direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C6.C42, C30.39C42, (C2×C12)⋊3C20, (C2×C60)⋊16C4, C6.5(C4×C20), (C2×C20)⋊8Dic3, C30.61(C4⋊C4), (C2×C30).21Q8, (C2×Dic3)⋊2C20, (C2×C10).52D12, (C2×C30).122D4, (C22×C20).6S3, (C22×C60).2C2, C2.5(Dic3×C20), C10.58(D6⋊C4), (C10×Dic3)⋊12C4, C23.32(S3×C10), (C22×C12).1C10, C22.12(S3×C20), C10.29(C4×Dic3), (C2×C10).15Dic6, C22.11(C5×D12), C22.3(C5×Dic6), C10.22(C4⋊Dic3), C15⋊7(C2.C42), (C22×C10).147D6, C30.100(C22⋊C4), C10.26(Dic3⋊C4), (C22×Dic3).1C10, C22.10(C10×Dic3), C10.32(C6.D4), (C22×C30).169C22, C6.9(C5×C4⋊C4), C2.2(C5×D6⋊C4), C3⋊(C5×C2.C42), (C2×C6).4(C5×Q8), (C2×C4)⋊2(C5×Dic3), (C2×C6).32(C5×D4), C2.2(C5×C4⋊Dic3), (C2×C10).84(C4×S3), (C2×C6).13(C2×C20), C6.12(C5×C22⋊C4), C2.2(C5×Dic3⋊C4), (C22×C4).4(C5×S3), (Dic3×C2×C10).9C2, (C2×C30).158(C2×C4), C2.2(C5×C6.D4), C22.16(C5×C3⋊D4), (C2×C10).88(C3⋊D4), (C22×C6).31(C2×C10), (C2×C10).62(C2×Dic3), SmallGroup(480,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C6.C42
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >
Subgroups: 292 in 152 conjugacy classes, 90 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C22×C4, C22×C4, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C30, C30, C2.C42, C2×C20, C2×C20, C22×C10, C22×Dic3, C22×C12, C5×Dic3, C60, C2×C30, C2×C30, C22×C20, C22×C20, C6.C42, C10×Dic3, C10×Dic3, C2×C60, C2×C60, C22×C30, C5×C2.C42, Dic3×C2×C10, C22×C60, C5×C6.C42
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C10, Dic3, D6, C42, C22⋊C4, C4⋊C4, C20, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C5×S3, C2.C42, C2×C20, C5×D4, C5×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C5×Dic3, S3×C10, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C6.C42, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C5×C2.C42, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×D6⋊C4, C5×C6.D4, C5×C6.C42
►Regular action on 480 points
Generators in S480
(1 317 428 419 291)(2 318 429 420 292)(3 319 430 409 293)(4 320 431 410 294)(5 321 432 411 295)(6 322 421 412 296)(7 323 422 413 297)(8 324 423 414 298)(9 313 424 415 299)(10 314 425 416 300)(11 315 426 417 289)(12 316 427 418 290)(13 404 172 306 53)(14 405 173 307 54)(15 406 174 308 55)(16 407 175 309 56)(17 408 176 310 57)(18 397 177 311 58)(19 398 178 312 59)(20 399 179 301 60)(21 400 180 302 49)(22 401 169 303 50)(23 402 170 304 51)(24 403 171 305 52)(25 329 240 196 458)(26 330 229 197 459)(27 331 230 198 460)(28 332 231 199 461)(29 333 232 200 462)(30 334 233 201 463)(31 335 234 202 464)(32 336 235 203 465)(33 325 236 204 466)(34 326 237 193 467)(35 327 238 194 468)(36 328 239 195 457)(37 474 382 214 258)(38 475 383 215 259)(39 476 384 216 260)(40 477 373 205 261)(41 478 374 206 262)(42 479 375 207 263)(43 480 376 208 264)(44 469 377 209 253)(45 470 378 210 254)(46 471 379 211 255)(47 472 380 212 256)(48 473 381 213 257)(61 149 353 110 434)(62 150 354 111 435)(63 151 355 112 436)(64 152 356 113 437)(65 153 357 114 438)(66 154 358 115 439)(67 155 359 116 440)(68 156 360 117 441)(69 145 349 118 442)(70 146 350 119 443)(71 147 351 120 444)(72 148 352 109 433)(73 126 166 362 102)(74 127 167 363 103)(75 128 168 364 104)(76 129 157 365 105)(77 130 158 366 106)(78 131 159 367 107)(79 132 160 368 108)(80 121 161 369 97)(81 122 162 370 98)(82 123 163 371 99)(83 124 164 372 100)(84 125 165 361 101)(85 192 393 141 222)(86 181 394 142 223)(87 182 395 143 224)(88 183 396 144 225)(89 184 385 133 226)(90 185 386 134 227)(91 186 387 135 228)(92 187 388 136 217)(93 188 389 137 218)(94 189 390 138 219)(95 190 391 139 220)(96 191 392 140 221)(241 456 285 348 274)(242 445 286 337 275)(243 446 287 338 276)(244 447 288 339 265)(245 448 277 340 266)(246 449 278 341 267)(247 450 279 342 268)(248 451 280 343 269)(249 452 281 344 270)(250 453 282 345 271)(251 454 283 346 272)(252 455 284 347 273)
(1 384)(2 373)(3 374)(4 375)(5 376)(6 377)(7 378)(8 379)(9 380)(10 381)(11 382)(12 383)(13 154)(14 155)(15 156)(16 145)(17 146)(18 147)(19 148)(20 149)(21 150)(22 151)(23 152)(24 153)(25 394)(26 395)(27 396)(28 385)(29 386)(30 387)(31 388)(32 389)(33 390)(34 391)(35 392)(36 393)(37 417)(38 418)(39 419)(40 420)(41 409)(42 410)(43 411)(44 412)(45 413)(46 414)(47 415)(48 416)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 61)(73 275)(74 276)(75 265)(76 266)(77 267)(78 268)(79 269)(80 270)(81 271)(82 272)(83 273)(84 274)(85 195)(86 196)(87 197)(88 198)(89 199)(90 200)(91 201)(92 202)(93 203)(94 204)(95 193)(96 194)(97 344)(98 345)(99 346)(100 347)(101 348)(102 337)(103 338)(104 339)(105 340)(106 341)(107 342)(108 343)(109 178)(110 179)(111 180)(112 169)(113 170)(114 171)(115 172)(116 173)(117 174)(118 175)(119 176)(120 177)(121 249)(122 250)(123 251)(124 252)(125 241)(126 242)(127 243)(128 244)(129 245)(130 246)(131 247)(132 248)(133 332)(134 333)(135 334)(136 335)(137 336)(138 325)(139 326)(140 327)(141 328)(142 329)(143 330)(144 331)(157 448)(158 449)(159 450)(160 451)(161 452)(162 453)(163 454)(164 455)(165 456)(166 445)(167 446)(168 447)(181 458)(182 459)(183 460)(184 461)(185 462)(186 463)(187 464)(188 465)(189 466)(190 467)(191 468)(192 457)(205 318)(206 319)(207 320)(208 321)(209 322)(210 323)(211 324)(212 313)(213 314)(214 315)(215 316)(216 317)(217 234)(218 235)(219 236)(220 237)(221 238)(222 239)(223 240)(224 229)(225 230)(226 231)(227 232)(228 233)(253 421)(254 422)(255 423)(256 424)(257 425)(258 426)(259 427)(260 428)(261 429)(262 430)(263 431)(264 432)(277 365)(278 366)(279 367)(280 368)(281 369)(282 370)(283 371)(284 372)(285 361)(286 362)(287 363)(288 364)(289 474)(290 475)(291 476)(292 477)(293 478)(294 479)(295 480)(296 469)(297 470)(298 471)(299 472)(300 473)(301 434)(302 435)(303 436)(304 437)(305 438)(306 439)(307 440)(308 441)(309 442)(310 443)(311 444)(312 433)(349 407)(350 408)(351 397)(352 398)(353 399)(354 400)(355 401)(356 402)(357 403)(358 404)(359 405)(360 406)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)(217 223)(218 224)(219 225)(220 226)(221 227)(222 228)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)(241 247)(242 248)(243 249)(244 250)(245 251)(246 252)(253 259)(254 260)(255 261)(256 262)(257 263)(258 264)(265 271)(266 272)(267 273)(268 274)(269 275)(270 276)(277 283)(278 284)(279 285)(280 286)(281 287)(282 288)(289 295)(290 296)(291 297)(292 298)(293 299)(294 300)(301 307)(302 308)(303 309)(304 310)(305 311)(306 312)(313 319)(314 320)(315 321)(316 322)(317 323)(318 324)(325 331)(326 332)(327 333)(328 334)(329 335)(330 336)(337 343)(338 344)(339 345)(340 346)(341 347)(342 348)(349 355)(350 356)(351 357)(352 358)(353 359)(354 360)(361 367)(362 368)(363 369)(364 370)(365 371)(366 372)(373 379)(374 380)(375 381)(376 382)(377 383)(378 384)(385 391)(386 392)(387 393)(388 394)(389 395)(390 396)(397 403)(398 404)(399 405)(400 406)(401 407)(402 408)(409 415)(410 416)(411 417)(412 418)(413 419)(414 420)(421 427)(422 428)(423 429)(424 430)(425 431)(426 432)(433 439)(434 440)(435 441)(436 442)(437 443)(438 444)(445 451)(446 452)(447 453)(448 454)(449 455)(450 456)(457 463)(458 464)(459 465)(460 466)(461 467)(462 468)(469 475)(470 476)(471 477)(472 478)(473 479)(474 480)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 269)(26 270)(27 271)(28 272)(29 273)(30 274)(31 275)(32 276)(33 265)(34 266)(35 267)(36 268)(37 439)(38 440)(39 441)(40 442)(41 443)(42 444)(43 433)(44 434)(45 435)(46 436)(47 437)(48 438)(49 297)(50 298)(51 299)(52 300)(53 289)(54 290)(55 291)(56 292)(57 293)(58 294)(59 295)(60 296)(61 469)(62 470)(63 471)(64 472)(65 473)(66 474)(67 475)(68 476)(69 477)(70 478)(71 479)(72 480)(73 388)(74 389)(75 390)(76 391)(77 392)(78 393)(79 394)(80 395)(81 396)(82 385)(83 386)(84 387)(85 367)(86 368)(87 369)(88 370)(89 371)(90 372)(91 361)(92 362)(93 363)(94 364)(95 365)(96 366)(97 182)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)(105 190)(106 191)(107 192)(108 181)(109 264)(110 253)(111 254)(112 255)(113 256)(114 257)(115 258)(116 259)(117 260)(118 261)(119 262)(120 263)(121 143)(122 144)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(131 141)(132 142)(145 373)(146 374)(147 375)(148 376)(149 377)(150 378)(151 379)(152 380)(153 381)(154 382)(155 383)(156 384)(157 220)(158 221)(159 222)(160 223)(161 224)(162 225)(163 226)(164 227)(165 228)(166 217)(167 218)(168 219)(169 423)(170 424)(171 425)(172 426)(173 427)(174 428)(175 429)(176 430)(177 431)(178 432)(179 421)(180 422)(193 277)(194 278)(195 279)(196 280)(197 281)(198 282)(199 283)(200 284)(201 285)(202 286)(203 287)(204 288)(205 349)(206 350)(207 351)(208 352)(209 353)(210 354)(211 355)(212 356)(213 357)(214 358)(215 359)(216 360)(229 452)(230 453)(231 454)(232 455)(233 456)(234 445)(235 446)(236 447)(237 448)(238 449)(239 450)(240 451)(241 334)(242 335)(243 336)(244 325)(245 326)(246 327)(247 328)(248 329)(249 330)(250 331)(251 332)(252 333)(301 412)(302 413)(303 414)(304 415)(305 416)(306 417)(307 418)(308 419)(309 420)(310 409)(311 410)(312 411)(313 402)(314 403)(315 404)(316 405)(317 406)(318 407)(319 408)(320 397)(321 398)(322 399)(323 400)(324 401)(337 464)(338 465)(339 466)(340 467)(341 468)(342 457)(343 458)(344 459)(345 460)(346 461)(347 462)(348 463)
(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 29 150 77)(2 266 151 385)(3 27 152 75)(4 276 153 395)(5 25 154 73)(6 274 155 393)(7 35 156 83)(8 272 145 391)(9 33 146 81)(10 270 147 389)(11 31 148 79)(12 268 149 387)(13 275 376 394)(14 36 377 84)(15 273 378 392)(16 34 379 82)(17 271 380 390)(18 32 381 80)(19 269 382 388)(20 30 383 78)(21 267 384 386)(22 28 373 76)(23 265 374 396)(24 26 375 74)(37 92 312 280)(38 367 301 201)(39 90 302 278)(40 365 303 199)(41 88 304 288)(42 363 305 197)(43 86 306 286)(44 361 307 195)(45 96 308 284)(46 371 309 193)(47 94 310 282)(48 369 311 203)(49 341 476 185)(50 461 477 105)(51 339 478 183)(52 459 479 103)(53 337 480 181)(54 457 469 101)(55 347 470 191)(56 467 471 99)(57 345 472 189)(58 465 473 97)(59 343 474 187)(60 463 475 107)(61 186 290 342)(62 106 291 462)(63 184 292 340)(64 104 293 460)(65 182 294 338)(66 102 295 458)(67 192 296 348)(68 100 297 468)(69 190 298 346)(70 98 299 466)(71 188 300 344)(72 108 289 464)(85 412 285 440)(87 410 287 438)(89 420 277 436)(91 418 279 434)(93 416 281 444)(95 414 283 442)(109 160 426 234)(110 228 427 450)(111 158 428 232)(112 226 429 448)(113 168 430 230)(114 224 431 446)(115 166 432 240)(116 222 421 456)(117 164 422 238)(118 220 423 454)(119 162 424 236)(120 218 425 452)(121 397 336 213)(122 313 325 350)(123 407 326 211)(124 323 327 360)(125 405 328 209)(126 321 329 358)(127 403 330 207)(128 319 331 356)(129 401 332 205)(130 317 333 354)(131 399 334 215)(132 315 335 352)(133 318 245 355)(134 400 246 216)(135 316 247 353)(136 398 248 214)(137 314 249 351)(138 408 250 212)(139 324 251 349)(140 406 252 210)(141 322 241 359)(142 404 242 208)(143 320 243 357)(144 402 244 206)(157 169 231 261)(159 179 233 259)(161 177 235 257)(163 175 237 255)(165 173 239 253)(167 171 229 263)(170 447 262 225)(172 445 264 223)(174 455 254 221)(176 453 256 219)(178 451 258 217)(180 449 260 227)(194 441 372 413)(196 439 362 411)(198 437 364 409)(200 435 366 419)(202 433 368 417)(204 443 370 415)
G:=sub<Sym(480)| (1,317,428,419,291)(2,318,429,420,292)(3,319,430,409,293)(4,320,431,410,294)(5,321,432,411,295)(6,322,421,412,296)(7,323,422,413,297)(8,324,423,414,298)(9,313,424,415,299)(10,314,425,416,300)(11,315,426,417,289)(12,316,427,418,290)(13,404,172,306,53)(14,405,173,307,54)(15,406,174,308,55)(16,407,175,309,56)(17,408,176,310,57)(18,397,177,311,58)(19,398,178,312,59)(20,399,179,301,60)(21,400,180,302,49)(22,401,169,303,50)(23,402,170,304,51)(24,403,171,305,52)(25,329,240,196,458)(26,330,229,197,459)(27,331,230,198,460)(28,332,231,199,461)(29,333,232,200,462)(30,334,233,201,463)(31,335,234,202,464)(32,336,235,203,465)(33,325,236,204,466)(34,326,237,193,467)(35,327,238,194,468)(36,328,239,195,457)(37,474,382,214,258)(38,475,383,215,259)(39,476,384,216,260)(40,477,373,205,261)(41,478,374,206,262)(42,479,375,207,263)(43,480,376,208,264)(44,469,377,209,253)(45,470,378,210,254)(46,471,379,211,255)(47,472,380,212,256)(48,473,381,213,257)(61,149,353,110,434)(62,150,354,111,435)(63,151,355,112,436)(64,152,356,113,437)(65,153,357,114,438)(66,154,358,115,439)(67,155,359,116,440)(68,156,360,117,441)(69,145,349,118,442)(70,146,350,119,443)(71,147,351,120,444)(72,148,352,109,433)(73,126,166,362,102)(74,127,167,363,103)(75,128,168,364,104)(76,129,157,365,105)(77,130,158,366,106)(78,131,159,367,107)(79,132,160,368,108)(80,121,161,369,97)(81,122,162,370,98)(82,123,163,371,99)(83,124,164,372,100)(84,125,165,361,101)(85,192,393,141,222)(86,181,394,142,223)(87,182,395,143,224)(88,183,396,144,225)(89,184,385,133,226)(90,185,386,134,227)(91,186,387,135,228)(92,187,388,136,217)(93,188,389,137,218)(94,189,390,138,219)(95,190,391,139,220)(96,191,392,140,221)(241,456,285,348,274)(242,445,286,337,275)(243,446,287,338,276)(244,447,288,339,265)(245,448,277,340,266)(246,449,278,341,267)(247,450,279,342,268)(248,451,280,343,269)(249,452,281,344,270)(250,453,282,345,271)(251,454,283,346,272)(252,455,284,347,273), (1,384)(2,373)(3,374)(4,375)(5,376)(6,377)(7,378)(8,379)(9,380)(10,381)(11,382)(12,383)(13,154)(14,155)(15,156)(16,145)(17,146)(18,147)(19,148)(20,149)(21,150)(22,151)(23,152)(24,153)(25,394)(26,395)(27,396)(28,385)(29,386)(30,387)(31,388)(32,389)(33,390)(34,391)(35,392)(36,393)(37,417)(38,418)(39,419)(40,420)(41,409)(42,410)(43,411)(44,412)(45,413)(46,414)(47,415)(48,416)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,61)(73,275)(74,276)(75,265)(76,266)(77,267)(78,268)(79,269)(80,270)(81,271)(82,272)(83,273)(84,274)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,193)(96,194)(97,344)(98,345)(99,346)(100,347)(101,348)(102,337)(103,338)(104,339)(105,340)(106,341)(107,342)(108,343)(109,178)(110,179)(111,180)(112,169)(113,170)(114,171)(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,249)(122,250)(123,251)(124,252)(125,241)(126,242)(127,243)(128,244)(129,245)(130,246)(131,247)(132,248)(133,332)(134,333)(135,334)(136,335)(137,336)(138,325)(139,326)(140,327)(141,328)(142,329)(143,330)(144,331)(157,448)(158,449)(159,450)(160,451)(161,452)(162,453)(163,454)(164,455)(165,456)(166,445)(167,446)(168,447)(181,458)(182,459)(183,460)(184,461)(185,462)(186,463)(187,464)(188,465)(189,466)(190,467)(191,468)(192,457)(205,318)(206,319)(207,320)(208,321)(209,322)(210,323)(211,324)(212,313)(213,314)(214,315)(215,316)(216,317)(217,234)(218,235)(219,236)(220,237)(221,238)(222,239)(223,240)(224,229)(225,230)(226,231)(227,232)(228,233)(253,421)(254,422)(255,423)(256,424)(257,425)(258,426)(259,427)(260,428)(261,429)(262,430)(263,431)(264,432)(277,365)(278,366)(279,367)(280,368)(281,369)(282,370)(283,371)(284,372)(285,361)(286,362)(287,363)(288,364)(289,474)(290,475)(291,476)(292,477)(293,478)(294,479)(295,480)(296,469)(297,470)(298,471)(299,472)(300,473)(301,434)(302,435)(303,436)(304,437)(305,438)(306,439)(307,440)(308,441)(309,442)(310,443)(311,444)(312,433)(349,407)(350,408)(351,397)(352,398)(353,399)(354,400)(355,401)(356,402)(357,403)(358,404)(359,405)(360,406), 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);
G=PermutationGroup([[(1,317,428,419,291),(2,318,429,420,292),(3,319,430,409,293),(4,320,431,410,294),(5,321,432,411,295),(6,322,421,412,296),(7,323,422,413,297),(8,324,423,414,298),(9,313,424,415,299),(10,314,425,416,300),(11,315,426,417,289),(12,316,427,418,290),(13,404,172,306,53),(14,405,173,307,54),(15,406,174,308,55),(16,407,175,309,56),(17,408,176,310,57),(18,397,177,311,58),(19,398,178,312,59),(20,399,179,301,60),(21,400,180,302,49),(22,401,169,303,50),(23,402,170,304,51),(24,403,171,305,52),(25,329,240,196,458),(26,330,229,197,459),(27,331,230,198,460),(28,332,231,199,461),(29,333,232,200,462),(30,334,233,201,463),(31,335,234,202,464),(32,336,235,203,465),(33,325,236,204,466),(34,326,237,193,467),(35,327,238,194,468),(36,328,239,195,457),(37,474,382,214,258),(38,475,383,215,259),(39,476,384,216,260),(40,477,373,205,261),(41,478,374,206,262),(42,479,375,207,263),(43,480,376,208,264),(44,469,377,209,253),(45,470,378,210,254),(46,471,379,211,255),(47,472,380,212,256),(48,473,381,213,257),(61,149,353,110,434),(62,150,354,111,435),(63,151,355,112,436),(64,152,356,113,437),(65,153,357,114,438),(66,154,358,115,439),(67,155,359,116,440),(68,156,360,117,441),(69,145,349,118,442),(70,146,350,119,443),(71,147,351,120,444),(72,148,352,109,433),(73,126,166,362,102),(74,127,167,363,103),(75,128,168,364,104),(76,129,157,365,105),(77,130,158,366,106),(78,131,159,367,107),(79,132,160,368,108),(80,121,161,369,97),(81,122,162,370,98),(82,123,163,371,99),(83,124,164,372,100),(84,125,165,361,101),(85,192,393,141,222),(86,181,394,142,223),(87,182,395,143,224),(88,183,396,144,225),(89,184,385,133,226),(90,185,386,134,227),(91,186,387,135,228),(92,187,388,136,217),(93,188,389,137,218),(94,189,390,138,219),(95,190,391,139,220),(96,191,392,140,221),(241,456,285,348,274),(242,445,286,337,275),(243,446,287,338,276),(244,447,288,339,265),(245,448,277,340,266),(246,449,278,341,267),(247,450,279,342,268),(248,451,280,343,269),(249,452,281,344,270),(250,453,282,345,271),(251,454,283,346,272),(252,455,284,347,273)], [(1,384),(2,373),(3,374),(4,375),(5,376),(6,377),(7,378),(8,379),(9,380),(10,381),(11,382),(12,383),(13,154),(14,155),(15,156),(16,145),(17,146),(18,147),(19,148),(20,149),(21,150),(22,151),(23,152),(24,153),(25,394),(26,395),(27,396),(28,385),(29,386),(30,387),(31,388),(32,389),(33,390),(34,391),(35,392),(36,393),(37,417),(38,418),(39,419),(40,420),(41,409),(42,410),(43,411),(44,412),(45,413),(46,414),(47,415),(48,416),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,61),(73,275),(74,276),(75,265),(76,266),(77,267),(78,268),(79,269),(80,270),(81,271),(82,272),(83,273),(84,274),(85,195),(86,196),(87,197),(88,198),(89,199),(90,200),(91,201),(92,202),(93,203),(94,204),(95,193),(96,194),(97,344),(98,345),(99,346),(100,347),(101,348),(102,337),(103,338),(104,339),(105,340),(106,341),(107,342),(108,343),(109,178),(110,179),(111,180),(112,169),(113,170),(114,171),(115,172),(116,173),(117,174),(118,175),(119,176),(120,177),(121,249),(122,250),(123,251),(124,252),(125,241),(126,242),(127,243),(128,244),(129,245),(130,246),(131,247),(132,248),(133,332),(134,333),(135,334),(136,335),(137,336),(138,325),(139,326),(140,327),(141,328),(142,329),(143,330),(144,331),(157,448),(158,449),(159,450),(160,451),(161,452),(162,453),(163,454),(164,455),(165,456),(166,445),(167,446),(168,447),(181,458),(182,459),(183,460),(184,461),(185,462),(186,463),(187,464),(188,465),(189,466),(190,467),(191,468),(192,457),(205,318),(206,319),(207,320),(208,321),(209,322),(210,323),(211,324),(212,313),(213,314),(214,315),(215,316),(216,317),(217,234),(218,235),(219,236),(220,237),(221,238),(222,239),(223,240),(224,229),(225,230),(226,231),(227,232),(228,233),(253,421),(254,422),(255,423),(256,424),(257,425),(258,426),(259,427),(260,428),(261,429),(262,430),(263,431),(264,432),(277,365),(278,366),(279,367),(280,368),(281,369),(282,370),(283,371),(284,372),(285,361),(286,362),(287,363),(288,364),(289,474),(290,475),(291,476),(292,477),(293,478),(294,479),(295,480),(296,469),(297,470),(298,471),(299,472),(300,473),(301,434),(302,435),(303,436),(304,437),(305,438),(306,439),(307,440),(308,441),(309,442),(310,443),(311,444),(312,433),(349,407),(350,408),(351,397),(352,398),(353,399),(354,400),(355,401),(356,402),(357,403),(358,404),(359,405),(360,406)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(133,139),(134,140),(135,141),(136,142),(137,143),(138,144),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,163),(158,164),(159,165),(160,166),(161,167),(162,168),(169,175),(170,176),(171,177),(172,178),(173,179),(174,180),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192),(193,199),(194,200),(195,201),(196,202),(197,203),(198,204),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216),(217,223),(218,224),(219,225),(220,226),(221,227),(222,228),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240),(241,247),(242,248),(243,249),(244,250),(245,251),(246,252),(253,259),(254,260),(255,261),(256,262),(257,263),(258,264),(265,271),(266,272),(267,273),(268,274),(269,275),(270,276),(277,283),(278,284),(279,285),(280,286),(281,287),(282,288),(289,295),(290,296),(291,297),(292,298),(293,299),(294,300),(301,307),(302,308),(303,309),(304,310),(305,311),(306,312),(313,319),(314,320),(315,321),(316,322),(317,323),(318,324),(325,331),(326,332),(327,333),(328,334),(329,335),(330,336),(337,343),(338,344),(339,345),(340,346),(341,347),(342,348),(349,355),(350,356),(351,357),(352,358),(353,359),(354,360),(361,367),(362,368),(363,369),(364,370),(365,371),(366,372),(373,379),(374,380),(375,381),(376,382),(377,383),(378,384),(385,391),(386,392),(387,393),(388,394),(389,395),(390,396),(397,403),(398,404),(399,405),(400,406),(401,407),(402,408),(409,415),(410,416),(411,417),(412,418),(413,419),(414,420),(421,427),(422,428),(423,429),(424,430),(425,431),(426,432),(433,439),(434,440),(435,441),(436,442),(437,443),(438,444),(445,451),(446,452),(447,453),(448,454),(449,455),(450,456),(457,463),(458,464),(459,465),(460,466),(461,467),(462,468),(469,475),(470,476),(471,477),(472,478),(473,479),(474,480)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,13),(12,14),(25,269),(26,270),(27,271),(28,272),(29,273),(30,274),(31,275),(32,276),(33,265),(34,266),(35,267),(36,268),(37,439),(38,440),(39,441),(40,442),(41,443),(42,444),(43,433),(44,434),(45,435),(46,436),(47,437),(48,438),(49,297),(50,298),(51,299),(52,300),(53,289),(54,290),(55,291),(56,292),(57,293),(58,294),(59,295),(60,296),(61,469),(62,470),(63,471),(64,472),(65,473),(66,474),(67,475),(68,476),(69,477),(70,478),(71,479),(72,480),(73,388),(74,389),(75,390),(76,391),(77,392),(78,393),(79,394),(80,395),(81,396),(82,385),(83,386),(84,387),(85,367),(86,368),(87,369),(88,370),(89,371),(90,372),(91,361),(92,362),(93,363),(94,364),(95,365),(96,366),(97,182),(98,183),(99,184),(100,185),(101,186),(102,187),(103,188),(104,189),(105,190),(106,191),(107,192),(108,181),(109,264),(110,253),(111,254),(112,255),(113,256),(114,257),(115,258),(116,259),(117,260),(118,261),(119,262),(120,263),(121,143),(122,144),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(131,141),(132,142),(145,373),(146,374),(147,375),(148,376),(149,377),(150,378),(151,379),(152,380),(153,381),(154,382),(155,383),(156,384),(157,220),(158,221),(159,222),(160,223),(161,224),(162,225),(163,226),(164,227),(165,228),(166,217),(167,218),(168,219),(169,423),(170,424),(171,425),(172,426),(173,427),(174,428),(175,429),(176,430),(177,431),(178,432),(179,421),(180,422),(193,277),(194,278),(195,279),(196,280),(197,281),(198,282),(199,283),(200,284),(201,285),(202,286),(203,287),(204,288),(205,349),(206,350),(207,351),(208,352),(209,353),(210,354),(211,355),(212,356),(213,357),(214,358),(215,359),(216,360),(229,452),(230,453),(231,454),(232,455),(233,456),(234,445),(235,446),(236,447),(237,448),(238,449),(239,450),(240,451),(241,334),(242,335),(243,336),(244,325),(245,326),(246,327),(247,328),(248,329),(249,330),(250,331),(251,332),(252,333),(301,412),(302,413),(303,414),(304,415),(305,416),(306,417),(307,418),(308,419),(309,420),(310,409),(311,410),(312,411),(313,402),(314,403),(315,404),(316,405),(317,406),(318,407),(319,408),(320,397),(321,398),(322,399),(323,400),(324,401),(337,464),(338,465),(339,466),(340,467),(341,468),(342,457),(343,458),(344,459),(345,460),(346,461),(347,462),(348,463)], [(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)], [(1,29,150,77),(2,266,151,385),(3,27,152,75),(4,276,153,395),(5,25,154,73),(6,274,155,393),(7,35,156,83),(8,272,145,391),(9,33,146,81),(10,270,147,389),(11,31,148,79),(12,268,149,387),(13,275,376,394),(14,36,377,84),(15,273,378,392),(16,34,379,82),(17,271,380,390),(18,32,381,80),(19,269,382,388),(20,30,383,78),(21,267,384,386),(22,28,373,76),(23,265,374,396),(24,26,375,74),(37,92,312,280),(38,367,301,201),(39,90,302,278),(40,365,303,199),(41,88,304,288),(42,363,305,197),(43,86,306,286),(44,361,307,195),(45,96,308,284),(46,371,309,193),(47,94,310,282),(48,369,311,203),(49,341,476,185),(50,461,477,105),(51,339,478,183),(52,459,479,103),(53,337,480,181),(54,457,469,101),(55,347,470,191),(56,467,471,99),(57,345,472,189),(58,465,473,97),(59,343,474,187),(60,463,475,107),(61,186,290,342),(62,106,291,462),(63,184,292,340),(64,104,293,460),(65,182,294,338),(66,102,295,458),(67,192,296,348),(68,100,297,468),(69,190,298,346),(70,98,299,466),(71,188,300,344),(72,108,289,464),(85,412,285,440),(87,410,287,438),(89,420,277,436),(91,418,279,434),(93,416,281,444),(95,414,283,442),(109,160,426,234),(110,228,427,450),(111,158,428,232),(112,226,429,448),(113,168,430,230),(114,224,431,446),(115,166,432,240),(116,222,421,456),(117,164,422,238),(118,220,423,454),(119,162,424,236),(120,218,425,452),(121,397,336,213),(122,313,325,350),(123,407,326,211),(124,323,327,360),(125,405,328,209),(126,321,329,358),(127,403,330,207),(128,319,331,356),(129,401,332,205),(130,317,333,354),(131,399,334,215),(132,315,335,352),(133,318,245,355),(134,400,246,216),(135,316,247,353),(136,398,248,214),(137,314,249,351),(138,408,250,212),(139,324,251,349),(140,406,252,210),(141,322,241,359),(142,404,242,208),(143,320,243,357),(144,402,244,206),(157,169,231,261),(159,179,233,259),(161,177,235,257),(163,175,237,255),(165,173,239,253),(167,171,229,263),(170,447,262,225),(172,445,264,223),(174,455,254,221),(176,453,256,219),(178,451,258,217),(180,449,260,227),(194,441,372,413),(196,439,362,411),(198,437,364,409),(200,435,366,419),(202,433,368,417),(204,443,370,415)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 6A | ··· | 6G | 10A | ··· | 10AB | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AV | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | C5×S3 | C5×D4 | C5×Q8 | C5×Dic3 | S3×C10 | C5×Dic6 | S3×C20 | C5×D12 | C5×C3⋊D4 |
| kernel | C5×C6.C42 | Dic3×C2×C10 | C22×C60 | C10×Dic3 | C2×C60 | C6.C42 | C22×Dic3 | C22×C12 | C2×Dic3 | C2×C12 | C22×C20 | C2×C30 | C2×C30 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C2×C10 | C2×C10 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 32 | 16 | 1 | 3 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 12 | 4 | 8 | 4 | 8 | 16 | 8 | 16 |
Matrix representation of C5×C6.C42 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 20 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 52 | 52 |
| 0 | 0 | 0 | 9 | 43 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 48 | 30 | 0 | 0 |
| 0 | 31 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 11 | 0 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,20,0,0,0,0,0,20],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[50,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,52,9,0,0,0,52,43],[60,0,0,0,0,0,48,31,0,0,0,30,13,0,0,0,0,0,0,11,0,0,0,11,0] >;
C5×C6.C42 in GAP, Magma, Sage, TeX
C_5\times C_6.C_4^2
% in TeX
G:=Group("C5xC6.C4^2"); // GroupNames label
G:=SmallGroup(480,150);
// by ID
G=gap.SmallGroup(480,150);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,1149,288,15686]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=c,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
// generators/relations