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G = C6×C4⋊Dic5order 480 = 25·3·5

Direct product of C6 and C4⋊Dic5

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

Aliases: C6×C4⋊Dic5, (C2×C60)⋊22C4, C6041(C2×C4), C3010(C4⋊C4), C42(C6×Dic5), C2.2(C6×D20), C2010(C2×C12), (C2×C20)⋊10C12, C10.8(C6×Q8), C129(C2×Dic5), (C2×C12)⋊8Dic5, C10.13(C6×D4), (C2×C6).56D20, C6.86(C2×D20), (C2×C30).19Q8, C30.81(C2×Q8), (C2×C30).116D4, C30.287(C2×D4), C2.3(C6×Dic10), C23.35(C6×D5), (C2×C12).435D10, (C22×C20).11C6, (C22×C60).18C2, (C2×C6).17Dic10, C6.49(C2×Dic10), C22.15(C3×D20), (C22×C12).16D5, C30.221(C22×C4), (C2×C30).360C23, C10.36(C22×C12), (C2×C60).512C22, (C22×C6).132D10, (C22×Dic5).7C6, C22.5(C3×Dic10), C22.14(C6×Dic5), C6.33(C22×Dic5), (C22×C30).156C22, (C6×Dic5).246C22, C54(C6×C4⋊C4), C103(C3×C4⋊C4), C1520(C2×C4⋊C4), C2.4(C2×C6×Dic5), (C2×C4)⋊3(C3×Dic5), (C2×C4).85(C6×D5), (C2×C10).6(C3×Q8), C22.21(D5×C2×C6), (C2×C20).95(C2×C6), (C2×C10).20(C3×D4), (C2×C6×Dic5).13C2, (C22×C4).8(C3×D5), (C2×C30).190(C2×C4), (C2×C10).54(C2×C12), (C2×C6).45(C2×Dic5), (C22×C10).43(C2×C6), (C2×C10).43(C22×C6), (C2×Dic5).36(C2×C6), (C2×C6).356(C22×D5), SmallGroup(480,718)

Series: Derived Chief Lower central Upper central

C1C10 — C6×C4⋊Dic5
C1C5C10C2×C10C2×C30C6×Dic5C2×C6×Dic5 — C6×C4⋊Dic5
C5C10 — C6×C4⋊Dic5
C1C22×C6C22×C12

Generators and relations for C6×C4⋊Dic5
 G = < a,b,c,d | a6=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 432 in 184 conjugacy classes, 130 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×4], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×8], C23, C10 [×3], C10 [×4], C12 [×4], C12 [×4], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C12 [×6], C2×C12 [×8], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C3×C4⋊C4 [×4], C22×C12, C22×C12 [×2], C3×Dic5 [×4], C60 [×4], C2×C30, C2×C30 [×6], C4⋊Dic5 [×4], C22×Dic5 [×2], C22×C20, C6×C4⋊C4, C6×Dic5 [×4], C6×Dic5 [×4], C2×C60 [×6], C22×C30, C2×C4⋊Dic5, C3×C4⋊Dic5 [×4], C2×C6×Dic5 [×2], C22×C60, C6×C4⋊Dic5
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C12 [×4], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×C6, C3×D5, C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C3×C4⋊C4 [×4], C22×C12, C6×D4, C6×Q8, C3×Dic5 [×4], C6×D5 [×3], C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C6×C4⋊C4, C3×Dic10 [×2], C3×D20 [×2], C6×Dic5 [×6], D5×C2×C6, C2×C4⋊Dic5, C3×C4⋊Dic5 [×4], C6×Dic10, C6×D20, C2×C6×Dic5, C6×C4⋊Dic5

Smallest permutation representation of C6×C4⋊Dic5
Regular action on 480 points
Generators in S480
(1 162 60 142 40 182)(2 163 51 143 31 183)(3 164 52 144 32 184)(4 165 53 145 33 185)(5 166 54 146 34 186)(6 167 55 147 35 187)(7 168 56 148 36 188)(8 169 57 149 37 189)(9 170 58 150 38 190)(10 161 59 141 39 181)(11 315 468 355 448 335)(12 316 469 356 449 336)(13 317 470 357 450 337)(14 318 461 358 441 338)(15 319 462 359 442 339)(16 320 463 360 443 340)(17 311 464 351 444 331)(18 312 465 352 445 332)(19 313 466 353 446 333)(20 314 467 354 447 334)(21 159 61 139 41 179)(22 160 62 140 42 180)(23 151 63 131 43 171)(24 152 64 132 44 172)(25 153 65 133 45 173)(26 154 66 134 46 174)(27 155 67 135 47 175)(28 156 68 136 48 176)(29 157 69 137 49 177)(30 158 70 138 50 178)(71 223 111 203 91 246)(72 224 112 204 92 247)(73 225 113 205 93 248)(74 226 114 206 94 249)(75 227 115 207 95 250)(76 228 116 208 96 241)(77 229 117 209 97 242)(78 230 118 210 98 243)(79 221 119 201 99 244)(80 222 120 202 100 245)(81 219 121 199 101 239)(82 220 122 200 102 240)(83 211 123 191 103 231)(84 212 124 192 104 232)(85 213 125 193 105 233)(86 214 126 194 106 234)(87 215 127 195 107 235)(88 216 128 196 108 236)(89 217 129 197 109 237)(90 218 130 198 110 238)(251 404 291 384 271 424)(252 405 292 385 272 425)(253 406 293 386 273 426)(254 407 294 387 274 427)(255 408 295 388 275 428)(256 409 296 389 276 429)(257 410 297 390 277 430)(258 401 298 381 278 421)(259 402 299 382 279 422)(260 403 300 383 280 423)(261 398 301 378 281 418)(262 399 302 379 282 419)(263 400 303 380 283 420)(264 391 304 371 284 411)(265 392 305 372 285 412)(266 393 306 373 286 413)(267 394 307 374 287 414)(268 395 308 375 288 415)(269 396 309 376 289 416)(270 397 310 377 290 417)(321 458 361 438 341 478)(322 459 362 439 342 479)(323 460 363 440 343 480)(324 451 364 431 344 471)(325 452 365 432 345 472)(326 453 366 433 346 473)(327 454 367 434 347 474)(328 455 368 435 348 475)(329 456 369 436 349 476)(330 457 370 437 350 477)
(1 87 27 80)(2 88 28 71)(3 89 29 72)(4 90 30 73)(5 81 21 74)(6 82 22 75)(7 83 23 76)(8 84 24 77)(9 85 25 78)(10 86 26 79)(11 420 480 428)(12 411 471 429)(13 412 472 430)(14 413 473 421)(15 414 474 422)(16 415 475 423)(17 416 476 424)(18 417 477 425)(19 418 478 426)(20 419 479 427)(31 108 48 91)(32 109 49 92)(33 110 50 93)(34 101 41 94)(35 102 42 95)(36 103 43 96)(37 104 44 97)(38 105 45 98)(39 106 46 99)(40 107 47 100)(51 128 68 111)(52 129 69 112)(53 130 70 113)(54 121 61 114)(55 122 62 115)(56 123 63 116)(57 124 64 117)(58 125 65 118)(59 126 66 119)(60 127 67 120)(131 208 148 191)(132 209 149 192)(133 210 150 193)(134 201 141 194)(135 202 142 195)(136 203 143 196)(137 204 144 197)(138 205 145 198)(139 206 146 199)(140 207 147 200)(151 228 168 211)(152 229 169 212)(153 230 170 213)(154 221 161 214)(155 222 162 215)(156 223 163 216)(157 224 164 217)(158 225 165 218)(159 226 166 219)(160 227 167 220)(171 241 188 231)(172 242 189 232)(173 243 190 233)(174 244 181 234)(175 245 182 235)(176 246 183 236)(177 247 184 237)(178 248 185 238)(179 249 186 239)(180 250 187 240)(251 311 269 329)(252 312 270 330)(253 313 261 321)(254 314 262 322)(255 315 263 323)(256 316 264 324)(257 317 265 325)(258 318 266 326)(259 319 267 327)(260 320 268 328)(271 331 289 349)(272 332 290 350)(273 333 281 341)(274 334 282 342)(275 335 283 343)(276 336 284 344)(277 337 285 345)(278 338 286 346)(279 339 287 347)(280 340 288 348)(291 351 309 369)(292 352 310 370)(293 353 301 361)(294 354 302 362)(295 355 303 363)(296 356 304 364)(297 357 305 365)(298 358 306 366)(299 359 307 367)(300 360 308 368)(371 431 389 449)(372 432 390 450)(373 433 381 441)(374 434 382 442)(375 435 383 443)(376 436 384 444)(377 437 385 445)(378 438 386 446)(379 439 387 447)(380 440 388 448)(391 451 409 469)(392 452 410 470)(393 453 401 461)(394 454 402 462)(395 455 403 463)(396 456 404 464)(397 457 405 465)(398 458 406 466)(399 459 407 467)(400 460 408 468)
(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)
(1 255 6 260)(2 254 7 259)(3 253 8 258)(4 252 9 257)(5 251 10 256)(11 250 16 245)(12 249 17 244)(13 248 18 243)(14 247 19 242)(15 246 20 241)(21 269 26 264)(22 268 27 263)(23 267 28 262)(24 266 29 261)(25 265 30 270)(31 274 36 279)(32 273 37 278)(33 272 38 277)(34 271 39 276)(35 280 40 275)(41 289 46 284)(42 288 47 283)(43 287 48 282)(44 286 49 281)(45 285 50 290)(51 294 56 299)(52 293 57 298)(53 292 58 297)(54 291 59 296)(55 300 60 295)(61 309 66 304)(62 308 67 303)(63 307 68 302)(64 306 69 301)(65 305 70 310)(71 314 76 319)(72 313 77 318)(73 312 78 317)(74 311 79 316)(75 320 80 315)(81 329 86 324)(82 328 87 323)(83 327 88 322)(84 326 89 321)(85 325 90 330)(91 334 96 339)(92 333 97 338)(93 332 98 337)(94 331 99 336)(95 340 100 335)(101 349 106 344)(102 348 107 343)(103 347 108 342)(104 346 109 341)(105 345 110 350)(111 354 116 359)(112 353 117 358)(113 352 118 357)(114 351 119 356)(115 360 120 355)(121 369 126 364)(122 368 127 363)(123 367 128 362)(124 366 129 361)(125 365 130 370)(131 374 136 379)(132 373 137 378)(133 372 138 377)(134 371 139 376)(135 380 140 375)(141 389 146 384)(142 388 147 383)(143 387 148 382)(144 386 149 381)(145 385 150 390)(151 394 156 399)(152 393 157 398)(153 392 158 397)(154 391 159 396)(155 400 160 395)(161 409 166 404)(162 408 167 403)(163 407 168 402)(164 406 169 401)(165 405 170 410)(171 414 176 419)(172 413 177 418)(173 412 178 417)(174 411 179 416)(175 420 180 415)(181 429 186 424)(182 428 187 423)(183 427 188 422)(184 426 189 421)(185 425 190 430)(191 434 196 439)(192 433 197 438)(193 432 198 437)(194 431 199 436)(195 440 200 435)(201 449 206 444)(202 448 207 443)(203 447 208 442)(204 446 209 441)(205 445 210 450)(211 454 216 459)(212 453 217 458)(213 452 218 457)(214 451 219 456)(215 460 220 455)(221 469 226 464)(222 468 227 463)(223 467 228 462)(224 466 229 461)(225 465 230 470)(231 474 236 479)(232 473 237 478)(233 472 238 477)(234 471 239 476)(235 480 240 475)

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

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

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

156 conjugacy classes

class 1 2A···2G3A3B4A4B4C4D4E···4L5A5B6A···6N10A···10N12A···12H12I···12X15A15B15C15D20A···20P30A···30AB60A···60AF
order12···23344444···4556···610···1012···1212···121515151520···2030···3060···60
size11···111222210···10221···12···22···210···1022222···22···22···2

156 irreducible representations

dim11111111112222222222222222
type+++++-+-++-+
imageC1C2C2C2C3C4C6C6C6C12D4Q8D5Dic5D10D10C3×D4C3×Q8C3×D5Dic10D20C3×Dic5C6×D5C6×D5C3×Dic10C3×D20
kernelC6×C4⋊Dic5C3×C4⋊Dic5C2×C6×Dic5C22×C60C2×C4⋊Dic5C2×C60C4⋊Dic5C22×Dic5C22×C20C2×C20C2×C30C2×C30C22×C12C2×C12C2×C12C22×C6C2×C10C2×C10C22×C4C2×C6C2×C6C2×C4C2×C4C23C22C22
# reps142128842162228424448816841616

Matrix representation of C6×C4⋊Dic5 in GL5(𝔽61)

600000
047000
004700
00010
00001
,
600000
060000
006000
0003657
000425
,
10000
044100
060000
00001
0006018
,
600000
052700
0515600
000529
0005856

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,36,4,0,0,0,57,25],[1,0,0,0,0,0,44,60,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,1,18],[60,0,0,0,0,0,5,51,0,0,0,27,56,0,0,0,0,0,5,58,0,0,0,29,56] >;

C6×C4⋊Dic5 in GAP, Magma, Sage, TeX

C_6\times C_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C6xC4:Dic5");
// GroupNames label

G:=SmallGroup(480,718);
// by ID

G=gap.SmallGroup(480,718);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,268,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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