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G = C2×C605C4order 480 = 25·3·5

Direct product of C2 and C605C4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C605C4, C23.36D30, C22.15D60, C22.5Dic30, C308(C4⋊C4), (C2×C60)⋊19C4, C6039(C2×C4), C2.2(C2×D60), C62(C4⋊Dic5), C42(C2×Dic15), (C2×C4)⋊3Dic15, (C2×C12)⋊5Dic5, C127(C2×Dic5), (C2×C6).22D20, (C2×C4).85D30, C6.43(C2×D20), (C2×C30).15Q8, C30.72(C2×Q8), C103(C4⋊Dic3), C2010(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, C1518(C2×C4⋊C4), C54(C2×C4⋊Dic3), C33(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

C1C30 — C2×C605C4
C1C5C15C30C2×C30C2×Dic15C22×Dic15 — C2×C605C4
C15C30 — C2×C605C4
C1C23C22×C4

Generators and relations for C2×C605C4
 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 [×3], C2 [×4], C3, C4 [×4], C4 [×4], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×8], C23, C10 [×3], C10 [×4], Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×Dic3 [×8], C2×C12 [×6], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×Dic5 [×8], C2×C20 [×6], C22×C10, C4⋊Dic3 [×4], C22×Dic3 [×2], C22×C12, Dic15 [×4], C60 [×4], C2×C30, C2×C30 [×6], C4⋊Dic5 [×4], C22×Dic5 [×2], C22×C20, C2×C4⋊Dic3, C2×Dic15 [×4], C2×Dic15 [×4], C2×C60 [×6], C22×C30, C2×C4⋊Dic5, C605C4 [×4], C22×Dic15 [×2], C22×C60, C2×C605C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, Dic3 [×4], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], Dic6 [×2], D12 [×2], C2×Dic3 [×6], C22×S3, D15, C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C4⋊Dic3 [×4], C2×Dic6, C2×D12, C22×Dic3, Dic15 [×4], D30 [×3], C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic3, Dic30 [×2], D60 [×2], C2×Dic15 [×6], C22×D15, C2×C4⋊Dic5, C605C4 [×4], C2×Dic30, C2×D60, C22×Dic15, C2×C605C4

Smallest permutation representation of C2×C605C4
Regular action on 480 points
Generators in S480
(1 443)(2 444)(3 445)(4 446)(5 447)(6 448)(7 449)(8 450)(9 451)(10 452)(11 453)(12 454)(13 455)(14 456)(15 457)(16 458)(17 459)(18 460)(19 461)(20 462)(21 463)(22 464)(23 465)(24 466)(25 467)(26 468)(27 469)(28 470)(29 471)(30 472)(31 473)(32 474)(33 475)(34 476)(35 477)(36 478)(37 479)(38 480)(39 421)(40 422)(41 423)(42 424)(43 425)(44 426)(45 427)(46 428)(47 429)(48 430)(49 431)(50 432)(51 433)(52 434)(53 435)(54 436)(55 437)(56 438)(57 439)(58 440)(59 441)(60 442)(61 287)(62 288)(63 289)(64 290)(65 291)(66 292)(67 293)(68 294)(69 295)(70 296)(71 297)(72 298)(73 299)(74 300)(75 241)(76 242)(77 243)(78 244)(79 245)(80 246)(81 247)(82 248)(83 249)(84 250)(85 251)(86 252)(87 253)(88 254)(89 255)(90 256)(91 257)(92 258)(93 259)(94 260)(95 261)(96 262)(97 263)(98 264)(99 265)(100 266)(101 267)(102 268)(103 269)(104 270)(105 271)(106 272)(107 273)(108 274)(109 275)(110 276)(111 277)(112 278)(113 279)(114 280)(115 281)(116 282)(117 283)(118 284)(119 285)(120 286)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 217)(137 218)(138 219)(139 220)(140 221)(141 222)(142 223)(143 224)(144 225)(145 226)(146 227)(147 228)(148 229)(149 230)(150 231)(151 232)(152 233)(153 234)(154 235)(155 236)(156 237)(157 238)(158 239)(159 240)(160 181)(161 182)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(169 190)(170 191)(171 192)(172 193)(173 194)(174 195)(175 196)(176 197)(177 198)(178 199)(179 200)(180 201)(301 411)(302 412)(303 413)(304 414)(305 415)(306 416)(307 417)(308 418)(309 419)(310 420)(311 361)(312 362)(313 363)(314 364)(315 365)(316 366)(317 367)(318 368)(319 369)(320 370)(321 371)(322 372)(323 373)(324 374)(325 375)(326 376)(327 377)(328 378)(329 379)(330 380)(331 381)(332 382)(333 383)(334 384)(335 385)(336 386)(337 387)(338 388)(339 389)(340 390)(341 391)(342 392)(343 393)(344 394)(345 395)(346 396)(347 397)(348 398)(349 399)(350 400)(351 401)(352 402)(353 403)(354 404)(355 405)(356 406)(357 407)(358 408)(359 409)(360 410)
(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)
(1 364 278 130)(2 363 279 129)(3 362 280 128)(4 361 281 127)(5 420 282 126)(6 419 283 125)(7 418 284 124)(8 417 285 123)(9 416 286 122)(10 415 287 121)(11 414 288 180)(12 413 289 179)(13 412 290 178)(14 411 291 177)(15 410 292 176)(16 409 293 175)(17 408 294 174)(18 407 295 173)(19 406 296 172)(20 405 297 171)(21 404 298 170)(22 403 299 169)(23 402 300 168)(24 401 241 167)(25 400 242 166)(26 399 243 165)(27 398 244 164)(28 397 245 163)(29 396 246 162)(30 395 247 161)(31 394 248 160)(32 393 249 159)(33 392 250 158)(34 391 251 157)(35 390 252 156)(36 389 253 155)(37 388 254 154)(38 387 255 153)(39 386 256 152)(40 385 257 151)(41 384 258 150)(42 383 259 149)(43 382 260 148)(44 381 261 147)(45 380 262 146)(46 379 263 145)(47 378 264 144)(48 377 265 143)(49 376 266 142)(50 375 267 141)(51 374 268 140)(52 373 269 139)(53 372 270 138)(54 371 271 137)(55 370 272 136)(56 369 273 135)(57 368 274 134)(58 367 275 133)(59 366 276 132)(60 365 277 131)(61 202 452 305)(62 201 453 304)(63 200 454 303)(64 199 455 302)(65 198 456 301)(66 197 457 360)(67 196 458 359)(68 195 459 358)(69 194 460 357)(70 193 461 356)(71 192 462 355)(72 191 463 354)(73 190 464 353)(74 189 465 352)(75 188 466 351)(76 187 467 350)(77 186 468 349)(78 185 469 348)(79 184 470 347)(80 183 471 346)(81 182 472 345)(82 181 473 344)(83 240 474 343)(84 239 475 342)(85 238 476 341)(86 237 477 340)(87 236 478 339)(88 235 479 338)(89 234 480 337)(90 233 421 336)(91 232 422 335)(92 231 423 334)(93 230 424 333)(94 229 425 332)(95 228 426 331)(96 227 427 330)(97 226 428 329)(98 225 429 328)(99 224 430 327)(100 223 431 326)(101 222 432 325)(102 221 433 324)(103 220 434 323)(104 219 435 322)(105 218 436 321)(106 217 437 320)(107 216 438 319)(108 215 439 318)(109 214 440 317)(110 213 441 316)(111 212 442 315)(112 211 443 314)(113 210 444 313)(114 209 445 312)(115 208 446 311)(116 207 447 310)(117 206 448 309)(118 205 449 308)(119 204 450 307)(120 203 451 306)

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

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

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

132 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12···2344444···4556···610···1012···121515151520···2030···3060···60
size11···12222230···30222···22···22···222222···22···22···2

132 irreducible representations

dim1111122222222222222222222
type++++++-+-++-++-++-+-++-+
imageC1C2C2C2C4S3D4Q8D5Dic3D6D6Dic5D10D10Dic6D12D15Dic10D20Dic15D30D30Dic30D60
kernelC2×C605C4C605C4C22×Dic15C22×C60C2×C60C22×C20C2×C30C2×C30C22×C12C2×C20C2×C20C22×C10C2×C12C2×C12C22×C6C2×C10C2×C10C22×C4C2×C6C2×C6C2×C4C2×C4C23C22C22
# reps1421812224218424448816841616

Matrix representation of C2×C605C4 in GL6(𝔽61)

6000000
0600000
0060000
0006000
0000600
0000060
,
30440000
32490000
0016000
00484200
0000526
00002113
,
53120000
1080000
00443400
00131700
0000438
00002818

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×C605C4 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

׿
×
𝔽