)(305 452 335 362)(306 447 336 369)(307 454 325 364)(308 449 326 371)(309 456 327 366)(310 451 328 361)(311 446 329 368)(312 453 330 363)
(1 99 434 166)(2 104 435 159)(3 97 436 164)(4 102 437 157)(5 107 438 162)(6 100 439 167)(7 105 440 160)(8 98 441 165)(9 103 442 158)(10 108 443 163)(11 101 444 168)(12 106 433 161)(13 467 294 117)(14 460 295 110)(15 465 296 115)(16 458 297 120)(17 463 298 113)(18 468 299 118)(19 461 300 111)(20 466 289 116)(21 459 290 109)(22 464 291 114)(23 457 292 119)(24 462 293 112)(25 258 348 43)(26 263 337 48)(27 256 338 41)(28 261 339 46)(29 254 340 39)(30 259 341 44)(31 264 342 37)(32 257 343 42)(33 262 344 47)(34 255 345 40)(35 260 346 45)(36 253 347 38)(49 361 127 451)(50 366 128 456)(51 371 129 449)(52 364 130 454)(53 369 131 447)(54 362 132 452)(55 367 121 445)(56 372 122 450)(57 365 123 455)(58 370 124 448)(59 363 125 453)(60 368 126 446)(61 196 135 271)(62 201 136 276)(63 194 137 269)(64 199 138 274)(65 204 139 267)(66 197 140 272)(67 202 141 265)(68 195 142 270)(69 200 143 275)(70 193 144 268)(71 198 133 273)(72 203 134 266)(73 244 474 410)(74 249 475 415)(75 242 476 420)(76 247 477 413)(77 252 478 418)(78 245 479 411)(79 250 480 416)(80 243 469 409)(81 248 470 414)(82 241 471 419)(83 246 472 412)(84 251 473 417)(85 153 350 381)(86 146 351 374)(87 151 352 379)(88 156 353 384)(89 149 354 377)(90 154 355 382)(91 147 356 375)(92 152 357 380)(93 145 358 373)(94 150 359 378)(95 155 360 383)(96 148 349 376)(169 239 314 278)(170 232 315 283)(171 237 316 288)(172 230 317 281)(173 235 318 286)(174 240 319 279)(175 233 320 284)(176 238 321 277)(177 231 322 282)(178 236 323 287)(179 229 324 280)(180 234 313 285)(181 222 211 426)(182 227 212 431)(183 220 213 424)(184 225 214 429)(185 218 215 422)(186 223 216 427)(187 228 205 432)(188 221 206 425)(189 226 207 430)(190 219 208 423)(191 224 209 428)(192 217 210 421)(301 408 331 387)(302 401 332 392)(303 406 333 385)(304 399 334 390)(305 404 335 395)(306 397 336 388)(307 402 325 393)(308 407 326 386)(309 400 327 391)(310 405 328 396)(311 398 329 389)(312 403 330 394)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 12A ··· 12F 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 20Q ··· 20AF 20AG ··· 20AN 30A ··· 30L 60A ··· 60X order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 4 4 6 6 6 6 12 12 1 1 1 1 2 2 2 1 ··· 1 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 6 ··· 6 12 ··· 12 2 ··· 2 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + - - + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 Q8 D6 C4○D4 Dic6 C5×S3 C5×Q8 S3×C10 C5×C4○D4 C5×Dic6 D4⋊2S3 Q8⋊3S3 C5×D4⋊2S3 C5×Q8⋊3S3 kernel C5×C4.Dic6 Dic3×C20 C5×Dic3⋊C4 C5×C4⋊Dic3 C15×C4⋊C4 C4.Dic6 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C3×C4⋊C4 C5×C4⋊C4 C60 C2×C20 C30 C20 C4⋊C4 C12 C2×C4 C6 C4 C10 C10 C2 C2 # reps 1 1 2 3 1 4 4 8 12 4 1 2 3 4 4 4 8 12 16 16 1 1 4 4

Matrix representation of C5×C4.Dic6 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 1 0 0 0 0 1
,
 1 1 0 0 60 0 0 0 0 0 25 54 0 0 11 36
,
 23 46 0 0 15 38 0 0 0 0 27 59 0 0 59 34
,
 25 51 0 0 26 36 0 0 0 0 30 16 0 0 1 31
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[1,60,0,0,1,0,0,0,0,0,25,11,0,0,54,36],[23,15,0,0,46,38,0,0,0,0,27,59,0,0,59,34],[25,26,0,0,51,36,0,0,0,0,30,1,0,0,16,31] >;

C5×C4.Dic6 in GAP, Magma, Sage, TeX

C_5\times C_4.{\rm Dic}_6
% in TeX

G:=Group("C5xC4.Dic6");
// GroupNames label

G:=SmallGroup(480,769);
// by ID

G=gap.SmallGroup(480,769);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,568,2606,891,226,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations

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