direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic60, C30⋊4Q16, C4.8D60, C6⋊1Dic20, C8.17D30, C40.68D6, C10⋊1Dic12, C20.33D12, C24.68D10, C60.164D4, C12.33D20, C22.14D60, C120.82C22, C60.247C23, Dic30.38C22, (C2×C40).6S3, C15⋊10(C2×Q16), (C2×C8).4D15, (C2×C24).6D5, C3⋊2(C2×Dic20), C5⋊2(C2×Dic12), (C2×C4).83D30, (C2×C6).21D20, C2.14(C2×D60), C6.40(C2×D20), (C2×C120).10C2, (C2×C20).392D6, C30.268(C2×D4), (C2×C10).21D12, (C2×C30).106D4, C10.41(C2×D12), (C2×C12).398D10, (C2×Dic30).5C2, C4.28(C22×D15), (C2×C60).479C22, C20.218(C22×S3), C12.220(C22×D5), SmallGroup(480,870)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C24, Dic6, C2×Dic3, C2×C12, C30, C30, C2×Q16, C40, Dic10, C2×Dic5, C2×C20, Dic12, C2×C24, C2×Dic6, Dic15, C60, C2×C30, Dic20, C2×C40, C2×Dic10, C2×Dic12, C120, Dic30, Dic30, C2×Dic15, C2×C60, C2×Dic20, Dic60, C2×C120, C2×Dic30, C2×Dic60
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, D12, C22×S3, D15, C2×Q16, D20, C22×D5, Dic12, C2×D12, D30, Dic20, C2×D20, C2×Dic12, D60, C22×D15, C2×Dic20, Dic60, C2×D60, C2×Dic60
►Regular action on 480 points
Generators in S480
(1 357)(2 358)(3 359)(4 360)(5 241)(6 242)(7 243)(8 244)(9 245)(10 246)(11 247)(12 248)(13 249)(14 250)(15 251)(16 252)(17 253)(18 254)(19 255)(20 256)(21 257)(22 258)(23 259)(24 260)(25 261)(26 262)(27 263)(28 264)(29 265)(30 266)(31 267)(32 268)(33 269)(34 270)(35 271)(36 272)(37 273)(38 274)(39 275)(40 276)(41 277)(42 278)(43 279)(44 280)(45 281)(46 282)(47 283)(48 284)(49 285)(50 286)(51 287)(52 288)(53 289)(54 290)(55 291)(56 292)(57 293)(58 294)(59 295)(60 296)(61 297)(62 298)(63 299)(64 300)(65 301)(66 302)(67 303)(68 304)(69 305)(70 306)(71 307)(72 308)(73 309)(74 310)(75 311)(76 312)(77 313)(78 314)(79 315)(80 316)(81 317)(82 318)(83 319)(84 320)(85 321)(86 322)(87 323)(88 324)(89 325)(90 326)(91 327)(92 328)(93 329)(94 330)(95 331)(96 332)(97 333)(98 334)(99 335)(100 336)(101 337)(102 338)(103 339)(104 340)(105 341)(106 342)(107 343)(108 344)(109 345)(110 346)(111 347)(112 348)(113 349)(114 350)(115 351)(116 352)(117 353)(118 354)(119 355)(120 356)(121 458)(122 459)(123 460)(124 461)(125 462)(126 463)(127 464)(128 465)(129 466)(130 467)(131 468)(132 469)(133 470)(134 471)(135 472)(136 473)(137 474)(138 475)(139 476)(140 477)(141 478)(142 479)(143 480)(144 361)(145 362)(146 363)(147 364)(148 365)(149 366)(150 367)(151 368)(152 369)(153 370)(154 371)(155 372)(156 373)(157 374)(158 375)(159 376)(160 377)(161 378)(162 379)(163 380)(164 381)(165 382)(166 383)(167 384)(168 385)(169 386)(170 387)(171 388)(172 389)(173 390)(174 391)(175 392)(176 393)(177 394)(178 395)(179 396)(180 397)(181 398)(182 399)(183 400)(184 401)(185 402)(186 403)(187 404)(188 405)(189 406)(190 407)(191 408)(192 409)(193 410)(194 411)(195 412)(196 413)(197 414)(198 415)(199 416)(200 417)(201 418)(202 419)(203 420)(204 421)(205 422)(206 423)(207 424)(208 425)(209 426)(210 427)(211 428)(212 429)(213 430)(214 431)(215 432)(216 433)(217 434)(218 435)(219 436)(220 437)(221 438)(222 439)(223 440)(224 441)(225 442)(226 443)(227 444)(228 445)(229 446)(230 447)(231 448)(232 449)(233 450)(234 451)(235 452)(236 453)(237 454)(238 455)(239 456)(240 457)
(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)
(1 422 61 362)(2 421 62 361)(3 420 63 480)(4 419 64 479)(5 418 65 478)(6 417 66 477)(7 416 67 476)(8 415 68 475)(9 414 69 474)(10 413 70 473)(11 412 71 472)(12 411 72 471)(13 410 73 470)(14 409 74 469)(15 408 75 468)(16 407 76 467)(17 406 77 466)(18 405 78 465)(19 404 79 464)(20 403 80 463)(21 402 81 462)(22 401 82 461)(23 400 83 460)(24 399 84 459)(25 398 85 458)(26 397 86 457)(27 396 87 456)(28 395 88 455)(29 394 89 454)(30 393 90 453)(31 392 91 452)(32 391 92 451)(33 390 93 450)(34 389 94 449)(35 388 95 448)(36 387 96 447)(37 386 97 446)(38 385 98 445)(39 384 99 444)(40 383 100 443)(41 382 101 442)(42 381 102 441)(43 380 103 440)(44 379 104 439)(45 378 105 438)(46 377 106 437)(47 376 107 436)(48 375 108 435)(49 374 109 434)(50 373 110 433)(51 372 111 432)(52 371 112 431)(53 370 113 430)(54 369 114 429)(55 368 115 428)(56 367 116 427)(57 366 117 426)(58 365 118 425)(59 364 119 424)(60 363 120 423)(121 261 181 321)(122 260 182 320)(123 259 183 319)(124 258 184 318)(125 257 185 317)(126 256 186 316)(127 255 187 315)(128 254 188 314)(129 253 189 313)(130 252 190 312)(131 251 191 311)(132 250 192 310)(133 249 193 309)(134 248 194 308)(135 247 195 307)(136 246 196 306)(137 245 197 305)(138 244 198 304)(139 243 199 303)(140 242 200 302)(141 241 201 301)(142 360 202 300)(143 359 203 299)(144 358 204 298)(145 357 205 297)(146 356 206 296)(147 355 207 295)(148 354 208 294)(149 353 209 293)(150 352 210 292)(151 351 211 291)(152 350 212 290)(153 349 213 289)(154 348 214 288)(155 347 215 287)(156 346 216 286)(157 345 217 285)(158 344 218 284)(159 343 219 283)(160 342 220 282)(161 341 221 281)(162 340 222 280)(163 339 223 279)(164 338 224 278)(165 337 225 277)(166 336 226 276)(167 335 227 275)(168 334 228 274)(169 333 229 273)(170 332 230 272)(171 331 231 271)(172 330 232 270)(173 329 233 269)(174 328 234 268)(175 327 235 267)(176 326 236 266)(177 325 237 265)(178 324 238 264)(179 323 239 263)(180 322 240 262)
G:=sub<Sym(480)| (1,357)(2,358)(3,359)(4,360)(5,241)(6,242)(7,243)(8,244)(9,245)(10,246)(11,247)(12,248)(13,249)(14,250)(15,251)(16,252)(17,253)(18,254)(19,255)(20,256)(21,257)(22,258)(23,259)(24,260)(25,261)(26,262)(27,263)(28,264)(29,265)(30,266)(31,267)(32,268)(33,269)(34,270)(35,271)(36,272)(37,273)(38,274)(39,275)(40,276)(41,277)(42,278)(43,279)(44,280)(45,281)(46,282)(47,283)(48,284)(49,285)(50,286)(51,287)(52,288)(53,289)(54,290)(55,291)(56,292)(57,293)(58,294)(59,295)(60,296)(61,297)(62,298)(63,299)(64,300)(65,301)(66,302)(67,303)(68,304)(69,305)(70,306)(71,307)(72,308)(73,309)(74,310)(75,311)(76,312)(77,313)(78,314)(79,315)(80,316)(81,317)(82,318)(83,319)(84,320)(85,321)(86,322)(87,323)(88,324)(89,325)(90,326)(91,327)(92,328)(93,329)(94,330)(95,331)(96,332)(97,333)(98,334)(99,335)(100,336)(101,337)(102,338)(103,339)(104,340)(105,341)(106,342)(107,343)(108,344)(109,345)(110,346)(111,347)(112,348)(113,349)(114,350)(115,351)(116,352)(117,353)(118,354)(119,355)(120,356)(121,458)(122,459)(123,460)(124,461)(125,462)(126,463)(127,464)(128,465)(129,466)(130,467)(131,468)(132,469)(133,470)(134,471)(135,472)(136,473)(137,474)(138,475)(139,476)(140,477)(141,478)(142,479)(143,480)(144,361)(145,362)(146,363)(147,364)(148,365)(149,366)(150,367)(151,368)(152,369)(153,370)(154,371)(155,372)(156,373)(157,374)(158,375)(159,376)(160,377)(161,378)(162,379)(163,380)(164,381)(165,382)(166,383)(167,384)(168,385)(169,386)(170,387)(171,388)(172,389)(173,390)(174,391)(175,392)(176,393)(177,394)(178,395)(179,396)(180,397)(181,398)(182,399)(183,400)(184,401)(185,402)(186,403)(187,404)(188,405)(189,406)(190,407)(191,408)(192,409)(193,410)(194,411)(195,412)(196,413)(197,414)(198,415)(199,416)(200,417)(201,418)(202,419)(203,420)(204,421)(205,422)(206,423)(207,424)(208,425)(209,426)(210,427)(211,428)(212,429)(213,430)(214,431)(215,432)(216,433)(217,434)(218,435)(219,436)(220,437)(221,438)(222,439)(223,440)(224,441)(225,442)(226,443)(227,444)(228,445)(229,446)(230,447)(231,448)(232,449)(233,450)(234,451)(235,452)(236,453)(237,454)(238,455)(239,456)(240,457), (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), (1,422,61,362)(2,421,62,361)(3,420,63,480)(4,419,64,479)(5,418,65,478)(6,417,66,477)(7,416,67,476)(8,415,68,475)(9,414,69,474)(10,413,70,473)(11,412,71,472)(12,411,72,471)(13,410,73,470)(14,409,74,469)(15,408,75,468)(16,407,76,467)(17,406,77,466)(18,405,78,465)(19,404,79,464)(20,403,80,463)(21,402,81,462)(22,401,82,461)(23,400,83,460)(24,399,84,459)(25,398,85,458)(26,397,86,457)(27,396,87,456)(28,395,88,455)(29,394,89,454)(30,393,90,453)(31,392,91,452)(32,391,92,451)(33,390,93,450)(34,389,94,449)(35,388,95,448)(36,387,96,447)(37,386,97,446)(38,385,98,445)(39,384,99,444)(40,383,100,443)(41,382,101,442)(42,381,102,441)(43,380,103,440)(44,379,104,439)(45,378,105,438)(46,377,106,437)(47,376,107,436)(48,375,108,435)(49,374,109,434)(50,373,110,433)(51,372,111,432)(52,371,112,431)(53,370,113,430)(54,369,114,429)(55,368,115,428)(56,367,116,427)(57,366,117,426)(58,365,118,425)(59,364,119,424)(60,363,120,423)(121,261,181,321)(122,260,182,320)(123,259,183,319)(124,258,184,318)(125,257,185,317)(126,256,186,316)(127,255,187,315)(128,254,188,314)(129,253,189,313)(130,252,190,312)(131,251,191,311)(132,250,192,310)(133,249,193,309)(134,248,194,308)(135,247,195,307)(136,246,196,306)(137,245,197,305)(138,244,198,304)(139,243,199,303)(140,242,200,302)(141,241,201,301)(142,360,202,300)(143,359,203,299)(144,358,204,298)(145,357,205,297)(146,356,206,296)(147,355,207,295)(148,354,208,294)(149,353,209,293)(150,352,210,292)(151,351,211,291)(152,350,212,290)(153,349,213,289)(154,348,214,288)(155,347,215,287)(156,346,216,286)(157,345,217,285)(158,344,218,284)(159,343,219,283)(160,342,220,282)(161,341,221,281)(162,340,222,280)(163,339,223,279)(164,338,224,278)(165,337,225,277)(166,336,226,276)(167,335,227,275)(168,334,228,274)(169,333,229,273)(170,332,230,272)(171,331,231,271)(172,330,232,270)(173,329,233,269)(174,328,234,268)(175,327,235,267)(176,326,236,266)(177,325,237,265)(178,324,238,264)(179,323,239,263)(180,322,240,262)>;
G:=Group( (1,357)(2,358)(3,359)(4,360)(5,241)(6,242)(7,243)(8,244)(9,245)(10,246)(11,247)(12,248)(13,249)(14,250)(15,251)(16,252)(17,253)(18,254)(19,255)(20,256)(21,257)(22,258)(23,259)(24,260)(25,261)(26,262)(27,263)(28,264)(29,265)(30,266)(31,267)(32,268)(33,269)(34,270)(35,271)(36,272)(37,273)(38,274)(39,275)(40,276)(41,277)(42,278)(43,279)(44,280)(45,281)(46,282)(47,283)(48,284)(49,285)(50,286)(51,287)(52,288)(53,289)(54,290)(55,291)(56,292)(57,293)(58,294)(59,295)(60,296)(61,297)(62,298)(63,299)(64,300)(65,301)(66,302)(67,303)(68,304)(69,305)(70,306)(71,307)(72,308)(73,309)(74,310)(75,311)(76,312)(77,313)(78,314)(79,315)(80,316)(81,317)(82,318)(83,319)(84,320)(85,321)(86,322)(87,323)(88,324)(89,325)(90,326)(91,327)(92,328)(93,329)(94,330)(95,331)(96,332)(97,333)(98,334)(99,335)(100,336)(101,337)(102,338)(103,339)(104,340)(105,341)(106,342)(107,343)(108,344)(109,345)(110,346)(111,347)(112,348)(113,349)(114,350)(115,351)(116,352)(117,353)(118,354)(119,355)(120,356)(121,458)(122,459)(123,460)(124,461)(125,462)(126,463)(127,464)(128,465)(129,466)(130,467)(131,468)(132,469)(133,470)(134,471)(135,472)(136,473)(137,474)(138,475)(139,476)(140,477)(141,478)(142,479)(143,480)(144,361)(145,362)(146,363)(147,364)(148,365)(149,366)(150,367)(151,368)(152,369)(153,370)(154,371)(155,372)(156,373)(157,374)(158,375)(159,376)(160,377)(161,378)(162,379)(163,380)(164,381)(165,382)(166,383)(167,384)(168,385)(169,386)(170,387)(171,388)(172,389)(173,390)(174,391)(175,392)(176,393)(177,394)(178,395)(179,396)(180,397)(181,398)(182,399)(183,400)(184,401)(185,402)(186,403)(187,404)(188,405)(189,406)(190,407)(191,408)(192,409)(193,410)(194,411)(195,412)(196,413)(197,414)(198,415)(199,416)(200,417)(201,418)(202,419)(203,420)(204,421)(205,422)(206,423)(207,424)(208,425)(209,426)(210,427)(211,428)(212,429)(213,430)(214,431)(215,432)(216,433)(217,434)(218,435)(219,436)(220,437)(221,438)(222,439)(223,440)(224,441)(225,442)(226,443)(227,444)(228,445)(229,446)(230,447)(231,448)(232,449)(233,450)(234,451)(235,452)(236,453)(237,454)(238,455)(239,456)(240,457), (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), (1,422,61,362)(2,421,62,361)(3,420,63,480)(4,419,64,479)(5,418,65,478)(6,417,66,477)(7,416,67,476)(8,415,68,475)(9,414,69,474)(10,413,70,473)(11,412,71,472)(12,411,72,471)(13,410,73,470)(14,409,74,469)(15,408,75,468)(16,407,76,467)(17,406,77,466)(18,405,78,465)(19,404,79,464)(20,403,80,463)(21,402,81,462)(22,401,82,461)(23,400,83,460)(24,399,84,459)(25,398,85,458)(26,397,86,457)(27,396,87,456)(28,395,88,455)(29,394,89,454)(30,393,90,453)(31,392,91,452)(32,391,92,451)(33,390,93,450)(34,389,94,449)(35,388,95,448)(36,387,96,447)(37,386,97,446)(38,385,98,445)(39,384,99,444)(40,383,100,443)(41,382,101,442)(42,381,102,441)(43,380,103,440)(44,379,104,439)(45,378,105,438)(46,377,106,437)(47,376,107,436)(48,375,108,435)(49,374,109,434)(50,373,110,433)(51,372,111,432)(52,371,112,431)(53,370,113,430)(54,369,114,429)(55,368,115,428)(56,367,116,427)(57,366,117,426)(58,365,118,425)(59,364,119,424)(60,363,120,423)(121,261,181,321)(122,260,182,320)(123,259,183,319)(124,258,184,318)(125,257,185,317)(126,256,186,316)(127,255,187,315)(128,254,188,314)(129,253,189,313)(130,252,190,312)(131,251,191,311)(132,250,192,310)(133,249,193,309)(134,248,194,308)(135,247,195,307)(136,246,196,306)(137,245,197,305)(138,244,198,304)(139,243,199,303)(140,242,200,302)(141,241,201,301)(142,360,202,300)(143,359,203,299)(144,358,204,298)(145,357,205,297)(146,356,206,296)(147,355,207,295)(148,354,208,294)(149,353,209,293)(150,352,210,292)(151,351,211,291)(152,350,212,290)(153,349,213,289)(154,348,214,288)(155,347,215,287)(156,346,216,286)(157,345,217,285)(158,344,218,284)(159,343,219,283)(160,342,220,282)(161,341,221,281)(162,340,222,280)(163,339,223,279)(164,338,224,278)(165,337,225,277)(166,336,226,276)(167,335,227,275)(168,334,228,274)(169,333,229,273)(170,332,230,272)(171,331,231,271)(172,330,232,270)(173,329,233,269)(174,328,234,268)(175,327,235,267)(176,326,236,266)(177,325,237,265)(178,324,238,264)(179,323,239,263)(180,322,240,262) );
G=PermutationGroup([[(1,357),(2,358),(3,359),(4,360),(5,241),(6,242),(7,243),(8,244),(9,245),(10,246),(11,247),(12,248),(13,249),(14,250),(15,251),(16,252),(17,253),(18,254),(19,255),(20,256),(21,257),(22,258),(23,259),(24,260),(25,261),(26,262),(27,263),(28,264),(29,265),(30,266),(31,267),(32,268),(33,269),(34,270),(35,271),(36,272),(37,273),(38,274),(39,275),(40,276),(41,277),(42,278),(43,279),(44,280),(45,281),(46,282),(47,283),(48,284),(49,285),(50,286),(51,287),(52,288),(53,289),(54,290),(55,291),(56,292),(57,293),(58,294),(59,295),(60,296),(61,297),(62,298),(63,299),(64,300),(65,301),(66,302),(67,303),(68,304),(69,305),(70,306),(71,307),(72,308),(73,309),(74,310),(75,311),(76,312),(77,313),(78,314),(79,315),(80,316),(81,317),(82,318),(83,319),(84,320),(85,321),(86,322),(87,323),(88,324),(89,325),(90,326),(91,327),(92,328),(93,329),(94,330),(95,331),(96,332),(97,333),(98,334),(99,335),(100,336),(101,337),(102,338),(103,339),(104,340),(105,341),(106,342),(107,343),(108,344),(109,345),(110,346),(111,347),(112,348),(113,349),(114,350),(115,351),(116,352),(117,353),(118,354),(119,355),(120,356),(121,458),(122,459),(123,460),(124,461),(125,462),(126,463),(127,464),(128,465),(129,466),(130,467),(131,468),(132,469),(133,470),(134,471),(135,472),(136,473),(137,474),(138,475),(139,476),(140,477),(141,478),(142,479),(143,480),(144,361),(145,362),(146,363),(147,364),(148,365),(149,366),(150,367),(151,368),(152,369),(153,370),(154,371),(155,372),(156,373),(157,374),(158,375),(159,376),(160,377),(161,378),(162,379),(163,380),(164,381),(165,382),(166,383),(167,384),(168,385),(169,386),(170,387),(171,388),(172,389),(173,390),(174,391),(175,392),(176,393),(177,394),(178,395),(179,396),(180,397),(181,398),(182,399),(183,400),(184,401),(185,402),(186,403),(187,404),(188,405),(189,406),(190,407),(191,408),(192,409),(193,410),(194,411),(195,412),(196,413),(197,414),(198,415),(199,416),(200,417),(201,418),(202,419),(203,420),(204,421),(205,422),(206,423),(207,424),(208,425),(209,426),(210,427),(211,428),(212,429),(213,430),(214,431),(215,432),(216,433),(217,434),(218,435),(219,436),(220,437),(221,438),(222,439),(223,440),(224,441),(225,442),(226,443),(227,444),(228,445),(229,446),(230,447),(231,448),(232,449),(233,450),(234,451),(235,452),(236,453),(237,454),(238,455),(239,456),(240,457)], [(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)], [(1,422,61,362),(2,421,62,361),(3,420,63,480),(4,419,64,479),(5,418,65,478),(6,417,66,477),(7,416,67,476),(8,415,68,475),(9,414,69,474),(10,413,70,473),(11,412,71,472),(12,411,72,471),(13,410,73,470),(14,409,74,469),(15,408,75,468),(16,407,76,467),(17,406,77,466),(18,405,78,465),(19,404,79,464),(20,403,80,463),(21,402,81,462),(22,401,82,461),(23,400,83,460),(24,399,84,459),(25,398,85,458),(26,397,86,457),(27,396,87,456),(28,395,88,455),(29,394,89,454),(30,393,90,453),(31,392,91,452),(32,391,92,451),(33,390,93,450),(34,389,94,449),(35,388,95,448),(36,387,96,447),(37,386,97,446),(38,385,98,445),(39,384,99,444),(40,383,100,443),(41,382,101,442),(42,381,102,441),(43,380,103,440),(44,379,104,439),(45,378,105,438),(46,377,106,437),(47,376,107,436),(48,375,108,435),(49,374,109,434),(50,373,110,433),(51,372,111,432),(52,371,112,431),(53,370,113,430),(54,369,114,429),(55,368,115,428),(56,367,116,427),(57,366,117,426),(58,365,118,425),(59,364,119,424),(60,363,120,423),(121,261,181,321),(122,260,182,320),(123,259,183,319),(124,258,184,318),(125,257,185,317),(126,256,186,316),(127,255,187,315),(128,254,188,314),(129,253,189,313),(130,252,190,312),(131,251,191,311),(132,250,192,310),(133,249,193,309),(134,248,194,308),(135,247,195,307),(136,246,196,306),(137,245,197,305),(138,244,198,304),(139,243,199,303),(140,242,200,302),(141,241,201,301),(142,360,202,300),(143,359,203,299),(144,358,204,298),(145,357,205,297),(146,356,206,296),(147,355,207,295),(148,354,208,294),(149,353,209,293),(150,352,210,292),(151,351,211,291),(152,350,212,290),(153,349,213,289),(154,348,214,288),(155,347,215,287),(156,346,216,286),(157,345,217,285),(158,344,218,284),(159,343,219,283),(160,342,220,282),(161,341,221,281),(162,340,222,280),(163,339,223,279),(164,338,224,278),(165,337,225,277),(166,336,226,276),(167,335,227,275),(168,334,228,274),(169,333,229,273),(170,332,230,272),(171,331,231,271),(172,330,232,270),(173,329,233,269),(174,328,234,268),(175,327,235,267),(176,326,236,266),(177,325,237,265),(178,324,238,264),(179,323,239,263),(180,322,240,262)]])
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