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## G = C6×C10.D4order 480 = 25·3·5

### Direct product of C6 and C10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×C10.D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C2×C6×Dic5 — C6×C10.D4
 Lower central C5 — C10 — C6×C10.D4
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C6×C10.D4
G = < a,b,c,d | a6=b10=c4=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 432 in 184 conjugacy classes, 114 normal (34 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C3×C4⋊C4, C22×C12, C22×C12, C3×Dic5, C3×Dic5, C60, C2×C30, C2×C30, C10.D4, C22×Dic5, C22×C20, C6×C4⋊C4, C6×Dic5, C6×Dic5, C2×C60, C2×C60, C22×C30, C2×C10.D4, C3×C10.D4, C2×C6×Dic5, C22×C60, C6×C10.D4
Quotients:

Smallest permutation representation of C6×C10.D4
Regular action on 480 points
Generators in S480
(1 170 58 150 38 190)(2 161 59 141 39 181)(3 162 60 142 40 182)(4 163 51 143 31 183)(5 164 52 144 32 184)(6 165 53 145 33 185)(7 166 54 146 34 186)(8 167 55 147 35 187)(9 168 56 148 36 188)(10 169 57 149 37 189)(11 319 464 359 444 339)(12 320 465 360 445 340)(13 311 466 351 446 331)(14 312 467 352 447 332)(15 313 468 353 448 333)(16 314 469 354 449 334)(17 315 470 355 450 335)(18 316 461 356 441 336)(19 317 462 357 442 337)(20 318 463 358 443 338)(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 91 242)(72 224 112 204 92 243)(73 225 113 205 93 244)(74 226 114 206 94 245)(75 227 115 207 95 246)(76 228 116 208 96 247)(77 229 117 209 97 248)(78 230 118 210 98 249)(79 221 119 201 99 250)(80 222 120 202 100 241)(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)(251 406 291 386 271 426)(252 407 292 387 272 427)(253 408 293 388 273 428)(254 409 294 389 274 429)(255 410 295 390 275 430)(256 401 296 381 276 421)(257 402 297 382 277 422)(258 403 298 383 278 423)(259 404 299 384 279 424)(260 405 300 385 280 425)(261 396 301 376 281 416)(262 397 302 377 282 417)(263 398 303 378 283 418)(264 399 304 379 284 419)(265 400 305 380 285 420)(266 391 306 371 286 411)(267 392 307 372 287 412)(268 393 308 373 288 413)(269 394 309 374 289 414)(270 395 310 375 290 415)(321 456 361 436 341 476)(322 457 362 437 342 477)(323 458 363 438 343 478)(324 459 364 439 344 479)(325 460 365 440 345 480)(326 451 366 431 346 471)(327 452 367 432 347 472)(328 453 368 433 348 473)(329 454 369 434 349 474)(330 455 370 435 350 475)
(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)
(1 270 25 260)(2 269 26 259)(3 268 27 258)(4 267 28 257)(5 266 29 256)(6 265 30 255)(7 264 21 254)(8 263 22 253)(9 262 23 252)(10 261 24 251)(11 250 474 234)(12 249 475 233)(13 248 476 232)(14 247 477 231)(15 246 478 240)(16 245 479 239)(17 244 480 238)(18 243 471 237)(19 242 472 236)(20 241 473 235)(31 287 48 277)(32 286 49 276)(33 285 50 275)(34 284 41 274)(35 283 42 273)(36 282 43 272)(37 281 44 271)(38 290 45 280)(39 289 46 279)(40 288 47 278)(51 307 68 297)(52 306 69 296)(53 305 70 295)(54 304 61 294)(55 303 62 293)(56 302 63 292)(57 301 64 291)(58 310 65 300)(59 309 66 299)(60 308 67 298)(71 327 88 317)(72 326 89 316)(73 325 90 315)(74 324 81 314)(75 323 82 313)(76 322 83 312)(77 321 84 311)(78 330 85 320)(79 329 86 319)(80 328 87 318)(91 347 108 337)(92 346 109 336)(93 345 110 335)(94 344 101 334)(95 343 102 333)(96 342 103 332)(97 341 104 331)(98 350 105 340)(99 349 106 339)(100 348 107 338)(111 367 128 357)(112 366 129 356)(113 365 130 355)(114 364 121 354)(115 363 122 353)(116 362 123 352)(117 361 124 351)(118 370 125 360)(119 369 126 359)(120 368 127 358)(131 387 148 377)(132 386 149 376)(133 385 150 375)(134 384 141 374)(135 383 142 373)(136 382 143 372)(137 381 144 371)(138 390 145 380)(139 389 146 379)(140 388 147 378)(151 407 168 397)(152 406 169 396)(153 405 170 395)(154 404 161 394)(155 403 162 393)(156 402 163 392)(157 401 164 391)(158 410 165 400)(159 409 166 399)(160 408 167 398)(171 427 188 417)(172 426 189 416)(173 425 190 415)(174 424 181 414)(175 423 182 413)(176 422 183 412)(177 421 184 411)(178 430 185 420)(179 429 186 419)(180 428 187 418)(191 447 208 437)(192 446 209 436)(193 445 210 435)(194 444 201 434)(195 443 202 433)(196 442 203 432)(197 441 204 431)(198 450 205 440)(199 449 206 439)(200 448 207 438)(211 467 228 457)(212 466 229 456)(213 465 230 455)(214 464 221 454)(215 463 222 453)(216 462 223 452)(217 461 224 451)(218 470 225 460)(219 469 226 459)(220 468 227 458)
(1 315 6 320)(2 314 7 319)(3 313 8 318)(4 312 9 317)(5 311 10 316)(11 181 16 186)(12 190 17 185)(13 189 18 184)(14 188 19 183)(15 187 20 182)(21 329 26 324)(22 328 27 323)(23 327 28 322)(24 326 29 321)(25 325 30 330)(31 332 36 337)(32 331 37 336)(33 340 38 335)(34 339 39 334)(35 338 40 333)(41 349 46 344)(42 348 47 343)(43 347 48 342)(44 346 49 341)(45 345 50 350)(51 352 56 357)(52 351 57 356)(53 360 58 355)(54 359 59 354)(55 358 60 353)(61 369 66 364)(62 368 67 363)(63 367 68 362)(64 366 69 361)(65 365 70 370)(71 257 76 252)(72 256 77 251)(73 255 78 260)(74 254 79 259)(75 253 80 258)(81 264 86 269)(82 263 87 268)(83 262 88 267)(84 261 89 266)(85 270 90 265)(91 277 96 272)(92 276 97 271)(93 275 98 280)(94 274 99 279)(95 273 100 278)(101 284 106 289)(102 283 107 288)(103 282 108 287)(104 281 109 286)(105 290 110 285)(111 297 116 292)(112 296 117 291)(113 295 118 300)(114 294 119 299)(115 293 120 298)(121 304 126 309)(122 303 127 308)(123 302 128 307)(124 301 129 306)(125 310 130 305)(131 432 136 437)(132 431 137 436)(133 440 138 435)(134 439 139 434)(135 438 140 433)(141 449 146 444)(142 448 147 443)(143 447 148 442)(144 446 149 441)(145 445 150 450)(151 452 156 457)(152 451 157 456)(153 460 158 455)(154 459 159 454)(155 458 160 453)(161 469 166 464)(162 468 167 463)(163 467 168 462)(164 466 169 461)(165 465 170 470)(171 472 176 477)(172 471 177 476)(173 480 178 475)(174 479 179 474)(175 478 180 473)(191 377 196 372)(192 376 197 371)(193 375 198 380)(194 374 199 379)(195 373 200 378)(201 384 206 389)(202 383 207 388)(203 382 208 387)(204 381 209 386)(205 390 210 385)(211 397 216 392)(212 396 217 391)(213 395 218 400)(214 394 219 399)(215 393 220 398)(221 404 226 409)(222 403 227 408)(223 402 228 407)(224 401 229 406)(225 410 230 405)(231 417 236 412)(232 416 237 411)(233 415 238 420)(234 414 239 419)(235 413 240 418)(241 423 246 428)(242 422 247 427)(243 421 248 426)(244 430 249 425)(245 429 250 424)

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

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

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

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 D10 D10 C3×D4 C3×Q8 C3×D5 Dic10 C4×D5 C5⋊D4 C6×D5 C6×D5 C3×Dic10 D5×C12 C3×C5⋊D4 kernel C6×C10.D4 C3×C10.D4 C2×C6×Dic5 C22×C60 C2×C10.D4 C6×Dic5 C10.D4 C22×Dic5 C22×C20 C2×Dic5 C2×C30 C2×C30 C22×C12 C2×C12 C22×C6 C2×C10 C2×C10 C22×C4 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 2 1 2 8 8 4 2 16 2 2 2 4 2 4 4 4 8 8 8 8 4 16 16 16

Matrix representation of C6×C10.D4 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 48 0 0 0 0 48
,
 60 0 0 0 0 1 0 0 0 0 43 18 0 0 43 1
,
 1 0 0 0 0 60 0 0 0 0 7 23 0 0 27 54
,
 11 0 0 0 0 60 0 0 0 0 22 2 0 0 32 39
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,48,0,0,0,0,48],[60,0,0,0,0,1,0,0,0,0,43,43,0,0,18,1],[1,0,0,0,0,60,0,0,0,0,7,27,0,0,23,54],[11,0,0,0,0,60,0,0,0,0,22,32,0,0,2,39] >;

C6×C10.D4 in GAP, Magma, Sage, TeX

C_6\times C_{10}.D_4
% in TeX

G:=Group("C6xC10.D4");
// GroupNames label

G:=SmallGroup(480,716);
// by ID

G=gap.SmallGroup(480,716);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,1094,142,18822]);
// Polycyclic

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

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