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G = C5×Dic6⋊C4order 480 = 25·3·5

Direct product of C5 and Dic6⋊C4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×Dic6⋊C4, Dic65C20, C32(Q8×C20), C1520(C4×Q8), C4.4(S3×C20), C20.79(C4×S3), Dic33(C5×Q8), C6.10(Q8×C10), C10.47(S3×Q8), C60.180(C2×C4), C12.10(C2×C20), (C5×Dic6)⋊17C4, (C5×Dic3)⋊10Q8, (C2×C20).353D6, C6.8(C22×C20), C30.108(C2×Q8), Dic3⋊C4.5C10, Dic3.2(C2×C20), (C2×Dic6).7C10, (C4×Dic3).1C10, C30.248(C4○D4), (C2×C60).350C22, C30.199(C22×C4), (C2×C30).407C23, (Dic3×C20).10C2, (C10×Dic6).17C2, C10.113(D42S3), (C10×Dic3).239C22, C2.1(C5×S3×Q8), C4⋊C4.7(C5×S3), C2.10(S3×C2×C20), (C3×C4⋊C4).4C10, (C5×C4⋊C4).14S3, C10.135(S3×C2×C4), C6.23(C5×C4○D4), (C15×C4⋊C4).18C2, (C2×C4).41(S3×C10), C2.3(C5×D42S3), C22.15(S3×C2×C10), (C2×C12).56(C2×C10), (C5×Dic3⋊C4).14C2, (C2×C6).28(C22×C10), (C5×Dic3).38(C2×C4), (C2×C10).341(C22×S3), (C2×Dic3).23(C2×C10), SmallGroup(480,766)

Series: Derived Chief Lower central Upper central

C1C6 — C5×Dic6⋊C4
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C20 — C5×Dic6⋊C4
C3C6 — C5×Dic6⋊C4
C1C2×C10C5×C4⋊C4

Generators and relations for C5×Dic6⋊C4
 G = < a,b,c,d | a5=b12=d4=1, c2=b6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b7, cd=dc >

Subgroups: 244 in 140 conjugacy classes, 86 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], Dic3 [×6], Dic3, C12 [×2], C12 [×2], C2×C6, C15, C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×Q8, C20 [×2], C20 [×9], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, C2×C20, C2×C20 [×2], C2×C20 [×4], C5×Q8 [×4], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C5×Dic3 [×6], C5×Dic3, C60 [×2], C60 [×2], C2×C30, C4×C20 [×3], C5×C4⋊C4, C5×C4⋊C4 [×2], Q8×C10, Dic6⋊C4, C5×Dic6 [×4], C10×Dic3 [×2], C10×Dic3 [×2], C2×C60, C2×C60 [×2], Q8×C20, Dic3×C20, Dic3×C20 [×2], C5×Dic3⋊C4 [×2], C15×C4⋊C4, C10×Dic6, C5×Dic6⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], Q8 [×2], C23, C10 [×7], D6 [×3], C22×C4, C2×Q8, C4○D4, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C4×Q8, C2×C20 [×6], C5×Q8 [×2], C22×C10, S3×C2×C4, D42S3, S3×Q8, S3×C10 [×3], C22×C20, Q8×C10, C5×C4○D4, Dic6⋊C4, S3×C20 [×2], S3×C2×C10, Q8×C20, S3×C2×C20, C5×D42S3, C5×S3×Q8, C5×Dic6⋊C4

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

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

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

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

150 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J4K···4P5A5B5C5D6A6B6C10A···10L12A···12F15A15B15C15D20A···20X20Y···20AN20AO···20BL30A···30L60A···60X
order122234···444444···4555566610···1012···121515151520···2020···2020···2030···3060···60
size111122···233336···611112221···14···422222···23···36···62···24···4

150 irreducible representations

dim11111111111122222222224444
type++++++-+--
imageC1C2C2C2C2C4C5C10C10C10C10C20S3Q8D6C4○D4C4×S3C5×S3C5×Q8S3×C10C5×C4○D4S3×C20D42S3S3×Q8C5×D42S3C5×S3×Q8
kernelC5×Dic6⋊C4Dic3×C20C5×Dic3⋊C4C15×C4⋊C4C10×Dic6C5×Dic6Dic6⋊C4C4×Dic3Dic3⋊C4C3×C4⋊C4C2×Dic6Dic6C5×C4⋊C4C5×Dic3C2×C20C30C20C4⋊C4Dic3C2×C4C6C4C10C10C2C2
# reps132118412844321232448128161144

Matrix representation of C5×Dic6⋊C4 in GL5(𝔽61)

10000
020000
002000
000200
000020
,
600000
0384300
0432300
000060
000160
,
600000
00100
060000
0003012
0004231
,
110000
00100
060000
000600
000060

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,20],[60,0,0,0,0,0,38,43,0,0,0,43,23,0,0,0,0,0,0,1,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,1,0,0,0,0,0,0,30,42,0,0,0,12,31],[11,0,0,0,0,0,0,60,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60] >;

C5×Dic6⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C5xDic6:C4");
// GroupNames label

G:=SmallGroup(480,766);
// by ID

G=gap.SmallGroup(480,766);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,288,471,646,15686]);
// Polycyclic

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

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