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G = C6×C4⋊Dic5order 480 = 25·3·5

Direct product of C6 and C4⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×C4⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C2×C6×Dic5 — C6×C4⋊Dic5
 Lower central C5 — C10 — C6×C4⋊Dic5
 Upper central C1 — C22×C6 — C22×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, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C3×C4⋊C4, C22×C12, C22×C12, C3×Dic5, C60, C2×C30, C2×C30, C4⋊Dic5, C22×Dic5, C22×C20, C6×C4⋊C4, C6×Dic5, C6×Dic5, C2×C60, C22×C30, C2×C4⋊Dic5, C3×C4⋊Dic5, C2×C6×Dic5, C22×C60, C6×C4⋊Dic5
Quotients:

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

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 Q8 D5 Dic5 D10 D10 C3×D4 C3×Q8 C3×D5 Dic10 D20 C3×Dic5 C6×D5 C6×D5 C3×Dic10 C3×D20 kernel C6×C4⋊Dic5 C3×C4⋊Dic5 C2×C6×Dic5 C22×C60 C2×C4⋊Dic5 C2×C60 C4⋊Dic5 C22×Dic5 C22×C20 C2×C20 C2×C30 C2×C30 C22×C12 C2×C12 C2×C12 C22×C6 C2×C10 C2×C10 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 2 8 8 4 2 16 2 2 2 8 4 2 4 4 4 8 8 16 8 4 16 16

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

 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 57 0 0 0 4 25
,
 1 0 0 0 0 0 44 1 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 60 18
,
 60 0 0 0 0 0 5 27 0 0 0 51 56 0 0 0 0 0 5 29 0 0 0 58 56

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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