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G = C5×C6.C42order 480 = 25·3·5

Direct product of C5 and C6.C42

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), C157(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

C1C6 — C5×C6.C42
C1C3C6C2×C6C22×C6C22×C30Dic3×C2×C10 — C5×C6.C42
C3C6 — C5×C6.C42
C1C22×C10C22×C20

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 [×3], C2 [×4], C3, C4 [×6], C22 [×3], C22 [×4], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C23, C10 [×3], C10 [×4], Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C15, C22×C4, C22×C4 [×2], C20 [×6], C2×C10 [×3], C2×C10 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×4], C2.C42, C2×C20 [×2], C2×C20 [×10], C22×C10, C22×Dic3 [×2], C22×C12, C5×Dic3 [×4], C60 [×2], C2×C30 [×3], C2×C30 [×4], C22×C20, C22×C20 [×2], C6.C42, C10×Dic3 [×4], C10×Dic3 [×4], C2×C60 [×2], C2×C60 [×2], C22×C30, C5×C2.C42, Dic3×C2×C10 [×2], C22×C60, C5×C6.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C5, S3, C2×C4 [×3], D4 [×3], Q8, C10 [×3], Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C5×S3, C2.C42, C2×C20 [×3], C5×D4 [×3], C5×Q8, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C5×Dic3 [×2], S3×C10, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C6.C42, C5×Dic6, S3×C20 [×2], C5×D12, C10×Dic3, C5×C3⋊D4 [×2], C5×C2.C42, Dic3×C20, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×D6⋊C4 [×2], C5×C6.D4, C5×C6.C42

Smallest permutation representation of C5×C6.C42
Regular action on 480 points
Generators in S480
(1 119 390 198 35)(2 120 391 199 36)(3 109 392 200 25)(4 110 393 201 26)(5 111 394 202 27)(6 112 395 203 28)(7 113 396 204 29)(8 114 385 193 30)(9 115 386 194 31)(10 116 387 195 32)(11 117 388 196 33)(12 118 389 197 34)(13 140 332 50 321)(14 141 333 51 322)(15 142 334 52 323)(16 143 335 53 324)(17 144 336 54 313)(18 133 325 55 314)(19 134 326 56 315)(20 135 327 57 316)(21 136 328 58 317)(22 137 329 59 318)(23 138 330 60 319)(24 139 331 49 320)(37 282 188 465 289)(38 283 189 466 290)(39 284 190 467 291)(40 285 191 468 292)(41 286 192 457 293)(42 287 181 458 294)(43 288 182 459 295)(44 277 183 460 296)(45 278 184 461 297)(46 279 185 462 298)(47 280 186 463 299)(48 281 187 464 300)(61 267 428 252 478)(62 268 429 241 479)(63 269 430 242 480)(64 270 431 243 469)(65 271 432 244 470)(66 272 421 245 471)(67 273 422 246 472)(68 274 423 247 473)(69 275 424 248 474)(70 276 425 249 475)(71 265 426 250 476)(72 266 427 251 477)(73 223 341 397 175)(74 224 342 398 176)(75 225 343 399 177)(76 226 344 400 178)(77 227 345 401 179)(78 228 346 402 180)(79 217 347 403 169)(80 218 348 404 170)(81 219 337 405 171)(82 220 338 406 172)(83 221 339 407 173)(84 222 340 408 174)(85 311 122 237 441)(86 312 123 238 442)(87 301 124 239 443)(88 302 125 240 444)(89 303 126 229 433)(90 304 127 230 434)(91 305 128 231 435)(92 306 129 232 436)(93 307 130 233 437)(94 308 131 234 438)(95 309 132 235 439)(96 310 121 236 440)(97 354 156 365 417)(98 355 145 366 418)(99 356 146 367 419)(100 357 147 368 420)(101 358 148 369 409)(102 359 149 370 410)(103 360 150 371 411)(104 349 151 372 412)(105 350 152 361 413)(106 351 153 362 414)(107 352 154 363 415)(108 353 155 364 416)(157 212 382 451 258)(158 213 383 452 259)(159 214 384 453 260)(160 215 373 454 261)(161 216 374 455 262)(162 205 375 456 263)(163 206 376 445 264)(164 207 377 446 253)(165 208 378 447 254)(166 209 379 448 255)(167 210 380 449 256)(168 211 381 450 257)
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 157)(10 158)(11 159)(12 160)(13 463)(14 464)(15 465)(16 466)(17 467)(18 468)(19 457)(20 458)(21 459)(22 460)(23 461)(24 462)(25 264)(26 253)(27 254)(28 255)(29 256)(30 257)(31 258)(32 259)(33 260)(34 261)(35 262)(36 263)(37 334)(38 335)(39 336)(40 325)(41 326)(42 327)(43 328)(44 329)(45 330)(46 331)(47 332)(48 333)(49 279)(50 280)(51 281)(52 282)(53 283)(54 284)(55 285)(56 286)(57 287)(58 288)(59 277)(60 278)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 169)(70 170)(71 171)(72 172)(73 269)(74 270)(75 271)(76 272)(77 273)(78 274)(79 275)(80 276)(81 265)(82 266)(83 267)(84 268)(85 412)(86 413)(87 414)(88 415)(89 416)(90 417)(91 418)(92 419)(93 420)(94 409)(95 410)(96 411)(97 304)(98 305)(99 306)(100 307)(101 308)(102 309)(103 310)(104 311)(105 312)(106 301)(107 302)(108 303)(109 206)(110 207)(111 208)(112 209)(113 210)(114 211)(115 212)(116 213)(117 214)(118 215)(119 216)(120 205)(121 360)(122 349)(123 350)(124 351)(125 352)(126 353)(127 354)(128 355)(129 356)(130 357)(131 358)(132 359)(133 292)(134 293)(135 294)(136 295)(137 296)(138 297)(139 298)(140 299)(141 300)(142 289)(143 290)(144 291)(145 231)(146 232)(147 233)(148 234)(149 235)(150 236)(151 237)(152 238)(153 239)(154 240)(155 229)(156 230)(181 316)(182 317)(183 318)(184 319)(185 320)(186 321)(187 322)(188 323)(189 324)(190 313)(191 314)(192 315)(193 450)(194 451)(195 452)(196 453)(197 454)(198 455)(199 456)(200 445)(201 446)(202 447)(203 448)(204 449)(217 424)(218 425)(219 426)(220 427)(221 428)(222 429)(223 430)(224 431)(225 432)(226 421)(227 422)(228 423)(241 340)(242 341)(243 342)(244 343)(245 344)(246 345)(247 346)(248 347)(249 348)(250 337)(251 338)(252 339)(361 442)(362 443)(363 444)(364 433)(365 434)(366 435)(367 436)(368 437)(369 438)(370 439)(371 440)(372 441)(373 389)(374 390)(375 391)(376 392)(377 393)(378 394)(379 395)(380 396)(381 385)(382 386)(383 387)(384 388)(397 480)(398 469)(399 470)(400 471)(401 472)(402 473)(403 474)(404 475)(405 476)(406 477)(407 478)(408 479)
(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)
(1 185)(2 186)(3 187)(4 188)(5 189)(6 190)(7 191)(8 192)(9 181)(10 182)(11 183)(12 184)(13 205)(14 206)(15 207)(16 208)(17 209)(18 210)(19 211)(20 212)(21 213)(22 214)(23 215)(24 216)(25 281)(26 282)(27 283)(28 284)(29 285)(30 286)(31 287)(32 288)(33 277)(34 278)(35 279)(36 280)(37 201)(38 202)(39 203)(40 204)(41 193)(42 194)(43 195)(44 196)(45 197)(46 198)(47 199)(48 200)(49 262)(50 263)(51 264)(52 253)(53 254)(54 255)(55 256)(56 257)(57 258)(58 259)(59 260)(60 261)(61 311)(62 312)(63 301)(64 302)(65 303)(66 304)(67 305)(68 306)(69 307)(70 308)(71 309)(72 310)(73 351)(74 352)(75 353)(76 354)(77 355)(78 356)(79 357)(80 358)(81 359)(82 360)(83 349)(84 350)(85 478)(86 479)(87 480)(88 469)(89 470)(90 471)(91 472)(92 473)(93 474)(94 475)(95 476)(96 477)(97 178)(98 179)(99 180)(100 169)(101 170)(102 171)(103 172)(104 173)(105 174)(106 175)(107 176)(108 177)(109 464)(110 465)(111 466)(112 467)(113 468)(114 457)(115 458)(116 459)(117 460)(118 461)(119 462)(120 463)(121 266)(122 267)(123 268)(124 269)(125 270)(126 271)(127 272)(128 273)(129 274)(130 275)(131 276)(132 265)(133 380)(134 381)(135 382)(136 383)(137 384)(138 373)(139 374)(140 375)(141 376)(142 377)(143 378)(144 379)(145 227)(146 228)(147 217)(148 218)(149 219)(150 220)(151 221)(152 222)(153 223)(154 224)(155 225)(156 226)(157 316)(158 317)(159 318)(160 319)(161 320)(162 321)(163 322)(164 323)(165 324)(166 313)(167 314)(168 315)(229 432)(230 421)(231 422)(232 423)(233 424)(234 425)(235 426)(236 427)(237 428)(238 429)(239 430)(240 431)(241 442)(242 443)(243 444)(244 433)(245 434)(246 435)(247 436)(248 437)(249 438)(250 439)(251 440)(252 441)(289 393)(290 394)(291 395)(292 396)(293 385)(294 386)(295 387)(296 388)(297 389)(298 390)(299 391)(300 392)(325 449)(326 450)(327 451)(328 452)(329 453)(330 454)(331 455)(332 456)(333 445)(334 446)(335 447)(336 448)(337 370)(338 371)(339 372)(340 361)(341 362)(342 363)(343 364)(344 365)(345 366)(346 367)(347 368)(348 369)(397 414)(398 415)(399 416)(400 417)(401 418)(402 419)(403 420)(404 409)(405 410)(406 411)(407 412)(408 413)
(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)
(1 397 314 93)(2 419 315 479)(3 407 316 91)(4 417 317 477)(5 405 318 89)(6 415 319 475)(7 403 320 87)(8 413 321 473)(9 401 322 85)(10 411 323 471)(11 399 324 95)(12 409 313 469)(13 68 114 105)(14 311 115 179)(15 66 116 103)(16 309 117 177)(17 64 118 101)(18 307 119 175)(19 62 120 99)(20 305 109 173)(21 72 110 97)(22 303 111 171)(23 70 112 107)(24 301 113 169)(25 339 57 435)(26 365 58 251)(27 337 59 433)(28 363 60 249)(29 347 49 443)(30 361 50 247)(31 345 51 441)(32 371 52 245)(33 343 53 439)(34 369 54 243)(35 341 55 437)(36 367 56 241)(37 226 452 236)(38 149 453 432)(39 224 454 234)(40 147 455 430)(41 222 456 232)(42 145 445 428)(43 220 446 230)(44 155 447 426)(45 218 448 240)(46 153 449 424)(47 228 450 238)(48 151 451 422)(61 458 98 206)(63 468 100 216)(65 466 102 214)(67 464 104 212)(69 462 106 210)(71 460 108 208)(73 133 130 390)(74 373 131 291)(75 143 132 388)(76 383 121 289)(77 141 122 386)(78 381 123 299)(79 139 124 396)(80 379 125 297)(81 137 126 394)(82 377 127 295)(83 135 128 392)(84 375 129 293)(86 186 402 168)(88 184 404 166)(90 182 406 164)(92 192 408 162)(94 190 398 160)(96 188 400 158)(134 268 391 356)(136 266 393 354)(138 276 395 352)(140 274 385 350)(142 272 387 360)(144 270 389 358)(146 326 429 199)(148 336 431 197)(150 334 421 195)(152 332 423 193)(154 330 425 203)(156 328 427 201)(157 472 187 412)(159 470 189 410)(161 480 191 420)(163 478 181 418)(165 476 183 416)(167 474 185 414)(170 209 302 461)(172 207 304 459)(174 205 306 457)(176 215 308 467)(178 213 310 465)(180 211 312 463)(194 227 333 237)(196 225 335 235)(198 223 325 233)(200 221 327 231)(202 219 329 229)(204 217 331 239)(242 285 368 262)(244 283 370 260)(246 281 372 258)(248 279 362 256)(250 277 364 254)(252 287 366 264)(253 434 288 338)(255 444 278 348)(257 442 280 346)(259 440 282 344)(261 438 284 342)(263 436 286 340)(265 296 353 378)(267 294 355 376)(269 292 357 374)(271 290 359 384)(273 300 349 382)(275 298 351 380)

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

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

180 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B5C5D6A···6G10A···10AB12A···12H15A15B15C15D20A···20P20Q···20AV30A···30AB60A···60AF
order12···2344444···455556···610···1012···121515151520···2020···2030···3060···60
size11···1222226···611112···21···12···222222···26···62···22···2

180 irreducible representations

dim1111111111222222222222222222
type+++++--+-+
imageC1C2C2C4C4C5C10C10C20C20S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4C5×S3C5×D4C5×Q8C5×Dic3S3×C10C5×Dic6S3×C20C5×D12C5×C3⋊D4
kernelC5×C6.C42Dic3×C2×C10C22×C60C10×Dic3C2×C60C6.C42C22×Dic3C22×C12C2×Dic3C2×C12C22×C20C2×C30C2×C30C2×C20C22×C10C2×C10C2×C10C2×C10C2×C10C22×C4C2×C6C2×C6C2×C4C23C22C22C22C22
# reps121844843216131212424412484816816

Matrix representation of C5×C6.C42 in GL5(𝔽61)

10000
09000
00900
000200
000020
,
600000
060000
006000
00010
00001
,
600000
01000
00100
00010
00001
,
10000
060000
006000
000600
000060
,
500000
00100
01000
0005252
000943
,
600000
0483000
0311300
000011
000110

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

׿
×
𝔽