direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C60⋊5C4, C23.36D30, C22.15D60, C22.5Dic30, C30⋊8(C4⋊C4), (C2×C60)⋊19C4, C60⋊39(C2×C4), C2.2(C2×D60), C6⋊2(C4⋊Dic5), C4⋊2(C2×Dic15), (C2×C4)⋊3Dic15, (C2×C12)⋊5Dic5, C12⋊7(C2×Dic5), (C2×C6).22D20, (C2×C4).85D30, C6.43(C2×D20), (C2×C30).15Q8, C30.72(C2×Q8), C10⋊3(C4⋊Dic3), C20⋊10(C2×Dic3), (C2×C20)⋊10Dic3, (C2×C30).107D4, C10.44(C2×D12), C30.271(C2×D4), (C2×C10).22D12, (C2×C20).394D6, C2.3(C2×Dic30), (C22×C4).8D15, (C22×C12).7D5, (C2×C12).399D10, (C22×C60).10C2, (C22×C20).11S3, (C2×C10).14Dic6, (C2×C6).14Dic10, C10.40(C2×Dic6), C6.40(C2×Dic10), (C2×C60).480C22, (C2×C30).299C23, C30.213(C22×C4), (C22×C10).134D6, (C22×C6).116D10, C6.24(C22×Dic5), C2.4(C22×Dic15), (C22×Dic15).5C2, C22.14(C2×Dic15), C10.37(C22×Dic3), C22.21(C22×D15), (C22×C30).139C22, (C2×Dic15).168C22, C15⋊18(C2×C4⋊C4), C5⋊4(C2×C4⋊Dic3), C3⋊3(C2×C4⋊Dic5), (C2×C30).179(C2×C4), (C2×C6).36(C2×Dic5), (C2×C10).56(C2×Dic3), (C2×C6).295(C22×D5), (C2×C10).294(C22×S3), SmallGroup(480,890)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C60⋊5C4
G = < a,b,c | a2=b60=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 756 in 184 conjugacy classes, 119 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×Dic5, C2×C20, C22×C10, C4⋊Dic3, C22×Dic3, C22×C12, Dic15, C60, C2×C30, C2×C30, C4⋊Dic5, C22×Dic5, C22×C20, C2×C4⋊Dic3, C2×Dic15, C2×Dic15, C2×C60, C22×C30, C2×C4⋊Dic5, C60⋊5C4, C22×Dic15, C22×C60, C2×C60⋊5C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, Dic6, D12, C2×Dic3, C22×S3, D15, C2×C4⋊C4, Dic10, D20, C2×Dic5, C22×D5, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, Dic15, D30, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic3, Dic30, D60, C2×Dic15, C22×D15, C2×C4⋊Dic5, C60⋊5C4, C2×Dic30, C2×D60, C22×Dic15, C2×C60⋊5C4
►Regular action on 480 points
Generators in S480
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(121 284)(122 285)(123 286)(124 287)(125 288)(126 289)(127 290)(128 291)(129 292)(130 293)(131 294)(132 295)(133 296)(134 297)(135 298)(136 299)(137 300)(138 241)(139 242)(140 243)(141 244)(142 245)(143 246)(144 247)(145 248)(146 249)(147 250)(148 251)(149 252)(150 253)(151 254)(152 255)(153 256)(154 257)(155 258)(156 259)(157 260)(158 261)(159 262)(160 263)(161 264)(162 265)(163 266)(164 267)(165 268)(166 269)(167 270)(168 271)(169 272)(170 273)(171 274)(172 275)(173 276)(174 277)(175 278)(176 279)(177 280)(178 281)(179 282)(180 283)(181 460)(182 461)(183 462)(184 463)(185 464)(186 465)(187 466)(188 467)(189 468)(190 469)(191 470)(192 471)(193 472)(194 473)(195 474)(196 475)(197 476)(198 477)(199 478)(200 479)(201 480)(202 421)(203 422)(204 423)(205 424)(206 425)(207 426)(208 427)(209 428)(210 429)(211 430)(212 431)(213 432)(214 433)(215 434)(216 435)(217 436)(218 437)(219 438)(220 439)(221 440)(222 441)(223 442)(224 443)(225 444)(226 445)(227 446)(228 447)(229 448)(230 449)(231 450)(232 451)(233 452)(234 453)(235 454)(236 455)(237 456)(238 457)(239 458)(240 459)(301 404)(302 405)(303 406)(304 407)(305 408)(306 409)(307 410)(308 411)(309 412)(310 413)(311 414)(312 415)(313 416)(314 417)(315 418)(316 419)(317 420)(318 361)(319 362)(320 363)(321 364)(322 365)(323 366)(324 367)(325 368)(326 369)(327 370)(328 371)(329 372)(330 373)(331 374)(332 375)(333 376)(334 377)(335 378)(336 379)(337 380)(338 381)(339 382)(340 383)(341 384)(342 385)(343 386)(344 387)(345 388)(346 389)(347 390)(348 391)(349 392)(350 393)(351 394)(352 395)(353 396)(354 397)(355 398)(356 399)(357 400)(358 401)(359 402)(360 403)
(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)
(1 273 471 389)(2 272 472 388)(3 271 473 387)(4 270 474 386)(5 269 475 385)(6 268 476 384)(7 267 477 383)(8 266 478 382)(9 265 479 381)(10 264 480 380)(11 263 421 379)(12 262 422 378)(13 261 423 377)(14 260 424 376)(15 259 425 375)(16 258 426 374)(17 257 427 373)(18 256 428 372)(19 255 429 371)(20 254 430 370)(21 253 431 369)(22 252 432 368)(23 251 433 367)(24 250 434 366)(25 249 435 365)(26 248 436 364)(27 247 437 363)(28 246 438 362)(29 245 439 361)(30 244 440 420)(31 243 441 419)(32 242 442 418)(33 241 443 417)(34 300 444 416)(35 299 445 415)(36 298 446 414)(37 297 447 413)(38 296 448 412)(39 295 449 411)(40 294 450 410)(41 293 451 409)(42 292 452 408)(43 291 453 407)(44 290 454 406)(45 289 455 405)(46 288 456 404)(47 287 457 403)(48 286 458 402)(49 285 459 401)(50 284 460 400)(51 283 461 399)(52 282 462 398)(53 281 463 397)(54 280 464 396)(55 279 465 395)(56 278 466 394)(57 277 467 393)(58 276 468 392)(59 275 469 391)(60 274 470 390)(61 164 198 340)(62 163 199 339)(63 162 200 338)(64 161 201 337)(65 160 202 336)(66 159 203 335)(67 158 204 334)(68 157 205 333)(69 156 206 332)(70 155 207 331)(71 154 208 330)(72 153 209 329)(73 152 210 328)(74 151 211 327)(75 150 212 326)(76 149 213 325)(77 148 214 324)(78 147 215 323)(79 146 216 322)(80 145 217 321)(81 144 218 320)(82 143 219 319)(83 142 220 318)(84 141 221 317)(85 140 222 316)(86 139 223 315)(87 138 224 314)(88 137 225 313)(89 136 226 312)(90 135 227 311)(91 134 228 310)(92 133 229 309)(93 132 230 308)(94 131 231 307)(95 130 232 306)(96 129 233 305)(97 128 234 304)(98 127 235 303)(99 126 236 302)(100 125 237 301)(101 124 238 360)(102 123 239 359)(103 122 240 358)(104 121 181 357)(105 180 182 356)(106 179 183 355)(107 178 184 354)(108 177 185 353)(109 176 186 352)(110 175 187 351)(111 174 188 350)(112 173 189 349)(113 172 190 348)(114 171 191 347)(115 170 192 346)(116 169 193 345)(117 168 194 344)(118 167 195 343)(119 166 196 342)(120 165 197 341)
G:=sub<Sym(480)| (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(121,284)(122,285)(123,286)(124,287)(125,288)(126,289)(127,290)(128,291)(129,292)(130,293)(131,294)(132,295)(133,296)(134,297)(135,298)(136,299)(137,300)(138,241)(139,242)(140,243)(141,244)(142,245)(143,246)(144,247)(145,248)(146,249)(147,250)(148,251)(149,252)(150,253)(151,254)(152,255)(153,256)(154,257)(155,258)(156,259)(157,260)(158,261)(159,262)(160,263)(161,264)(162,265)(163,266)(164,267)(165,268)(166,269)(167,270)(168,271)(169,272)(170,273)(171,274)(172,275)(173,276)(174,277)(175,278)(176,279)(177,280)(178,281)(179,282)(180,283)(181,460)(182,461)(183,462)(184,463)(185,464)(186,465)(187,466)(188,467)(189,468)(190,469)(191,470)(192,471)(193,472)(194,473)(195,474)(196,475)(197,476)(198,477)(199,478)(200,479)(201,480)(202,421)(203,422)(204,423)(205,424)(206,425)(207,426)(208,427)(209,428)(210,429)(211,430)(212,431)(213,432)(214,433)(215,434)(216,435)(217,436)(218,437)(219,438)(220,439)(221,440)(222,441)(223,442)(224,443)(225,444)(226,445)(227,446)(228,447)(229,448)(230,449)(231,450)(232,451)(233,452)(234,453)(235,454)(236,455)(237,456)(238,457)(239,458)(240,459)(301,404)(302,405)(303,406)(304,407)(305,408)(306,409)(307,410)(308,411)(309,412)(310,413)(311,414)(312,415)(313,416)(314,417)(315,418)(316,419)(317,420)(318,361)(319,362)(320,363)(321,364)(322,365)(323,366)(324,367)(325,368)(326,369)(327,370)(328,371)(329,372)(330,373)(331,374)(332,375)(333,376)(334,377)(335,378)(336,379)(337,380)(338,381)(339,382)(340,383)(341,384)(342,385)(343,386)(344,387)(345,388)(346,389)(347,390)(348,391)(349,392)(350,393)(351,394)(352,395)(353,396)(354,397)(355,398)(356,399)(357,400)(358,401)(359,402)(360,403), (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), (1,273,471,389)(2,272,472,388)(3,271,473,387)(4,270,474,386)(5,269,475,385)(6,268,476,384)(7,267,477,383)(8,266,478,382)(9,265,479,381)(10,264,480,380)(11,263,421,379)(12,262,422,378)(13,261,423,377)(14,260,424,376)(15,259,425,375)(16,258,426,374)(17,257,427,373)(18,256,428,372)(19,255,429,371)(20,254,430,370)(21,253,431,369)(22,252,432,368)(23,251,433,367)(24,250,434,366)(25,249,435,365)(26,248,436,364)(27,247,437,363)(28,246,438,362)(29,245,439,361)(30,244,440,420)(31,243,441,419)(32,242,442,418)(33,241,443,417)(34,300,444,416)(35,299,445,415)(36,298,446,414)(37,297,447,413)(38,296,448,412)(39,295,449,411)(40,294,450,410)(41,293,451,409)(42,292,452,408)(43,291,453,407)(44,290,454,406)(45,289,455,405)(46,288,456,404)(47,287,457,403)(48,286,458,402)(49,285,459,401)(50,284,460,400)(51,283,461,399)(52,282,462,398)(53,281,463,397)(54,280,464,396)(55,279,465,395)(56,278,466,394)(57,277,467,393)(58,276,468,392)(59,275,469,391)(60,274,470,390)(61,164,198,340)(62,163,199,339)(63,162,200,338)(64,161,201,337)(65,160,202,336)(66,159,203,335)(67,158,204,334)(68,157,205,333)(69,156,206,332)(70,155,207,331)(71,154,208,330)(72,153,209,329)(73,152,210,328)(74,151,211,327)(75,150,212,326)(76,149,213,325)(77,148,214,324)(78,147,215,323)(79,146,216,322)(80,145,217,321)(81,144,218,320)(82,143,219,319)(83,142,220,318)(84,141,221,317)(85,140,222,316)(86,139,223,315)(87,138,224,314)(88,137,225,313)(89,136,226,312)(90,135,227,311)(91,134,228,310)(92,133,229,309)(93,132,230,308)(94,131,231,307)(95,130,232,306)(96,129,233,305)(97,128,234,304)(98,127,235,303)(99,126,236,302)(100,125,237,301)(101,124,238,360)(102,123,239,359)(103,122,240,358)(104,121,181,357)(105,180,182,356)(106,179,183,355)(107,178,184,354)(108,177,185,353)(109,176,186,352)(110,175,187,351)(111,174,188,350)(112,173,189,349)(113,172,190,348)(114,171,191,347)(115,170,192,346)(116,169,193,345)(117,168,194,344)(118,167,195,343)(119,166,196,342)(120,165,197,341)>;
G:=Group( (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(121,284)(122,285)(123,286)(124,287)(125,288)(126,289)(127,290)(128,291)(129,292)(130,293)(131,294)(132,295)(133,296)(134,297)(135,298)(136,299)(137,300)(138,241)(139,242)(140,243)(141,244)(142,245)(143,246)(144,247)(145,248)(146,249)(147,250)(148,251)(149,252)(150,253)(151,254)(152,255)(153,256)(154,257)(155,258)(156,259)(157,260)(158,261)(159,262)(160,263)(161,264)(162,265)(163,266)(164,267)(165,268)(166,269)(167,270)(168,271)(169,272)(170,273)(171,274)(172,275)(173,276)(174,277)(175,278)(176,279)(177,280)(178,281)(179,282)(180,283)(181,460)(182,461)(183,462)(184,463)(185,464)(186,465)(187,466)(188,467)(189,468)(190,469)(191,470)(192,471)(193,472)(194,473)(195,474)(196,475)(197,476)(198,477)(199,478)(200,479)(201,480)(202,421)(203,422)(204,423)(205,424)(206,425)(207,426)(208,427)(209,428)(210,429)(211,430)(212,431)(213,432)(214,433)(215,434)(216,435)(217,436)(218,437)(219,438)(220,439)(221,440)(222,441)(223,442)(224,443)(225,444)(226,445)(227,446)(228,447)(229,448)(230,449)(231,450)(232,451)(233,452)(234,453)(235,454)(236,455)(237,456)(238,457)(239,458)(240,459)(301,404)(302,405)(303,406)(304,407)(305,408)(306,409)(307,410)(308,411)(309,412)(310,413)(311,414)(312,415)(313,416)(314,417)(315,418)(316,419)(317,420)(318,361)(319,362)(320,363)(321,364)(322,365)(323,366)(324,367)(325,368)(326,369)(327,370)(328,371)(329,372)(330,373)(331,374)(332,375)(333,376)(334,377)(335,378)(336,379)(337,380)(338,381)(339,382)(340,383)(341,384)(342,385)(343,386)(344,387)(345,388)(346,389)(347,390)(348,391)(349,392)(350,393)(351,394)(352,395)(353,396)(354,397)(355,398)(356,399)(357,400)(358,401)(359,402)(360,403), (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), (1,273,471,389)(2,272,472,388)(3,271,473,387)(4,270,474,386)(5,269,475,385)(6,268,476,384)(7,267,477,383)(8,266,478,382)(9,265,479,381)(10,264,480,380)(11,263,421,379)(12,262,422,378)(13,261,423,377)(14,260,424,376)(15,259,425,375)(16,258,426,374)(17,257,427,373)(18,256,428,372)(19,255,429,371)(20,254,430,370)(21,253,431,369)(22,252,432,368)(23,251,433,367)(24,250,434,366)(25,249,435,365)(26,248,436,364)(27,247,437,363)(28,246,438,362)(29,245,439,361)(30,244,440,420)(31,243,441,419)(32,242,442,418)(33,241,443,417)(34,300,444,416)(35,299,445,415)(36,298,446,414)(37,297,447,413)(38,296,448,412)(39,295,449,411)(40,294,450,410)(41,293,451,409)(42,292,452,408)(43,291,453,407)(44,290,454,406)(45,289,455,405)(46,288,456,404)(47,287,457,403)(48,286,458,402)(49,285,459,401)(50,284,460,400)(51,283,461,399)(52,282,462,398)(53,281,463,397)(54,280,464,396)(55,279,465,395)(56,278,466,394)(57,277,467,393)(58,276,468,392)(59,275,469,391)(60,274,470,390)(61,164,198,340)(62,163,199,339)(63,162,200,338)(64,161,201,337)(65,160,202,336)(66,159,203,335)(67,158,204,334)(68,157,205,333)(69,156,206,332)(70,155,207,331)(71,154,208,330)(72,153,209,329)(73,152,210,328)(74,151,211,327)(75,150,212,326)(76,149,213,325)(77,148,214,324)(78,147,215,323)(79,146,216,322)(80,145,217,321)(81,144,218,320)(82,143,219,319)(83,142,220,318)(84,141,221,317)(85,140,222,316)(86,139,223,315)(87,138,224,314)(88,137,225,313)(89,136,226,312)(90,135,227,311)(91,134,228,310)(92,133,229,309)(93,132,230,308)(94,131,231,307)(95,130,232,306)(96,129,233,305)(97,128,234,304)(98,127,235,303)(99,126,236,302)(100,125,237,301)(101,124,238,360)(102,123,239,359)(103,122,240,358)(104,121,181,357)(105,180,182,356)(106,179,183,355)(107,178,184,354)(108,177,185,353)(109,176,186,352)(110,175,187,351)(111,174,188,350)(112,173,189,349)(113,172,190,348)(114,171,191,347)(115,170,192,346)(116,169,193,345)(117,168,194,344)(118,167,195,343)(119,166,196,342)(120,165,197,341) );
G=PermutationGroup([[(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(121,284),(122,285),(123,286),(124,287),(125,288),(126,289),(127,290),(128,291),(129,292),(130,293),(131,294),(132,295),(133,296),(134,297),(135,298),(136,299),(137,300),(138,241),(139,242),(140,243),(141,244),(142,245),(143,246),(144,247),(145,248),(146,249),(147,250),(148,251),(149,252),(150,253),(151,254),(152,255),(153,256),(154,257),(155,258),(156,259),(157,260),(158,261),(159,262),(160,263),(161,264),(162,265),(163,266),(164,267),(165,268),(166,269),(167,270),(168,271),(169,272),(170,273),(171,274),(172,275),(173,276),(174,277),(175,278),(176,279),(177,280),(178,281),(179,282),(180,283),(181,460),(182,461),(183,462),(184,463),(185,464),(186,465),(187,466),(188,467),(189,468),(190,469),(191,470),(192,471),(193,472),(194,473),(195,474),(196,475),(197,476),(198,477),(199,478),(200,479),(201,480),(202,421),(203,422),(204,423),(205,424),(206,425),(207,426),(208,427),(209,428),(210,429),(211,430),(212,431),(213,432),(214,433),(215,434),(216,435),(217,436),(218,437),(219,438),(220,439),(221,440),(222,441),(223,442),(224,443),(225,444),(226,445),(227,446),(228,447),(229,448),(230,449),(231,450),(232,451),(233,452),(234,453),(235,454),(236,455),(237,456),(238,457),(239,458),(240,459),(301,404),(302,405),(303,406),(304,407),(305,408),(306,409),(307,410),(308,411),(309,412),(310,413),(311,414),(312,415),(313,416),(314,417),(315,418),(316,419),(317,420),(318,361),(319,362),(320,363),(321,364),(322,365),(323,366),(324,367),(325,368),(326,369),(327,370),(328,371),(329,372),(330,373),(331,374),(332,375),(333,376),(334,377),(335,378),(336,379),(337,380),(338,381),(339,382),(340,383),(341,384),(342,385),(343,386),(344,387),(345,388),(346,389),(347,390),(348,391),(349,392),(350,393),(351,394),(352,395),(353,396),(354,397),(355,398),(356,399),(357,400),(358,401),(359,402),(360,403)], [(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)], [(1,273,471,389),(2,272,472,388),(3,271,473,387),(4,270,474,386),(5,269,475,385),(6,268,476,384),(7,267,477,383),(8,266,478,382),(9,265,479,381),(10,264,480,380),(11,263,421,379),(12,262,422,378),(13,261,423,377),(14,260,424,376),(15,259,425,375),(16,258,426,374),(17,257,427,373),(18,256,428,372),(19,255,429,371),(20,254,430,370),(21,253,431,369),(22,252,432,368),(23,251,433,367),(24,250,434,366),(25,249,435,365),(26,248,436,364),(27,247,437,363),(28,246,438,362),(29,245,439,361),(30,244,440,420),(31,243,441,419),(32,242,442,418),(33,241,443,417),(34,300,444,416),(35,299,445,415),(36,298,446,414),(37,297,447,413),(38,296,448,412),(39,295,449,411),(40,294,450,410),(41,293,451,409),(42,292,452,408),(43,291,453,407),(44,290,454,406),(45,289,455,405),(46,288,456,404),(47,287,457,403),(48,286,458,402),(49,285,459,401),(50,284,460,400),(51,283,461,399),(52,282,462,398),(53,281,463,397),(54,280,464,396),(55,279,465,395),(56,278,466,394),(57,277,467,393),(58,276,468,392),(59,275,469,391),(60,274,470,390),(61,164,198,340),(62,163,199,339),(63,162,200,338),(64,161,201,337),(65,160,202,336),(66,159,203,335),(67,158,204,334),(68,157,205,333),(69,156,206,332),(70,155,207,331),(71,154,208,330),(72,153,209,329),(73,152,210,328),(74,151,211,327),(75,150,212,326),(76,149,213,325),(77,148,214,324),(78,147,215,323),(79,146,216,322),(80,145,217,321),(81,144,218,320),(82,143,219,319),(83,142,220,318),(84,141,221,317),(85,140,222,316),(86,139,223,315),(87,138,224,314),(88,137,225,313),(89,136,226,312),(90,135,227,311),(91,134,228,310),(92,133,229,309),(93,132,230,308),(94,131,231,307),(95,130,232,306),(96,129,233,305),(97,128,234,304),(98,127,235,303),(99,126,236,302),(100,125,237,301),(101,124,238,360),(102,123,239,359),(103,122,240,358),(104,121,181,357),(105,180,182,356),(106,179,183,355),(107,178,184,354),(108,177,185,353),(109,176,186,352),(110,175,187,351),(111,174,188,350),(112,173,189,349),(113,172,190,348),(114,171,191,347),(115,170,192,346),(116,169,193,345),(117,168,194,344),(118,167,195,343),(119,166,196,342),(120,165,197,341)]])
132 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C4 | S3 | D4 | Q8 | D5 | Dic3 | D6 | D6 | Dic5 | D10 | D10 | Dic6 | D12 | D15 | Dic10 | D20 | Dic15 | D30 | D30 | Dic30 | D60 |
| kernel | C2×C60⋊5C4 | C60⋊5C4 | C22×Dic15 | C22×C60 | C2×C60 | C22×C20 | C2×C30 | C2×C30 | C22×C12 | C2×C20 | C2×C20 | C22×C10 | C2×C12 | C2×C12 | C22×C6 | C2×C10 | C2×C10 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 8 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 8 | 4 | 16 | 16 |
Matrix representation of C2×C60⋊5C4 ►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 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 30 | 44 | 0 | 0 | 0 | 0 |
| 32 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 48 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 6 |
| 0 | 0 | 0 | 0 | 21 | 13 |
| 53 | 12 | 0 | 0 | 0 | 0 |
| 10 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 34 | 0 | 0 |
| 0 | 0 | 13 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 8 |
| 0 | 0 | 0 | 0 | 28 | 18 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[30,32,0,0,0,0,44,49,0,0,0,0,0,0,16,48,0,0,0,0,0,42,0,0,0,0,0,0,52,21,0,0,0,0,6,13],[53,10,0,0,0,0,12,8,0,0,0,0,0,0,44,13,0,0,0,0,34,17,0,0,0,0,0,0,43,28,0,0,0,0,8,18] >;
C2×C60⋊5C4 in GAP, Magma, Sage, TeX
C_2\times C_{60}\rtimes_5C_4 % in TeX
G:=Group("C2xC60:5C4"); // GroupNames label
G:=SmallGroup(480,890);
// by ID
G=gap.SmallGroup(480,890);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^60=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations