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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 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×14], C23, C10 [×3], C10 [×4], Dic3 [×4], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×Dic3 [×6], C2×Dic3 [×4], C2×C12 [×4], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20 [×6], C22×C10, Dic3⋊C4 [×4], C22×Dic3, C22×Dic3, C22×C12, C5×Dic3 [×4], C3×Dic5 [×2], Dic15 [×2], C2×C30, C2×C30 [×6], C4⋊Dic5 [×4], C22×Dic5, C22×Dic5, C22×C20, C2×Dic3⋊C4, C6×Dic5 [×2], C6×Dic5 [×2], C10×Dic3 [×6], C2×Dic15 [×2], C2×Dic15 [×2], C22×C30, C2×C4⋊Dic5, C6.Dic10 [×4], C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C2×C6.Dic10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, S3×D5, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C2×Dic3⋊C4, S3×Dic5 [×2], C3⋊D20 [×2], C15⋊Q8 [×2], C2×S3×D5, C2×C4⋊Dic5, C6.Dic10 [×4], C2×S3×Dic5, C2×C3⋊D20, C2×C15⋊Q8, C2×C6.Dic10

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

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

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

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

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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