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G = C6×C10.D4order 480 = 25·3·5

Direct product of C6 and C10.D4

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

Aliases: C6×C10.D4, C309(C4⋊C4), C10.6(C6×Q8), C10.42(C6×D4), (C2×C30).18Q8, C30.79(C2×Q8), (C6×Dic5)⋊11C4, Dic55(C2×C12), (C2×Dic5)⋊5C12, (C2×C30).161D4, C30.396(C2×D4), (C22×C20).8C6, (C22×C60).6C2, C2.2(C6×Dic10), C23.34(C6×D5), (C22×C12).4D5, (C2×C12).377D10, C6.47(C2×Dic10), (C2×C6).16Dic10, C22.16(D5×C12), (C2×C60).447C22, C30.189(C22×C4), C10.31(C22×C12), (C2×C30).358C23, (C22×C6).131D10, (C22×Dic5).6C6, C22.4(C3×Dic10), (C22×C30).154C22, (C6×Dic5).245C22, C53(C6×C4⋊C4), C102(C3×C4⋊C4), C1519(C2×C4⋊C4), C2.18(D5×C2×C12), C6.114(C2×C4×D5), C2.1(C6×C5⋊D4), (C2×C6).65(C4×D5), (C2×C4).64(C6×D5), (C2×C10).5(C3×Q8), C22.20(D5×C2×C6), (C2×C20).78(C2×C6), (C2×C10).36(C3×D4), C6.123(C2×C5⋊D4), (C2×C6×Dic5).12C2, (C22×C4).5(C3×D5), (C2×C10).37(C2×C12), (C2×C30).152(C2×C4), (C3×Dic5)⋊24(C2×C4), (C2×C6).91(C5⋊D4), C22.19(C3×C5⋊D4), (C22×C10).41(C2×C6), (C2×C10).41(C22×C6), (C2×Dic5).35(C2×C6), (C2×C6).354(C22×D5), SmallGroup(480,716)

Series: Derived Chief Lower central Upper central

C1C10 — C6×C10.D4
C1C5C10C2×C10C2×C30C6×Dic5C2×C6×Dic5 — C6×C10.D4
C5C10 — C6×C10.D4
C1C22×C6C22×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 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×4], C12 [×8], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×6], C2×C12 [×2], C2×C12 [×12], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C3×C4⋊C4 [×4], C22×C12, C22×C12 [×2], C3×Dic5 [×4], C3×Dic5 [×2], C60 [×2], C2×C30, C2×C30 [×6], C10.D4 [×4], C22×Dic5 [×2], C22×C20, C6×C4⋊C4, C6×Dic5 [×8], C6×Dic5 [×2], C2×C60 [×2], C2×C60 [×2], C22×C30, C2×C10.D4, C3×C10.D4 [×4], C2×C6×Dic5 [×2], C22×C60, C6×C10.D4
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, D10 [×3], C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×C6, C3×D5, C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C3×C4⋊C4 [×4], C22×C12, C6×D4, C6×Q8, C6×D5 [×3], C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C6×C4⋊C4, C3×Dic10 [×2], D5×C12 [×2], C3×C5⋊D4 [×2], D5×C2×C6, C2×C10.D4, C3×C10.D4 [×4], C6×Dic10, D5×C2×C12, C6×C5⋊D4, C6×C10.D4

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

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

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

G=PermutationGroup([(1,170,54,150,34,190),(2,161,55,141,35,181),(3,162,56,142,36,182),(4,163,57,143,37,183),(5,164,58,144,38,184),(6,165,59,145,39,185),(7,166,60,146,40,186),(8,167,51,147,31,187),(9,168,52,148,32,188),(10,169,53,149,33,189),(11,315,464,355,444,335),(12,316,465,356,445,336),(13,317,466,357,446,337),(14,318,467,358,447,338),(15,319,468,359,448,339),(16,320,469,360,449,340),(17,311,470,351,450,331),(18,312,461,352,441,332),(19,313,462,353,442,333),(20,314,463,354,443,334),(21,155,61,135,41,175),(22,156,62,136,42,176),(23,157,63,137,43,177),(24,158,64,138,44,178),(25,159,65,139,45,179),(26,160,66,140,46,180),(27,151,67,131,47,171),(28,152,68,132,48,172),(29,153,69,133,49,173),(30,154,70,134,50,174),(71,227,111,207,91,246),(72,228,112,208,92,247),(73,229,113,209,93,248),(74,230,114,210,94,249),(75,221,115,201,95,250),(76,222,116,202,96,241),(77,223,117,203,97,242),(78,224,118,204,98,243),(79,225,119,205,99,244),(80,226,120,206,100,245),(81,215,121,195,101,235),(82,216,122,196,102,236),(83,217,123,197,103,237),(84,218,124,198,104,238),(85,219,125,199,105,239),(86,220,126,200,106,240),(87,211,127,191,107,231),(88,212,128,192,108,232),(89,213,129,193,109,233),(90,214,130,194,110,234),(251,410,291,390,271,430),(252,401,292,381,272,421),(253,402,293,382,273,422),(254,403,294,383,274,423),(255,404,295,384,275,424),(256,405,296,385,276,425),(257,406,297,386,277,426),(258,407,298,387,278,427),(259,408,299,388,279,428),(260,409,300,389,280,429),(261,392,301,372,281,412),(262,393,302,373,282,413),(263,394,303,374,283,414),(264,395,304,375,284,415),(265,396,305,376,285,416),(266,397,306,377,286,417),(267,398,307,378,287,418),(268,399,308,379,288,419),(269,400,309,380,289,420),(270,391,310,371,290,411),(321,452,361,432,341,472),(322,453,362,433,342,473),(323,454,363,434,343,474),(324,455,364,435,344,475),(325,456,365,436,345,476),(326,457,366,437,346,477),(327,458,367,438,347,478),(328,459,368,439,348,479),(329,460,369,440,349,480),(330,451,370,431,350,471)], [(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)], [(1,270,25,256),(2,269,26,255),(3,268,27,254),(4,267,28,253),(5,266,29,252),(6,265,30,251),(7,264,21,260),(8,263,22,259),(9,262,23,258),(10,261,24,257),(11,250,480,240),(12,249,471,239),(13,248,472,238),(14,247,473,237),(15,246,474,236),(16,245,475,235),(17,244,476,234),(18,243,477,233),(19,242,478,232),(20,241,479,231),(31,283,42,279),(32,282,43,278),(33,281,44,277),(34,290,45,276),(35,289,46,275),(36,288,47,274),(37,287,48,273),(38,286,49,272),(39,285,50,271),(40,284,41,280),(51,303,62,299),(52,302,63,298),(53,301,64,297),(54,310,65,296),(55,309,66,295),(56,308,67,294),(57,307,68,293),(58,306,69,292),(59,305,70,291),(60,304,61,300),(71,323,82,319),(72,322,83,318),(73,321,84,317),(74,330,85,316),(75,329,86,315),(76,328,87,314),(77,327,88,313),(78,326,89,312),(79,325,90,311),(80,324,81,320),(91,343,102,339),(92,342,103,338),(93,341,104,337),(94,350,105,336),(95,349,106,335),(96,348,107,334),(97,347,108,333),(98,346,109,332),(99,345,110,331),(100,344,101,340),(111,363,122,359),(112,362,123,358),(113,361,124,357),(114,370,125,356),(115,369,126,355),(116,368,127,354),(117,367,128,353),(118,366,129,352),(119,365,130,351),(120,364,121,360),(131,383,142,379),(132,382,143,378),(133,381,144,377),(134,390,145,376),(135,389,146,375),(136,388,147,374),(137,387,148,373),(138,386,149,372),(139,385,150,371),(140,384,141,380),(151,403,162,399),(152,402,163,398),(153,401,164,397),(154,410,165,396),(155,409,166,395),(156,408,167,394),(157,407,168,393),(158,406,169,392),(159,405,170,391),(160,404,161,400),(171,423,182,419),(172,422,183,418),(173,421,184,417),(174,430,185,416),(175,429,186,415),(176,428,187,414),(177,427,188,413),(178,426,189,412),(179,425,190,411),(180,424,181,420),(191,443,202,439),(192,442,203,438),(193,441,204,437),(194,450,205,436),(195,449,206,435),(196,448,207,434),(197,447,208,433),(198,446,209,432),(199,445,210,431),(200,444,201,440),(211,463,222,459),(212,462,223,458),(213,461,224,457),(214,470,225,456),(215,469,226,455),(216,468,227,454),(217,467,228,453),(218,466,229,452),(219,465,230,451),(220,464,221,460)], [(1,311,6,316),(2,320,7,315),(3,319,8,314),(4,318,9,313),(5,317,10,312),(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),(31,334,36,339),(32,333,37,338),(33,332,38,337),(34,331,39,336),(35,340,40,335),(41,349,46,344),(42,348,47,343),(43,347,48,342),(44,346,49,341),(45,345,50,350),(51,354,56,359),(52,353,57,358),(53,352,58,357),(54,351,59,356),(55,360,60,355),(61,369,66,364),(62,368,67,363),(63,367,68,362),(64,366,69,361),(65,365,70,370),(71,259,76,254),(72,258,77,253),(73,257,78,252),(74,256,79,251),(75,255,80,260),(81,264,86,269),(82,263,87,268),(83,262,88,267),(84,261,89,266),(85,270,90,265),(91,279,96,274),(92,278,97,273),(93,277,98,272),(94,276,99,271),(95,275,100,280),(101,284,106,289),(102,283,107,288),(103,282,108,287),(104,281,109,286),(105,290,110,285),(111,299,116,294),(112,298,117,293),(113,297,118,292),(114,296,119,291),(115,295,120,300),(121,304,126,309),(122,303,127,308),(123,302,128,307),(124,301,129,306),(125,310,130,305),(131,434,136,439),(132,433,137,438),(133,432,138,437),(134,431,139,436),(135,440,140,435),(141,449,146,444),(142,448,147,443),(143,447,148,442),(144,446,149,441),(145,445,150,450),(151,454,156,459),(152,453,157,458),(153,452,158,457),(154,451,159,456),(155,460,160,455),(161,469,166,464),(162,468,167,463),(163,467,168,462),(164,466,169,461),(165,465,170,470),(171,474,176,479),(172,473,177,478),(173,472,178,477),(174,471,179,476),(175,480,180,475),(191,379,196,374),(192,378,197,373),(193,377,198,372),(194,376,199,371),(195,375,200,380),(201,384,206,389),(202,383,207,388),(203,382,208,387),(204,381,209,386),(205,390,210,385),(211,399,216,394),(212,398,217,393),(213,397,218,392),(214,396,219,391),(215,395,220,400),(221,404,226,409),(222,403,227,408),(223,402,228,407),(224,401,229,406),(225,410,230,405),(231,419,236,414),(232,418,237,413),(233,417,238,412),(234,416,239,411),(235,415,240,420),(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···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+++++-+++-
imageC1C2C2C2C3C4C6C6C6C12D4Q8D5D10D10C3×D4C3×Q8C3×D5Dic10C4×D5C5⋊D4C6×D5C6×D5C3×Dic10D5×C12C3×C5⋊D4
kernelC6×C10.D4C3×C10.D4C2×C6×Dic5C22×C60C2×C10.D4C6×Dic5C10.D4C22×Dic5C22×C20C2×Dic5C2×C30C2×C30C22×C12C2×C12C22×C6C2×C10C2×C10C22×C4C2×C6C2×C6C2×C6C2×C4C23C22C22C22
# reps142128842162224244488884161616

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

1000
06000
00480
00048
,
60000
0100
004318
00431
,
1000
06000
00723
002754
,
11000
06000
00222
003239
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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