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## G = C2×Dic60order 480 = 25·3·5

### Direct product of C2 and Dic60

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×Dic60
 Chief series C1 — C5 — C15 — C30 — C60 — Dic30 — C2×Dic30 — C2×Dic60
 Lower central C15 — C30 — C60 — C2×Dic60
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×Dic60
G = < a,b,c | a2=b120=1, c2=b60, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 692 in 120 conjugacy classes, 55 normal (29 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×2], C2×C10, C24 [×2], Dic6 [×6], C2×Dic3 [×2], C2×C12, C30, C30 [×2], C2×Q16, C40 [×2], Dic10 [×6], C2×Dic5 [×2], C2×C20, Dic12 [×4], C2×C24, C2×Dic6 [×2], Dic15 [×4], C60 [×2], C2×C30, Dic20 [×4], C2×C40, C2×Dic10 [×2], C2×Dic12, C120 [×2], Dic30 [×4], Dic30 [×2], C2×Dic15 [×2], C2×C60, C2×Dic20, Dic60 [×4], C2×C120, C2×Dic30 [×2], C2×Dic60
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, C2×Q16, D20 [×2], C22×D5, Dic12 [×2], C2×D12, D30 [×3], Dic20 [×2], C2×D20, C2×Dic12, D60 [×2], C22×D15, C2×Dic20, Dic60 [×2], C2×D60, C2×Dic60

Smallest permutation representation of C2×Dic60
Regular action on 480 points
Generators in S480
(1 256)(2 257)(3 258)(4 259)(5 260)(6 261)(7 262)(8 263)(9 264)(10 265)(11 266)(12 267)(13 268)(14 269)(15 270)(16 271)(17 272)(18 273)(19 274)(20 275)(21 276)(22 277)(23 278)(24 279)(25 280)(26 281)(27 282)(28 283)(29 284)(30 285)(31 286)(32 287)(33 288)(34 289)(35 290)(36 291)(37 292)(38 293)(39 294)(40 295)(41 296)(42 297)(43 298)(44 299)(45 300)(46 301)(47 302)(48 303)(49 304)(50 305)(51 306)(52 307)(53 308)(54 309)(55 310)(56 311)(57 312)(58 313)(59 314)(60 315)(61 316)(62 317)(63 318)(64 319)(65 320)(66 321)(67 322)(68 323)(69 324)(70 325)(71 326)(72 327)(73 328)(74 329)(75 330)(76 331)(77 332)(78 333)(79 334)(80 335)(81 336)(82 337)(83 338)(84 339)(85 340)(86 341)(87 342)(88 343)(89 344)(90 345)(91 346)(92 347)(93 348)(94 349)(95 350)(96 351)(97 352)(98 353)(99 354)(100 355)(101 356)(102 357)(103 358)(104 359)(105 360)(106 241)(107 242)(108 243)(109 244)(110 245)(111 246)(112 247)(113 248)(114 249)(115 250)(116 251)(117 252)(118 253)(119 254)(120 255)(121 394)(122 395)(123 396)(124 397)(125 398)(126 399)(127 400)(128 401)(129 402)(130 403)(131 404)(132 405)(133 406)(134 407)(135 408)(136 409)(137 410)(138 411)(139 412)(140 413)(141 414)(142 415)(143 416)(144 417)(145 418)(146 419)(147 420)(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)(161 434)(162 435)(163 436)(164 437)(165 438)(166 439)(167 440)(168 441)(169 442)(170 443)(171 444)(172 445)(173 446)(174 447)(175 448)(176 449)(177 450)(178 451)(179 452)(180 453)(181 454)(182 455)(183 456)(184 457)(185 458)(186 459)(187 460)(188 461)(189 462)(190 463)(191 464)(192 465)(193 466)(194 467)(195 468)(196 469)(197 470)(198 471)(199 472)(200 473)(201 474)(202 475)(203 476)(204 477)(205 478)(206 479)(207 480)(208 361)(209 362)(210 363)(211 364)(212 365)(213 366)(214 367)(215 368)(216 369)(217 370)(218 371)(219 372)(220 373)(221 374)(222 375)(223 376)(224 377)(225 378)(226 379)(227 380)(228 381)(229 382)(230 383)(231 384)(232 385)(233 386)(234 387)(235 388)(236 389)(237 390)(238 391)(239 392)(240 393)
(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)
(1 452 61 392)(2 451 62 391)(3 450 63 390)(4 449 64 389)(5 448 65 388)(6 447 66 387)(7 446 67 386)(8 445 68 385)(9 444 69 384)(10 443 70 383)(11 442 71 382)(12 441 72 381)(13 440 73 380)(14 439 74 379)(15 438 75 378)(16 437 76 377)(17 436 77 376)(18 435 78 375)(19 434 79 374)(20 433 80 373)(21 432 81 372)(22 431 82 371)(23 430 83 370)(24 429 84 369)(25 428 85 368)(26 427 86 367)(27 426 87 366)(28 425 88 365)(29 424 89 364)(30 423 90 363)(31 422 91 362)(32 421 92 361)(33 420 93 480)(34 419 94 479)(35 418 95 478)(36 417 96 477)(37 416 97 476)(38 415 98 475)(39 414 99 474)(40 413 100 473)(41 412 101 472)(42 411 102 471)(43 410 103 470)(44 409 104 469)(45 408 105 468)(46 407 106 467)(47 406 107 466)(48 405 108 465)(49 404 109 464)(50 403 110 463)(51 402 111 462)(52 401 112 461)(53 400 113 460)(54 399 114 459)(55 398 115 458)(56 397 116 457)(57 396 117 456)(58 395 118 455)(59 394 119 454)(60 393 120 453)(121 254 181 314)(122 253 182 313)(123 252 183 312)(124 251 184 311)(125 250 185 310)(126 249 186 309)(127 248 187 308)(128 247 188 307)(129 246 189 306)(130 245 190 305)(131 244 191 304)(132 243 192 303)(133 242 193 302)(134 241 194 301)(135 360 195 300)(136 359 196 299)(137 358 197 298)(138 357 198 297)(139 356 199 296)(140 355 200 295)(141 354 201 294)(142 353 202 293)(143 352 203 292)(144 351 204 291)(145 350 205 290)(146 349 206 289)(147 348 207 288)(148 347 208 287)(149 346 209 286)(150 345 210 285)(151 344 211 284)(152 343 212 283)(153 342 213 282)(154 341 214 281)(155 340 215 280)(156 339 216 279)(157 338 217 278)(158 337 218 277)(159 336 219 276)(160 335 220 275)(161 334 221 274)(162 333 222 273)(163 332 223 272)(164 331 224 271)(165 330 225 270)(166 329 226 269)(167 328 227 268)(168 327 228 267)(169 326 229 266)(170 325 230 265)(171 324 231 264)(172 323 232 263)(173 322 233 262)(174 321 234 261)(175 320 235 260)(176 319 236 259)(177 318 237 258)(178 317 238 257)(179 316 239 256)(180 315 240 255)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 2 2 2 60 60 60 60 2 2 2 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + - + + + + + + + - + + - + + - image C1 C2 C2 C2 S3 D4 D4 D5 D6 D6 Q16 D10 D10 D12 D12 D15 D20 D20 Dic12 D30 D30 Dic20 D60 D60 Dic60 kernel C2×Dic60 Dic60 C2×C120 C2×Dic30 C2×C40 C60 C2×C30 C2×C24 C40 C2×C20 C30 C24 C2×C12 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 1 1 1 2 2 1 4 4 2 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C2×Dic60 in GL5(𝔽241)

 240 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 1 0 0 0 240 240 0 0 0 0 0 42 232 0 0 0 236 47
,
 240 0 0 0 0 0 62 175 0 0 0 113 179 0 0 0 0 0 169 31 0 0 0 237 72

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,1,240,0,0,0,0,0,42,236,0,0,0,232,47],[240,0,0,0,0,0,62,113,0,0,0,175,179,0,0,0,0,0,169,237,0,0,0,31,72] >;

C2×Dic60 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{60}
% in TeX

G:=Group("C2xDic60");
// GroupNames label

G:=SmallGroup(480,870);
// by ID

G=gap.SmallGroup(480,870);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,142,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^2=b^120=1,c^2=b^60,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽