direct product, abelian, monomial, 3-elementary
Aliases: C34×C6, SmallGroup(486,261)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C34×C6 |
| C1 — C34×C6 |
| C1 — C34×C6 |
Generators and relations for C34×C6
G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >
Subgroups: 5328, all normal (4 characteristic)
C1, C2, C3, C6, C32, C3×C6, C33, C32×C6, C34, C33×C6, C35, C34×C6
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, C32×C6, C34, C33×C6, C35, C34×C6
►Regular action on 486 points
Generators in S486
(1 206 47)(2 207 48)(3 208 43)(4 209 44)(5 210 45)(6 205 46)(7 100 172)(8 101 173)(9 102 174)(10 97 169)(11 98 170)(12 99 171)(13 326 408)(14 327 403)(15 328 404)(16 329 405)(17 330 406)(18 325 407)(19 112 184)(20 113 185)(21 114 186)(22 109 181)(23 110 182)(24 111 183)(25 338 420)(26 339 415)(27 340 416)(28 341 417)(29 342 418)(30 337 419)(31 124 196)(32 125 197)(33 126 198)(34 121 193)(35 122 194)(36 123 195)(37 350 432)(38 351 427)(39 352 428)(40 353 429)(41 354 430)(42 349 431)(49 362 444)(50 363 439)(51 364 440)(52 365 441)(53 366 442)(54 361 443)(55 375 220)(56 376 221)(57 377 222)(58 378 217)(59 373 218)(60 374 219)(61 133 456)(62 134 451)(63 135 452)(64 136 453)(65 137 454)(66 138 455)(67 387 232)(68 388 233)(69 389 234)(70 390 229)(71 385 230)(72 386 231)(73 145 468)(74 146 463)(75 147 464)(76 148 465)(77 149 466)(78 150 467)(79 399 483)(80 400 484)(81 401 485)(82 402 486)(83 397 481)(84 398 482)(85 157 480)(86 158 475)(87 159 476)(88 160 477)(89 161 478)(90 162 479)(91 411 252)(92 412 247)(93 413 248)(94 414 249)(95 409 250)(96 410 251)(103 423 264)(104 424 259)(105 425 260)(106 426 261)(107 421 262)(108 422 263)(115 435 276)(116 436 271)(117 437 272)(118 438 273)(119 433 274)(120 434 275)(127 450 285)(128 445 286)(129 446 287)(130 447 288)(131 448 283)(132 449 284)(139 462 297)(140 457 298)(141 458 299)(142 459 300)(143 460 295)(144 461 296)(151 474 309)(152 469 310)(153 470 311)(154 471 312)(155 472 307)(156 473 308)(163 245 321)(164 246 322)(165 241 323)(166 242 324)(167 243 319)(168 244 320)(175 254 333)(176 255 334)(177 256 335)(178 257 336)(179 258 331)(180 253 332)(187 266 345)(188 267 346)(189 268 347)(190 269 348)(191 270 343)(192 265 344)(199 278 357)(200 279 358)(201 280 359)(202 281 360)(203 282 355)(204 277 356)(211 290 372)(212 291 367)(213 292 368)(214 293 369)(215 294 370)(216 289 371)(223 302 384)(224 303 379)(225 304 380)(226 305 381)(227 306 382)(228 301 383)(235 314 396)(236 315 391)(237 316 392)(238 317 393)(239 318 394)(240 313 395)
(1 152 26)(2 153 27)(3 154 28)(4 155 29)(5 156 30)(6 151 25)(7 276 381)(8 271 382)(9 272 383)(10 273 384)(11 274 379)(12 275 380)(13 34 139)(14 35 140)(15 36 141)(16 31 142)(17 32 143)(18 33 144)(19 130 393)(20 131 394)(21 132 395)(22 127 396)(23 128 391)(24 129 392)(37 378 163)(38 373 164)(39 374 165)(40 375 166)(41 376 167)(42 377 168)(43 312 417)(44 307 418)(45 308 419)(46 309 420)(47 310 415)(48 311 416)(49 390 254)(50 385 255)(51 386 256)(52 387 257)(53 388 258)(54 389 253)(55 324 429)(56 319 430)(57 320 431)(58 321 432)(59 322 427)(60 323 428)(61 82 187)(62 83 188)(63 84 189)(64 79 190)(65 80 191)(66 81 192)(67 178 441)(68 179 442)(69 180 443)(70 175 444)(71 176 439)(72 177 440)(73 414 199)(74 409 200)(75 410 201)(76 411 202)(77 412 203)(78 413 204)(85 426 290)(86 421 291)(87 422 292)(88 423 293)(89 424 294)(90 425 289)(91 360 465)(92 355 466)(93 356 467)(94 357 468)(95 358 463)(96 359 464)(97 118 223)(98 119 224)(99 120 225)(100 115 226)(101 116 227)(102 117 228)(103 214 477)(104 215 478)(105 216 479)(106 211 480)(107 212 475)(108 213 476)(109 450 235)(110 445 236)(111 446 237)(112 447 238)(113 448 239)(114 449 240)(121 462 326)(122 457 327)(123 458 328)(124 459 329)(125 460 330)(126 461 325)(133 402 266)(134 397 267)(135 398 268)(136 399 269)(137 400 270)(138 401 265)(145 249 278)(146 250 279)(147 251 280)(148 252 281)(149 247 282)(150 248 277)(157 261 372)(158 262 367)(159 263 368)(160 264 369)(161 259 370)(162 260 371)(169 438 302)(170 433 303)(171 434 304)(172 435 305)(173 436 306)(174 437 301)(181 285 314)(182 286 315)(183 287 316)(184 288 317)(185 283 318)(186 284 313)(193 297 408)(194 298 403)(195 299 404)(196 300 405)(197 295 406)(198 296 407)(205 474 338)(206 469 339)(207 470 340)(208 471 341)(209 472 342)(210 473 337)(217 245 350)(218 246 351)(219 241 352)(220 242 353)(221 243 354)(222 244 349)(229 333 362)(230 334 363)(231 335 364)(232 336 365)(233 331 366)(234 332 361)(343 454 484)(344 455 485)(345 456 486)(346 451 481)(347 452 482)(348 453 483)
(1 134 11)(2 135 12)(3 136 7)(4 137 8)(5 138 9)(6 133 10)(13 254 372)(14 255 367)(15 256 368)(16 257 369)(17 258 370)(18 253 371)(19 40 148)(20 41 149)(21 42 150)(22 37 145)(23 38 146)(24 39 147)(25 266 384)(26 267 379)(27 268 380)(28 269 381)(29 270 382)(30 265 383)(31 52 160)(32 53 161)(33 54 162)(34 49 157)(35 50 158)(36 51 159)(43 64 172)(44 65 173)(45 66 174)(46 61 169)(47 62 170)(48 63 171)(55 411 288)(56 412 283)(57 413 284)(58 414 285)(59 409 286)(60 410 287)(67 423 300)(68 424 295)(69 425 296)(70 426 297)(71 421 298)(72 422 299)(73 181 432)(74 182 427)(75 183 428)(76 184 429)(77 185 430)(78 186 431)(79 435 312)(80 436 307)(81 437 308)(82 438 309)(83 433 310)(84 434 311)(85 193 444)(86 194 439)(87 195 440)(88 196 441)(89 197 442)(90 198 443)(91 447 220)(92 448 221)(93 449 222)(94 450 217)(95 445 218)(96 446 219)(97 205 456)(98 206 451)(99 207 452)(100 208 453)(101 209 454)(102 210 455)(103 459 232)(104 460 233)(105 461 234)(106 462 229)(107 457 230)(108 458 231)(109 350 468)(110 351 463)(111 352 464)(112 353 465)(113 354 466)(114 349 467)(115 471 483)(116 472 484)(117 473 485)(118 474 486)(119 469 481)(120 470 482)(121 362 480)(122 363 475)(123 364 476)(124 365 477)(125 366 478)(126 361 479)(127 378 249)(128 373 250)(129 374 251)(130 375 252)(131 376 247)(132 377 248)(139 390 261)(140 385 262)(141 386 263)(142 387 264)(143 388 259)(144 389 260)(151 402 273)(152 397 274)(153 398 275)(154 399 276)(155 400 271)(156 401 272)(163 278 396)(164 279 391)(165 280 392)(166 281 393)(167 282 394)(168 277 395)(175 290 408)(176 291 403)(177 292 404)(178 293 405)(179 294 406)(180 289 407)(187 302 420)(188 303 415)(189 304 416)(190 305 417)(191 306 418)(192 301 419)(199 314 321)(200 315 322)(201 316 323)(202 317 324)(203 318 319)(204 313 320)(211 326 333)(212 327 334)(213 328 335)(214 329 336)(215 330 331)(216 325 332)(223 338 345)(224 339 346)(225 340 347)(226 341 348)(227 342 343)(228 337 344)(235 245 357)(236 246 358)(237 241 359)(238 242 360)(239 243 355)(240 244 356)
(1 128 14)(2 129 15)(3 130 16)(4 131 17)(5 132 18)(6 127 13)(7 252 369)(8 247 370)(9 248 371)(10 249 372)(11 250 367)(12 251 368)(19 142 28)(20 143 29)(21 144 30)(22 139 25)(23 140 26)(24 141 27)(31 154 393)(32 155 394)(33 156 395)(34 151 396)(35 152 391)(36 153 392)(37 390 266)(38 385 267)(39 386 268)(40 387 269)(41 388 270)(42 389 265)(43 288 405)(44 283 406)(45 284 407)(46 285 408)(47 286 403)(48 287 404)(49 402 163)(50 397 164)(51 398 165)(52 399 166)(53 400 167)(54 401 168)(55 178 64)(56 179 65)(57 180 66)(58 175 61)(59 176 62)(60 177 63)(67 190 429)(68 191 430)(69 192 431)(70 187 432)(71 188 427)(72 189 428)(73 426 302)(74 421 303)(75 422 304)(76 423 305)(77 424 306)(78 425 301)(79 324 441)(80 319 442)(81 320 443)(82 321 444)(83 322 439)(84 323 440)(85 438 199)(86 433 200)(87 434 201)(88 435 202)(89 436 203)(90 437 204)(91 214 100)(92 215 101)(93 216 102)(94 211 97)(95 212 98)(96 213 99)(103 226 465)(104 227 466)(105 228 467)(106 223 468)(107 224 463)(108 225 464)(109 462 338)(110 457 339)(111 458 340)(112 459 341)(113 460 342)(114 461 337)(115 360 477)(116 355 478)(117 356 479)(118 357 480)(119 358 475)(120 359 476)(121 474 235)(122 469 236)(123 470 237)(124 471 238)(125 472 239)(126 473 240)(133 378 254)(134 373 255)(135 374 256)(136 375 257)(137 376 258)(138 377 253)(145 261 384)(146 262 379)(147 263 380)(148 264 381)(149 259 382)(150 260 383)(157 273 278)(158 274 279)(159 275 280)(160 276 281)(161 271 282)(162 272 277)(169 414 290)(170 409 291)(171 410 292)(172 411 293)(173 412 294)(174 413 289)(181 297 420)(182 298 415)(183 299 416)(184 300 417)(185 295 418)(186 296 419)(193 309 314)(194 310 315)(195 311 316)(196 312 317)(197 307 318)(198 308 313)(205 450 326)(206 445 327)(207 446 328)(208 447 329)(209 448 330)(210 449 325)(217 333 456)(218 334 451)(219 335 452)(220 336 453)(221 331 454)(222 332 455)(229 345 350)(230 346 351)(231 347 352)(232 348 353)(233 343 354)(234 344 349)(241 364 482)(242 365 483)(243 366 484)(244 361 485)(245 362 486)(246 363 481)
(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)(481 482 483 484 485 486)
G:=sub<Sym(486)| (1,206,47)(2,207,48)(3,208,43)(4,209,44)(5,210,45)(6,205,46)(7,100,172)(8,101,173)(9,102,174)(10,97,169)(11,98,170)(12,99,171)(13,326,408)(14,327,403)(15,328,404)(16,329,405)(17,330,406)(18,325,407)(19,112,184)(20,113,185)(21,114,186)(22,109,181)(23,110,182)(24,111,183)(25,338,420)(26,339,415)(27,340,416)(28,341,417)(29,342,418)(30,337,419)(31,124,196)(32,125,197)(33,126,198)(34,121,193)(35,122,194)(36,123,195)(37,350,432)(38,351,427)(39,352,428)(40,353,429)(41,354,430)(42,349,431)(49,362,444)(50,363,439)(51,364,440)(52,365,441)(53,366,442)(54,361,443)(55,375,220)(56,376,221)(57,377,222)(58,378,217)(59,373,218)(60,374,219)(61,133,456)(62,134,451)(63,135,452)(64,136,453)(65,137,454)(66,138,455)(67,387,232)(68,388,233)(69,389,234)(70,390,229)(71,385,230)(72,386,231)(73,145,468)(74,146,463)(75,147,464)(76,148,465)(77,149,466)(78,150,467)(79,399,483)(80,400,484)(81,401,485)(82,402,486)(83,397,481)(84,398,482)(85,157,480)(86,158,475)(87,159,476)(88,160,477)(89,161,478)(90,162,479)(91,411,252)(92,412,247)(93,413,248)(94,414,249)(95,409,250)(96,410,251)(103,423,264)(104,424,259)(105,425,260)(106,426,261)(107,421,262)(108,422,263)(115,435,276)(116,436,271)(117,437,272)(118,438,273)(119,433,274)(120,434,275)(127,450,285)(128,445,286)(129,446,287)(130,447,288)(131,448,283)(132,449,284)(139,462,297)(140,457,298)(141,458,299)(142,459,300)(143,460,295)(144,461,296)(151,474,309)(152,469,310)(153,470,311)(154,471,312)(155,472,307)(156,473,308)(163,245,321)(164,246,322)(165,241,323)(166,242,324)(167,243,319)(168,244,320)(175,254,333)(176,255,334)(177,256,335)(178,257,336)(179,258,331)(180,253,332)(187,266,345)(188,267,346)(189,268,347)(190,269,348)(191,270,343)(192,265,344)(199,278,357)(200,279,358)(201,280,359)(202,281,360)(203,282,355)(204,277,356)(211,290,372)(212,291,367)(213,292,368)(214,293,369)(215,294,370)(216,289,371)(223,302,384)(224,303,379)(225,304,380)(226,305,381)(227,306,382)(228,301,383)(235,314,396)(236,315,391)(237,316,392)(238,317,393)(239,318,394)(240,313,395), (1,152,26)(2,153,27)(3,154,28)(4,155,29)(5,156,30)(6,151,25)(7,276,381)(8,271,382)(9,272,383)(10,273,384)(11,274,379)(12,275,380)(13,34,139)(14,35,140)(15,36,141)(16,31,142)(17,32,143)(18,33,144)(19,130,393)(20,131,394)(21,132,395)(22,127,396)(23,128,391)(24,129,392)(37,378,163)(38,373,164)(39,374,165)(40,375,166)(41,376,167)(42,377,168)(43,312,417)(44,307,418)(45,308,419)(46,309,420)(47,310,415)(48,311,416)(49,390,254)(50,385,255)(51,386,256)(52,387,257)(53,388,258)(54,389,253)(55,324,429)(56,319,430)(57,320,431)(58,321,432)(59,322,427)(60,323,428)(61,82,187)(62,83,188)(63,84,189)(64,79,190)(65,80,191)(66,81,192)(67,178,441)(68,179,442)(69,180,443)(70,175,444)(71,176,439)(72,177,440)(73,414,199)(74,409,200)(75,410,201)(76,411,202)(77,412,203)(78,413,204)(85,426,290)(86,421,291)(87,422,292)(88,423,293)(89,424,294)(90,425,289)(91,360,465)(92,355,466)(93,356,467)(94,357,468)(95,358,463)(96,359,464)(97,118,223)(98,119,224)(99,120,225)(100,115,226)(101,116,227)(102,117,228)(103,214,477)(104,215,478)(105,216,479)(106,211,480)(107,212,475)(108,213,476)(109,450,235)(110,445,236)(111,446,237)(112,447,238)(113,448,239)(114,449,240)(121,462,326)(122,457,327)(123,458,328)(124,459,329)(125,460,330)(126,461,325)(133,402,266)(134,397,267)(135,398,268)(136,399,269)(137,400,270)(138,401,265)(145,249,278)(146,250,279)(147,251,280)(148,252,281)(149,247,282)(150,248,277)(157,261,372)(158,262,367)(159,263,368)(160,264,369)(161,259,370)(162,260,371)(169,438,302)(170,433,303)(171,434,304)(172,435,305)(173,436,306)(174,437,301)(181,285,314)(182,286,315)(183,287,316)(184,288,317)(185,283,318)(186,284,313)(193,297,408)(194,298,403)(195,299,404)(196,300,405)(197,295,406)(198,296,407)(205,474,338)(206,469,339)(207,470,340)(208,471,341)(209,472,342)(210,473,337)(217,245,350)(218,246,351)(219,241,352)(220,242,353)(221,243,354)(222,244,349)(229,333,362)(230,334,363)(231,335,364)(232,336,365)(233,331,366)(234,332,361)(343,454,484)(344,455,485)(345,456,486)(346,451,481)(347,452,482)(348,453,483), (1,134,11)(2,135,12)(3,136,7)(4,137,8)(5,138,9)(6,133,10)(13,254,372)(14,255,367)(15,256,368)(16,257,369)(17,258,370)(18,253,371)(19,40,148)(20,41,149)(21,42,150)(22,37,145)(23,38,146)(24,39,147)(25,266,384)(26,267,379)(27,268,380)(28,269,381)(29,270,382)(30,265,383)(31,52,160)(32,53,161)(33,54,162)(34,49,157)(35,50,158)(36,51,159)(43,64,172)(44,65,173)(45,66,174)(46,61,169)(47,62,170)(48,63,171)(55,411,288)(56,412,283)(57,413,284)(58,414,285)(59,409,286)(60,410,287)(67,423,300)(68,424,295)(69,425,296)(70,426,297)(71,421,298)(72,422,299)(73,181,432)(74,182,427)(75,183,428)(76,184,429)(77,185,430)(78,186,431)(79,435,312)(80,436,307)(81,437,308)(82,438,309)(83,433,310)(84,434,311)(85,193,444)(86,194,439)(87,195,440)(88,196,441)(89,197,442)(90,198,443)(91,447,220)(92,448,221)(93,449,222)(94,450,217)(95,445,218)(96,446,219)(97,205,456)(98,206,451)(99,207,452)(100,208,453)(101,209,454)(102,210,455)(103,459,232)(104,460,233)(105,461,234)(106,462,229)(107,457,230)(108,458,231)(109,350,468)(110,351,463)(111,352,464)(112,353,465)(113,354,466)(114,349,467)(115,471,483)(116,472,484)(117,473,485)(118,474,486)(119,469,481)(120,470,482)(121,362,480)(122,363,475)(123,364,476)(124,365,477)(125,366,478)(126,361,479)(127,378,249)(128,373,250)(129,374,251)(130,375,252)(131,376,247)(132,377,248)(139,390,261)(140,385,262)(141,386,263)(142,387,264)(143,388,259)(144,389,260)(151,402,273)(152,397,274)(153,398,275)(154,399,276)(155,400,271)(156,401,272)(163,278,396)(164,279,391)(165,280,392)(166,281,393)(167,282,394)(168,277,395)(175,290,408)(176,291,403)(177,292,404)(178,293,405)(179,294,406)(180,289,407)(187,302,420)(188,303,415)(189,304,416)(190,305,417)(191,306,418)(192,301,419)(199,314,321)(200,315,322)(201,316,323)(202,317,324)(203,318,319)(204,313,320)(211,326,333)(212,327,334)(213,328,335)(214,329,336)(215,330,331)(216,325,332)(223,338,345)(224,339,346)(225,340,347)(226,341,348)(227,342,343)(228,337,344)(235,245,357)(236,246,358)(237,241,359)(238,242,360)(239,243,355)(240,244,356), (1,128,14)(2,129,15)(3,130,16)(4,131,17)(5,132,18)(6,127,13)(7,252,369)(8,247,370)(9,248,371)(10,249,372)(11,250,367)(12,251,368)(19,142,28)(20,143,29)(21,144,30)(22,139,25)(23,140,26)(24,141,27)(31,154,393)(32,155,394)(33,156,395)(34,151,396)(35,152,391)(36,153,392)(37,390,266)(38,385,267)(39,386,268)(40,387,269)(41,388,270)(42,389,265)(43,288,405)(44,283,406)(45,284,407)(46,285,408)(47,286,403)(48,287,404)(49,402,163)(50,397,164)(51,398,165)(52,399,166)(53,400,167)(54,401,168)(55,178,64)(56,179,65)(57,180,66)(58,175,61)(59,176,62)(60,177,63)(67,190,429)(68,191,430)(69,192,431)(70,187,432)(71,188,427)(72,189,428)(73,426,302)(74,421,303)(75,422,304)(76,423,305)(77,424,306)(78,425,301)(79,324,441)(80,319,442)(81,320,443)(82,321,444)(83,322,439)(84,323,440)(85,438,199)(86,433,200)(87,434,201)(88,435,202)(89,436,203)(90,437,204)(91,214,100)(92,215,101)(93,216,102)(94,211,97)(95,212,98)(96,213,99)(103,226,465)(104,227,466)(105,228,467)(106,223,468)(107,224,463)(108,225,464)(109,462,338)(110,457,339)(111,458,340)(112,459,341)(113,460,342)(114,461,337)(115,360,477)(116,355,478)(117,356,479)(118,357,480)(119,358,475)(120,359,476)(121,474,235)(122,469,236)(123,470,237)(124,471,238)(125,472,239)(126,473,240)(133,378,254)(134,373,255)(135,374,256)(136,375,257)(137,376,258)(138,377,253)(145,261,384)(146,262,379)(147,263,380)(148,264,381)(149,259,382)(150,260,383)(157,273,278)(158,274,279)(159,275,280)(160,276,281)(161,271,282)(162,272,277)(169,414,290)(170,409,291)(171,410,292)(172,411,293)(173,412,294)(174,413,289)(181,297,420)(182,298,415)(183,299,416)(184,300,417)(185,295,418)(186,296,419)(193,309,314)(194,310,315)(195,311,316)(196,312,317)(197,307,318)(198,308,313)(205,450,326)(206,445,327)(207,446,328)(208,447,329)(209,448,330)(210,449,325)(217,333,456)(218,334,451)(219,335,452)(220,336,453)(221,331,454)(222,332,455)(229,345,350)(230,346,351)(231,347,352)(232,348,353)(233,343,354)(234,344,349)(241,364,482)(242,365,483)(243,366,484)(244,361,485)(245,362,486)(246,363,481), 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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)(481,482,483,484,485,486) );
G=PermutationGroup([[(1,206,47),(2,207,48),(3,208,43),(4,209,44),(5,210,45),(6,205,46),(7,100,172),(8,101,173),(9,102,174),(10,97,169),(11,98,170),(12,99,171),(13,326,408),(14,327,403),(15,328,404),(16,329,405),(17,330,406),(18,325,407),(19,112,184),(20,113,185),(21,114,186),(22,109,181),(23,110,182),(24,111,183),(25,338,420),(26,339,415),(27,340,416),(28,341,417),(29,342,418),(30,337,419),(31,124,196),(32,125,197),(33,126,198),(34,121,193),(35,122,194),(36,123,195),(37,350,432),(38,351,427),(39,352,428),(40,353,429),(41,354,430),(42,349,431),(49,362,444),(50,363,439),(51,364,440),(52,365,441),(53,366,442),(54,361,443),(55,375,220),(56,376,221),(57,377,222),(58,378,217),(59,373,218),(60,374,219),(61,133,456),(62,134,451),(63,135,452),(64,136,453),(65,137,454),(66,138,455),(67,387,232),(68,388,233),(69,389,234),(70,390,229),(71,385,230),(72,386,231),(73,145,468),(74,146,463),(75,147,464),(76,148,465),(77,149,466),(78,150,467),(79,399,483),(80,400,484),(81,401,485),(82,402,486),(83,397,481),(84,398,482),(85,157,480),(86,158,475),(87,159,476),(88,160,477),(89,161,478),(90,162,479),(91,411,252),(92,412,247),(93,413,248),(94,414,249),(95,409,250),(96,410,251),(103,423,264),(104,424,259),(105,425,260),(106,426,261),(107,421,262),(108,422,263),(115,435,276),(116,436,271),(117,437,272),(118,438,273),(119,433,274),(120,434,275),(127,450,285),(128,445,286),(129,446,287),(130,447,288),(131,448,283),(132,449,284),(139,462,297),(140,457,298),(141,458,299),(142,459,300),(143,460,295),(144,461,296),(151,474,309),(152,469,310),(153,470,311),(154,471,312),(155,472,307),(156,473,308),(163,245,321),(164,246,322),(165,241,323),(166,242,324),(167,243,319),(168,244,320),(175,254,333),(176,255,334),(177,256,335),(178,257,336),(179,258,331),(180,253,332),(187,266,345),(188,267,346),(189,268,347),(190,269,348),(191,270,343),(192,265,344),(199,278,357),(200,279,358),(201,280,359),(202,281,360),(203,282,355),(204,277,356),(211,290,372),(212,291,367),(213,292,368),(214,293,369),(215,294,370),(216,289,371),(223,302,384),(224,303,379),(225,304,380),(226,305,381),(227,306,382),(228,301,383),(235,314,396),(236,315,391),(237,316,392),(238,317,393),(239,318,394),(240,313,395)], [(1,152,26),(2,153,27),(3,154,28),(4,155,29),(5,156,30),(6,151,25),(7,276,381),(8,271,382),(9,272,383),(10,273,384),(11,274,379),(12,275,380),(13,34,139),(14,35,140),(15,36,141),(16,31,142),(17,32,143),(18,33,144),(19,130,393),(20,131,394),(21,132,395),(22,127,396),(23,128,391),(24,129,392),(37,378,163),(38,373,164),(39,374,165),(40,375,166),(41,376,167),(42,377,168),(43,312,417),(44,307,418),(45,308,419),(46,309,420),(47,310,415),(48,311,416),(49,390,254),(50,385,255),(51,386,256),(52,387,257),(53,388,258),(54,389,253),(55,324,429),(56,319,430),(57,320,431),(58,321,432),(59,322,427),(60,323,428),(61,82,187),(62,83,188),(63,84,189),(64,79,190),(65,80,191),(66,81,192),(67,178,441),(68,179,442),(69,180,443),(70,175,444),(71,176,439),(72,177,440),(73,414,199),(74,409,200),(75,410,201),(76,411,202),(77,412,203),(78,413,204),(85,426,290),(86,421,291),(87,422,292),(88,423,293),(89,424,294),(90,425,289),(91,360,465),(92,355,466),(93,356,467),(94,357,468),(95,358,463),(96,359,464),(97,118,223),(98,119,224),(99,120,225),(100,115,226),(101,116,227),(102,117,228),(103,214,477),(104,215,478),(105,216,479),(106,211,480),(107,212,475),(108,213,476),(109,450,235),(110,445,236),(111,446,237),(112,447,238),(113,448,239),(114,449,240),(121,462,326),(122,457,327),(123,458,328),(124,459,329),(125,460,330),(126,461,325),(133,402,266),(134,397,267),(135,398,268),(136,399,269),(137,400,270),(138,401,265),(145,249,278),(146,250,279),(147,251,280),(148,252,281),(149,247,282),(150,248,277),(157,261,372),(158,262,367),(159,263,368),(160,264,369),(161,259,370),(162,260,371),(169,438,302),(170,433,303),(171,434,304),(172,435,305),(173,436,306),(174,437,301),(181,285,314),(182,286,315),(183,287,316),(184,288,317),(185,283,318),(186,284,313),(193,297,408),(194,298,403),(195,299,404),(196,300,405),(197,295,406),(198,296,407),(205,474,338),(206,469,339),(207,470,340),(208,471,341),(209,472,342),(210,473,337),(217,245,350),(218,246,351),(219,241,352),(220,242,353),(221,243,354),(222,244,349),(229,333,362),(230,334,363),(231,335,364),(232,336,365),(233,331,366),(234,332,361),(343,454,484),(344,455,485),(345,456,486),(346,451,481),(347,452,482),(348,453,483)], [(1,134,11),(2,135,12),(3,136,7),(4,137,8),(5,138,9),(6,133,10),(13,254,372),(14,255,367),(15,256,368),(16,257,369),(17,258,370),(18,253,371),(19,40,148),(20,41,149),(21,42,150),(22,37,145),(23,38,146),(24,39,147),(25,266,384),(26,267,379),(27,268,380),(28,269,381),(29,270,382),(30,265,383),(31,52,160),(32,53,161),(33,54,162),(34,49,157),(35,50,158),(36,51,159),(43,64,172),(44,65,173),(45,66,174),(46,61,169),(47,62,170),(48,63,171),(55,411,288),(56,412,283),(57,413,284),(58,414,285),(59,409,286),(60,410,287),(67,423,300),(68,424,295),(69,425,296),(70,426,297),(71,421,298),(72,422,299),(73,181,432),(74,182,427),(75,183,428),(76,184,429),(77,185,430),(78,186,431),(79,435,312),(80,436,307),(81,437,308),(82,438,309),(83,433,310),(84,434,311),(85,193,444),(86,194,439),(87,195,440),(88,196,441),(89,197,442),(90,198,443),(91,447,220),(92,448,221),(93,449,222),(94,450,217),(95,445,218),(96,446,219),(97,205,456),(98,206,451),(99,207,452),(100,208,453),(101,209,454),(102,210,455),(103,459,232),(104,460,233),(105,461,234),(106,462,229),(107,457,230),(108,458,231),(109,350,468),(110,351,463),(111,352,464),(112,353,465),(113,354,466),(114,349,467),(115,471,483),(116,472,484),(117,473,485),(118,474,486),(119,469,481),(120,470,482),(121,362,480),(122,363,475),(123,364,476),(124,365,477),(125,366,478),(126,361,479),(127,378,249),(128,373,250),(129,374,251),(130,375,252),(131,376,247),(132,377,248),(139,390,261),(140,385,262),(141,386,263),(142,387,264),(143,388,259),(144,389,260),(151,402,273),(152,397,274),(153,398,275),(154,399,276),(155,400,271),(156,401,272),(163,278,396),(164,279,391),(165,280,392),(166,281,393),(167,282,394),(168,277,395),(175,290,408),(176,291,403),(177,292,404),(178,293,405),(179,294,406),(180,289,407),(187,302,420),(188,303,415),(189,304,416),(190,305,417),(191,306,418),(192,301,419),(199,314,321),(200,315,322),(201,316,323),(202,317,324),(203,318,319),(204,313,320),(211,326,333),(212,327,334),(213,328,335),(214,329,336),(215,330,331),(216,325,332),(223,338,345),(224,339,346),(225,340,347),(226,341,348),(227,342,343),(228,337,344),(235,245,357),(236,246,358),(237,241,359),(238,242,360),(239,243,355),(240,244,356)], [(1,128,14),(2,129,15),(3,130,16),(4,131,17),(5,132,18),(6,127,13),(7,252,369),(8,247,370),(9,248,371),(10,249,372),(11,250,367),(12,251,368),(19,142,28),(20,143,29),(21,144,30),(22,139,25),(23,140,26),(24,141,27),(31,154,393),(32,155,394),(33,156,395),(34,151,396),(35,152,391),(36,153,392),(37,390,266),(38,385,267),(39,386,268),(40,387,269),(41,388,270),(42,389,265),(43,288,405),(44,283,406),(45,284,407),(46,285,408),(47,286,403),(48,287,404),(49,402,163),(50,397,164),(51,398,165),(52,399,166),(53,400,167),(54,401,168),(55,178,64),(56,179,65),(57,180,66),(58,175,61),(59,176,62),(60,177,63),(67,190,429),(68,191,430),(69,192,431),(70,187,432),(71,188,427),(72,189,428),(73,426,302),(74,421,303),(75,422,304),(76,423,305),(77,424,306),(78,425,301),(79,324,441),(80,319,442),(81,320,443),(82,321,444),(83,322,439),(84,323,440),(85,438,199),(86,433,200),(87,434,201),(88,435,202),(89,436,203),(90,437,204),(91,214,100),(92,215,101),(93,216,102),(94,211,97),(95,212,98),(96,213,99),(103,226,465),(104,227,466),(105,228,467),(106,223,468),(107,224,463),(108,225,464),(109,462,338),(110,457,339),(111,458,340),(112,459,341),(113,460,342),(114,461,337),(115,360,477),(116,355,478),(117,356,479),(118,357,480),(119,358,475),(120,359,476),(121,474,235),(122,469,236),(123,470,237),(124,471,238),(125,472,239),(126,473,240),(133,378,254),(134,373,255),(135,374,256),(136,375,257),(137,376,258),(138,377,253),(145,261,384),(146,262,379),(147,263,380),(148,264,381),(149,259,382),(150,260,383),(157,273,278),(158,274,279),(159,275,280),(160,276,281),(161,271,282),(162,272,277),(169,414,290),(170,409,291),(171,410,292),(172,411,293),(173,412,294),(174,413,289),(181,297,420),(182,298,415),(183,299,416),(184,300,417),(185,295,418),(186,296,419),(193,309,314),(194,310,315),(195,311,316),(196,312,317),(197,307,318),(198,308,313),(205,450,326),(206,445,327),(207,446,328),(208,447,329),(209,448,330),(210,449,325),(217,333,456),(218,334,451),(219,335,452),(220,336,453),(221,331,454),(222,332,455),(229,345,350),(230,346,351),(231,347,352),(232,348,353),(233,343,354),(234,344,349),(241,364,482),(242,365,483),(243,366,484),(244,361,485),(245,362,486),(246,363,481)], [(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),(481,482,483,484,485,486)]])
486 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3IH | 6A | ··· | 6IH |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
486 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C3 | C6 |
| kernel | C34×C6 | C35 | C33×C6 | C34 |
| # reps | 1 | 1 | 242 | 242 |
Matrix representation of C34×C6 ►in GL5(𝔽7)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2],[2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4],[2,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1] >;
C34×C6 in GAP, Magma, Sage, TeX
C_3^4\times C_6
% in TeX
G:=Group("C3^4xC6"); // GroupNames label
G:=SmallGroup(486,261);
// by ID
G=gap.SmallGroup(486,261);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations