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G = C5×C4.Dic6order 480 = 25·3·5

Direct product of C5 and C4.Dic6

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

Aliases: C5×C4.Dic6, C60.21Q8, C20.18Dic6, C12.3(C5×Q8), C6.6(Q8×C10), C30.87(C2×Q8), C4.3(C5×Dic6), (C2×C20).237D6, C4⋊Dic3.7C10, C2.8(C10×Dic6), Dic3⋊C4.4C10, (C4×Dic3).3C10, C10.46(C2×Dic6), C1521(C42.C2), C30.249(C4○D4), (C2×C30).410C23, (C2×C60).351C22, (Dic3×C20).12C2, C10.47(Q83S3), C10.115(D42S3), (C10×Dic3).217C22, C4⋊C4.6(C5×S3), (C3×C4⋊C4).7C10, (C5×C4⋊C4).13S3, C33(C5×C42.C2), C6.24(C5×C4○D4), (C15×C4⋊C4).21C2, (C2×C4).43(S3×C10), C22.48(S3×C2×C10), C2.4(C5×Q83S3), (C2×C12).20(C2×C10), C2.12(C5×D42S3), (C5×C4⋊Dic3).21C2, (C5×Dic3⋊C4).11C2, (C2×C6).31(C22×C10), (C2×C10).344(C22×S3), (C2×Dic3).25(C2×C10), SmallGroup(480,769)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C4.Dic6
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C20 — C5×C4.Dic6
C3C2×C6 — C5×C4.Dic6
C1C2×C10C5×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 [×3], C3, C4 [×2], C4 [×6], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×5], C20 [×2], C20 [×6], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C42.C2, C2×C20, C2×C20 [×2], C2×C20 [×4], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], C3×C4⋊C4, C5×Dic3 [×4], C60 [×2], C60 [×2], C2×C30, C4×C20, C5×C4⋊C4, C5×C4⋊C4 [×5], C4.Dic6, C10×Dic3 [×2], C10×Dic3 [×2], C2×C60, C2×C60 [×2], C5×C42.C2, Dic3×C20, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×C4⋊Dic3 [×2], C15×C4⋊C4, C5×C4.Dic6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, Q8 [×2], C23, C10 [×7], D6 [×3], C2×Q8, C4○D4 [×2], C2×C10 [×7], Dic6 [×2], C22×S3, C5×S3, C42.C2, C5×Q8 [×2], C22×C10, C2×Dic6, D42S3, Q83S3, S3×C10 [×3], Q8×C10, C5×C4○D4 [×2], C4.Dic6, C5×Dic6 [×2], 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
(1 318 325 376 167)(2 319 326 377 168)(3 320 327 378 157)(4 321 328 379 158)(5 322 329 380 159)(6 323 330 381 160)(7 324 331 382 161)(8 313 332 383 162)(9 314 333 384 163)(10 315 334 373 164)(11 316 335 374 165)(12 317 336 375 166)(13 341 79 430 233)(14 342 80 431 234)(15 343 81 432 235)(16 344 82 421 236)(17 345 83 422 237)(18 346 84 423 238)(19 347 73 424 239)(20 348 74 425 240)(21 337 75 426 229)(22 338 76 427 230)(23 339 77 428 231)(24 340 78 429 232)(25 444 115 474 407)(26 433 116 475 408)(27 434 117 476 397)(28 435 118 477 398)(29 436 119 478 399)(30 437 120 479 400)(31 438 109 480 401)(32 439 110 469 402)(33 440 111 470 403)(34 441 112 471 404)(35 442 113 472 405)(36 443 114 473 406)(37 300 192 54 248)(38 289 181 55 249)(39 290 182 56 250)(40 291 183 57 251)(41 292 184 58 252)(42 293 185 59 241)(43 294 186 60 242)(44 295 187 49 243)(45 296 188 50 244)(46 297 189 51 245)(47 298 190 52 246)(48 299 191 53 247)(61 220 349 172 130)(62 221 350 173 131)(63 222 351 174 132)(64 223 352 175 121)(65 224 353 176 122)(66 225 354 177 123)(67 226 355 178 124)(68 227 356 179 125)(69 228 357 180 126)(70 217 358 169 127)(71 218 359 170 128)(72 219 360 171 129)(85 367 274 97 200)(86 368 275 98 201)(87 369 276 99 202)(88 370 265 100 203)(89 371 266 101 204)(90 372 267 102 193)(91 361 268 103 194)(92 362 269 104 195)(93 363 270 105 196)(94 364 271 106 197)(95 365 272 107 198)(96 366 273 108 199)(133 279 205 253 146)(134 280 206 254 147)(135 281 207 255 148)(136 282 208 256 149)(137 283 209 257 150)(138 284 210 258 151)(139 285 211 259 152)(140 286 212 260 153)(141 287 213 261 154)(142 288 214 262 155)(143 277 215 263 156)(144 278 216 264 145)(301 451 411 468 391)(302 452 412 457 392)(303 453 413 458 393)(304 454 414 459 394)(305 455 415 460 395)(306 456 416 461 396)(307 445 417 462 385)(308 446 418 463 386)(309 447 419 464 387)(310 448 420 465 388)(311 449 409 466 389)(312 450 410 467 390)
(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)
(1 304 101 220)(2 311 102 227)(3 306 103 222)(4 301 104 217)(5 308 105 224)(6 303 106 219)(7 310 107 226)(8 305 108 221)(9 312 97 228)(10 307 98 223)(11 302 99 218)(12 309 100 225)(13 251 475 137)(14 246 476 144)(15 241 477 139)(16 248 478 134)(17 243 479 141)(18 250 480 136)(19 245 469 143)(20 252 470 138)(21 247 471 133)(22 242 472 140)(23 249 473 135)(24 244 474 142)(25 214 78 296)(26 209 79 291)(27 216 80 298)(28 211 81 293)(29 206 82 300)(30 213 83 295)(31 208 84 290)(32 215 73 297)(33 210 74 292)(34 205 75 299)(35 212 76 294)(36 207 77 289)(37 399 280 344)(38 406 281 339)(39 401 282 346)(40 408 283 341)(41 403 284 348)(42 398 285 343)(43 405 286 338)(44 400 287 345)(45 407 288 340)(46 402 277 347)(47 397 278 342)(48 404 279 337)(49 120 154 237)(50 115 155 232)(51 110 156 239)(52 117 145 234)(53 112 146 229)(54 119 147 236)(55 114 148 231)(56 109 149 238)(57 116 150 233)(58 111 151 240)(59 118 152 235)(60 113 153 230)(61 167 394 266)(62 162 395 273)(63 157 396 268)(64 164 385 275)(65 159 386 270)(66 166 387 265)(67 161 388 272)(68 168 389 267)(69 163 390 274)(70 158 391 269)(71 165 392 276)(72 160 393 271)(85 180 333 410)(86 175 334 417)(87 170 335 412)(88 177 336 419)(89 172 325 414)(90 179 326 409)(91 174 327 416)(92 169 328 411)(93 176 329 418)(94 171 330 413)(95 178 331 420)(96 173 332 415)(121 373 462 368)(122 380 463 363)(123 375 464 370)(124 382 465 365)(125 377 466 372)(126 384 467 367)(127 379 468 362)(128 374 457 369)(129 381 458 364)(130 376 459 371)(131 383 460 366)(132 378 461 361)(181 443 255 428)(182 438 256 423)(183 433 257 430)(184 440 258 425)(185 435 259 432)(186 442 260 427)(187 437 261 422)(188 444 262 429)(189 439 263 424)(190 434 264 431)(191 441 253 426)(192 436 254 421)(193 356 319 449)(194 351 320 456)(195 358 321 451)(196 353 322 446)(197 360 323 453)(198 355 324 448)(199 350 313 455)(200 357 314 450)(201 352 315 445)(202 359 316 452)(203 354 317 447)(204 349 318 454)
(1 299 101 205)(2 292 102 210)(3 297 103 215)(4 290 104 208)(5 295 105 213)(6 300 106 206)(7 293 107 211)(8 298 108 216)(9 291 97 209)(10 296 98 214)(11 289 99 207)(12 294 100 212)(13 132 475 461)(14 125 476 466)(15 130 477 459)(16 123 478 464)(17 128 479 457)(18 121 480 462)(19 126 469 467)(20 131 470 460)(21 124 471 465)(22 129 472 458)(23 122 473 463)(24 127 474 468)(25 301 78 217)(26 306 79 222)(27 311 80 227)(28 304 81 220)(29 309 82 225)(30 302 83 218)(31 307 84 223)(32 312 73 228)(33 305 74 221)(34 310 75 226)(35 303 76 219)(36 308 77 224)(37 271 280 160)(38 276 281 165)(39 269 282 158)(40 274 283 163)(41 267 284 168)(42 272 285 161)(43 265 286 166)(44 270 287 159)(45 275 288 164)(46 268 277 157)(47 273 278 162)(48 266 279 167)(49 93 154 329)(50 86 155 334)(51 91 156 327)(52 96 145 332)(53 89 146 325)(54 94 147 330)(55 87 148 335)(56 92 149 328)(57 85 150 333)(58 90 151 326)(59 95 152 331)(60 88 153 336)(61 398 394 343)(62 403 395 348)(63 408 396 341)(64 401 385 346)(65 406 386 339)(66 399 387 344)(67 404 388 337)(68 397 389 342)(69 402 390 347)(70 407 391 340)(71 400 392 345)(72 405 393 338)(109 417 238 175)(110 410 239 180)(111 415 240 173)(112 420 229 178)(113 413 230 171)(114 418 231 176)(115 411 232 169)(116 416 233 174)(117 409 234 179)(118 414 235 172)(119 419 236 177)(120 412 237 170)(133 376 247 371)(134 381 248 364)(135 374 249 369)(136 379 250 362)(137 384 251 367)(138 377 252 372)(139 382 241 365)(140 375 242 370)(141 380 243 363)(142 373 244 368)(143 378 245 361)(144 383 246 366)(181 202 255 316)(182 195 256 321)(183 200 257 314)(184 193 258 319)(185 198 259 324)(186 203 260 317)(187 196 261 322)(188 201 262 315)(189 194 263 320)(190 199 264 313)(191 204 253 318)(192 197 254 323)(349 435 454 432)(350 440 455 425)(351 433 456 430)(352 438 445 423)(353 443 446 428)(354 436 447 421)(355 441 448 426)(356 434 449 431)(357 439 450 424)(358 444 451 429)(359 437 452 422)(360 442 453 427)

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B5C5D6A6B6C10A···10L12A···12F15A15B15C15D20A···20H20I···20P20Q···20AF20AG···20AN30A···30L60A···60X
order122234444444444555566610···1012···121515151520···2020···2020···2020···2030···3060···60
size1111222446666121211112221···14···422222···24···46···612···122···24···4

120 irreducible representations

dim111111111122222222224444
type++++++-+--+
imageC1C2C2C2C2C5C10C10C10C10S3Q8D6C4○D4Dic6C5×S3C5×Q8S3×C10C5×C4○D4C5×Dic6D42S3Q83S3C5×D42S3C5×Q83S3
kernelC5×C4.Dic6Dic3×C20C5×Dic3⋊C4C5×C4⋊Dic3C15×C4⋊C4C4.Dic6C4×Dic3Dic3⋊C4C4⋊Dic3C3×C4⋊C4C5×C4⋊C4C60C2×C20C30C20C4⋊C4C12C2×C4C6C4C10C10C2C2
# reps1123144812412344481216161144

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

34000
03400
0010
0001
,
1100
60000
002554
001136
,
234600
153800
002759
005934
,
255100
263600
003016
00131
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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