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## G = C2×C6.Dic10order 480 = 25·3·5

### Direct product of C2 and C6.Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C6.Dic10
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C6.Dic10 — C2×C6.Dic10
 Lower central C15 — C30 — C2×C6.Dic10
 Upper central C1 — C23

Generators and relations for C2×C6.Dic10
G = < a,b,c,d | a2=b6=c20=1, d2=b3c10, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 668 in 184 conjugacy classes, 100 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, C10, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, Dic3⋊C4, C22×Dic3, C22×Dic3, C22×C12, C5×Dic3, C3×Dic5, Dic15, C2×C30, C2×C30, C4⋊Dic5, C22×Dic5, C22×Dic5, C22×C20, C2×Dic3⋊C4, C6×Dic5, C6×Dic5, C10×Dic3, C2×Dic15, C2×Dic15, C22×C30, C2×C4⋊Dic5, C6.Dic10, C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C2×C6.Dic10
Quotients:

Smallest permutation representation of C2×C6.Dic10
Regular action on 480 points
Generators in S480
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 21)(17 22)(18 23)(19 24)(20 25)(41 180)(42 161)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 169)(51 170)(52 171)(53 172)(54 173)(55 174)(56 175)(57 176)(58 177)(59 178)(60 179)(61 369)(62 370)(63 371)(64 372)(65 373)(66 374)(67 375)(68 376)(69 377)(70 378)(71 379)(72 380)(73 361)(74 362)(75 363)(76 364)(77 365)(78 366)(79 367)(80 368)(81 288)(82 289)(83 290)(84 291)(85 292)(86 293)(87 294)(88 295)(89 296)(90 297)(91 298)(92 299)(93 300)(94 281)(95 282)(96 283)(97 284)(98 285)(99 286)(100 287)(101 469)(102 470)(103 471)(104 472)(105 473)(106 474)(107 475)(108 476)(109 477)(110 478)(111 479)(112 480)(113 461)(114 462)(115 463)(116 464)(117 465)(118 466)(119 467)(120 468)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 196)(130 197)(131 198)(132 199)(133 200)(134 181)(135 182)(136 183)(137 184)(138 185)(139 186)(140 187)(141 434)(142 435)(143 436)(144 437)(145 438)(146 439)(147 440)(148 421)(149 422)(150 423)(151 424)(152 425)(153 426)(154 427)(155 428)(156 429)(157 430)(158 431)(159 432)(160 433)(201 333)(202 334)(203 335)(204 336)(205 337)(206 338)(207 339)(208 340)(209 321)(210 322)(211 323)(212 324)(213 325)(214 326)(215 327)(216 328)(217 329)(218 330)(219 331)(220 332)(221 443)(222 444)(223 445)(224 446)(225 447)(226 448)(227 449)(228 450)(229 451)(230 452)(231 453)(232 454)(233 455)(234 456)(235 457)(236 458)(237 459)(238 460)(239 441)(240 442)(241 353)(242 354)(243 355)(244 356)(245 357)(246 358)(247 359)(248 360)(249 341)(250 342)(251 343)(252 344)(253 345)(254 346)(255 347)(256 348)(257 349)(258 350)(259 351)(260 352)(261 407)(262 408)(263 409)(264 410)(265 411)(266 412)(267 413)(268 414)(269 415)(270 416)(271 417)(272 418)(273 419)(274 420)(275 401)(276 402)(277 403)(278 404)(279 405)(280 406)(301 387)(302 388)(303 389)(304 390)(305 391)(306 392)(307 393)(308 394)(309 395)(310 396)(311 397)(312 398)(313 399)(314 400)(315 381)(316 382)(317 383)(318 384)(319 385)(320 386)
(1 110 265 67 322 249)(2 250 323 68 266 111)(3 112 267 69 324 251)(4 252 325 70 268 113)(5 114 269 71 326 253)(6 254 327 72 270 115)(7 116 271 73 328 255)(8 256 329 74 272 117)(9 118 273 75 330 257)(10 258 331 76 274 119)(11 120 275 77 332 259)(12 260 333 78 276 101)(13 102 277 79 334 241)(14 242 335 80 278 103)(15 104 279 61 336 243)(16 244 337 62 280 105)(17 106 261 63 338 245)(18 246 339 64 262 107)(19 108 263 65 340 247)(20 248 321 66 264 109)(21 356 205 370 406 473)(22 474 407 371 206 357)(23 358 207 372 408 475)(24 476 409 373 208 359)(25 360 209 374 410 477)(26 478 411 375 210 341)(27 342 211 376 412 479)(28 480 413 377 212 343)(29 344 213 378 414 461)(30 462 415 379 214 345)(31 346 215 380 416 463)(32 464 417 361 216 347)(33 348 217 362 418 465)(34 466 419 363 218 349)(35 350 219 364 420 467)(36 468 401 365 220 351)(37 352 201 366 402 469)(38 470 403 367 202 353)(39 354 203 368 404 471)(40 472 405 369 204 355)(41 311 134 158 291 223)(42 224 292 159 135 312)(43 313 136 160 293 225)(44 226 294 141 137 314)(45 315 138 142 295 227)(46 228 296 143 139 316)(47 317 140 144 297 229)(48 230 298 145 121 318)(49 319 122 146 299 231)(50 232 300 147 123 320)(51 301 124 148 281 233)(52 234 282 149 125 302)(53 303 126 150 283 235)(54 236 284 151 127 304)(55 305 128 152 285 237)(56 238 286 153 129 306)(57 307 130 154 287 239)(58 240 288 155 131 308)(59 309 132 156 289 221)(60 222 290 157 133 310)(81 428 198 394 177 442)(82 443 178 395 199 429)(83 430 200 396 179 444)(84 445 180 397 181 431)(85 432 182 398 161 446)(86 447 162 399 183 433)(87 434 184 400 163 448)(88 449 164 381 185 435)(89 436 186 382 165 450)(90 451 166 383 187 437)(91 438 188 384 167 452)(92 453 168 385 189 439)(93 440 190 386 169 454)(94 455 170 387 191 421)(95 422 192 388 171 456)(96 457 172 389 193 423)(97 424 194 390 173 458)(98 459 174 391 195 425)(99 426 196 392 175 460)(100 441 176 393 197 427)
(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)
(1 307 77 297)(2 306 78 296)(3 305 79 295)(4 304 80 294)(5 303 61 293)(6 302 62 292)(7 301 63 291)(8 320 64 290)(9 319 65 289)(10 318 66 288)(11 317 67 287)(12 316 68 286)(13 315 69 285)(14 314 70 284)(15 313 71 283)(16 312 72 282)(17 311 73 281)(18 310 74 300)(19 309 75 299)(20 308 76 298)(21 398 380 95)(22 397 361 94)(23 396 362 93)(24 395 363 92)(25 394 364 91)(26 393 365 90)(27 392 366 89)(28 391 367 88)(29 390 368 87)(30 389 369 86)(31 388 370 85)(32 387 371 84)(33 386 372 83)(34 385 373 82)(35 384 374 81)(36 383 375 100)(37 382 376 99)(38 381 377 98)(39 400 378 97)(40 399 379 96)(41 271 148 245)(42 270 149 244)(43 269 150 243)(44 268 151 242)(45 267 152 241)(46 266 153 260)(47 265 154 259)(48 264 155 258)(49 263 156 257)(50 262 157 256)(51 261 158 255)(52 280 159 254)(53 279 160 253)(54 278 141 252)(55 277 142 251)(56 276 143 250)(57 275 144 249)(58 274 145 248)(59 273 146 247)(60 272 147 246)(101 139 323 238)(102 138 324 237)(103 137 325 236)(104 136 326 235)(105 135 327 234)(106 134 328 233)(107 133 329 232)(108 132 330 231)(109 131 331 230)(110 130 332 229)(111 129 333 228)(112 128 334 227)(113 127 335 226)(114 126 336 225)(115 125 337 224)(116 124 338 223)(117 123 339 222)(118 122 340 221)(119 121 321 240)(120 140 322 239)(161 416 422 356)(162 415 423 355)(163 414 424 354)(164 413 425 353)(165 412 426 352)(166 411 427 351)(167 410 428 350)(168 409 429 349)(169 408 430 348)(170 407 431 347)(171 406 432 346)(172 405 433 345)(173 404 434 344)(174 403 435 343)(175 402 436 342)(176 401 437 341)(177 420 438 360)(178 419 439 359)(179 418 440 358)(180 417 421 357)(181 216 455 474)(182 215 456 473)(183 214 457 472)(184 213 458 471)(185 212 459 470)(186 211 460 469)(187 210 441 468)(188 209 442 467)(189 208 443 466)(190 207 444 465)(191 206 445 464)(192 205 446 463)(193 204 447 462)(194 203 448 461)(195 202 449 480)(196 201 450 479)(197 220 451 478)(198 219 452 477)(199 218 453 476)(200 217 454 475)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A ··· 6G 10A ··· 10N 12A ··· 12H 15A 15B 20A ··· 20P 30A ··· 30N order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 20 ··· 20 30 ··· 30 size 1 1 ··· 1 2 6 6 6 6 10 10 10 10 30 30 30 30 2 2 2 ··· 2 2 ··· 2 10 ··· 10 4 4 6 ··· 6 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + - + + - - + + - + - + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D5 D6 D6 Dic5 D10 D10 Dic6 C4×S3 C3⋊D4 Dic10 D20 S3×D5 S3×Dic5 C3⋊D20 C15⋊Q8 C2×S3×D5 kernel C2×C6.Dic10 C6.Dic10 C2×C6×Dic5 Dic3×C2×C10 C22×Dic15 C10×Dic3 C22×Dic5 C2×C30 C2×C30 C22×Dic3 C2×Dic5 C22×C10 C2×Dic3 C2×Dic3 C22×C6 C2×C10 C2×C10 C2×C10 C2×C6 C2×C6 C23 C22 C22 C22 C22 # reps 1 4 1 1 1 8 1 2 2 2 2 1 8 4 2 4 4 4 8 8 2 4 4 4 2

Matrix representation of C2×C6.Dic10 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 60 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 44 60 0 0 0 0 45 60 0 0 0 0 0 0 27 25 0 0 0 0 36 4
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 20 34 0 0 0 0 51 41 0 0 0 0 0 0 16 52 0 0 0 0 8 45

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,44,45,0,0,0,0,60,60,0,0,0,0,0,0,27,36,0,0,0,0,25,4],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,20,51,0,0,0,0,34,41,0,0,0,0,0,0,16,8,0,0,0,0,52,45] >;

C2×C6.Dic10 in GAP, Magma, Sage, TeX

C_2\times C_6.{\rm Dic}_{10}
% in TeX

G:=Group("C2xC6.Dic10");
// GroupNames label

G:=SmallGroup(480,621);
// by ID

G=gap.SmallGroup(480,621);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,120,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^20=1,d^2=b^3*c^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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