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## G = C34×C6order 486 = 2·35

### Abelian group of type [3,3,3,3,6]

Aliases: C34×C6, SmallGroup(486,261)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34×C6
 Chief series C1 — C3 — C32 — C33 — C34 — C35 — C34×C6
 Lower central C1 — C34×C6
 Upper central 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 [×121], C6 [×121], C32 [×1210], C3×C6 [×1210], C33 [×1210], C32×C6 [×1210], C34 [×121], C33×C6 [×121], C35, C34×C6
Quotients: C1, C2, C3 [×121], C6 [×121], C32 [×1210], C3×C6 [×1210], C33 [×1210], C32×C6 [×1210], C34 [×121], C33×C6 [×121], C35, C34×C6

Smallest permutation representation of 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)(7 330 172)(8 325 173)(9 326 174)(10 327 169)(11 328 170)(12 329 171)(13 343 178)(14 344 179)(15 345 180)(16 346 175)(17 347 176)(18 348 177)(19 98 418)(20 99 419)(21 100 420)(22 101 415)(23 102 416)(24 97 417)(25 114 190)(26 109 191)(27 110 192)(28 111 187)(29 112 188)(30 113 189)(31 341 183)(32 342 184)(33 337 185)(34 338 186)(35 339 181)(36 340 182)(37 360 202)(38 355 203)(39 356 204)(40 357 199)(41 358 200)(42 359 201)(49 131 448)(50 132 449)(51 127 450)(52 128 445)(53 129 446)(54 130 447)(55 375 220)(56 376 221)(57 377 222)(58 378 217)(59 373 218)(60 374 219)(61 368 213)(62 369 214)(63 370 215)(64 371 216)(65 372 211)(66 367 212)(67 387 232)(68 388 233)(69 389 234)(70 390 229)(71 385 230)(72 386 231)(73 393 238)(74 394 239)(75 395 240)(76 396 235)(77 391 236)(78 392 237)(79 161 478)(80 162 479)(81 157 480)(82 158 475)(83 159 476)(84 160 477)(85 405 253)(86 406 254)(87 407 255)(88 408 256)(89 403 257)(90 404 258)(91 398 482)(92 399 483)(93 400 484)(94 401 485)(95 402 486)(96 397 481)(103 423 264)(104 424 259)(105 425 260)(106 426 261)(107 421 262)(108 422 263)(115 435 283)(116 436 284)(117 437 285)(118 438 286)(119 433 287)(120 434 288)(121 428 269)(122 429 270)(123 430 265)(124 431 266)(125 432 267)(126 427 268)(133 456 291)(134 451 292)(135 452 293)(136 453 294)(137 454 289)(138 455 290)(139 462 297)(140 457 298)(141 458 299)(142 459 300)(143 460 295)(144 461 296)(145 468 316)(146 463 317)(147 464 318)(148 465 313)(149 466 314)(150 467 315)(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)(193 272 351)(194 273 352)(195 274 353)(196 275 354)(197 276 349)(198 271 350)(223 302 384)(224 303 379)(225 304 380)(226 305 381)(227 306 382)(228 301 383)(247 336 412)(248 331 413)(249 332 414)(250 333 409)(251 334 410)(252 335 411)(277 366 442)(278 361 443)(279 362 444)(280 363 439)(281 364 440)(282 365 441)
(1 152 22)(2 153 23)(3 154 24)(4 155 19)(5 156 20)(6 151 21)(7 283 381)(8 284 382)(9 285 383)(10 286 384)(11 287 379)(12 288 380)(13 282 387)(14 277 388)(15 278 389)(16 279 390)(17 280 385)(18 281 386)(25 136 399)(26 137 400)(27 138 401)(28 133 402)(29 134 397)(30 135 398)(31 129 392)(32 130 393)(33 131 394)(34 132 395)(35 127 396)(36 128 391)(37 148 252)(38 149 247)(39 150 248)(40 145 249)(41 146 250)(42 147 251)(43 312 417)(44 307 418)(45 308 419)(46 309 420)(47 310 415)(48 311 416)(49 74 185)(50 75 186)(51 76 181)(52 77 182)(53 78 183)(54 73 184)(55 324 429)(56 319 430)(57 320 431)(58 321 432)(59 322 427)(60 323 428)(61 83 422)(62 84 423)(63 79 424)(64 80 425)(65 81 426)(66 82 421)(67 178 441)(68 179 442)(69 180 443)(70 175 444)(71 176 439)(72 177 440)(85 354 459)(86 349 460)(87 350 461)(88 351 462)(89 352 457)(90 353 458)(91 189 293)(92 190 294)(93 191 289)(94 192 290)(95 187 291)(96 188 292)(97 208 471)(98 209 472)(99 210 473)(100 205 474)(101 206 469)(102 207 470)(103 214 477)(104 215 478)(105 216 479)(106 211 480)(107 212 475)(108 213 476)(109 454 484)(110 455 485)(111 456 486)(112 451 481)(113 452 482)(114 453 483)(115 226 330)(116 227 325)(117 228 326)(118 223 327)(119 224 328)(120 225 329)(121 219 241)(122 220 242)(123 221 243)(124 222 244)(125 217 245)(126 218 246)(139 256 272)(140 257 273)(141 258 274)(142 253 275)(143 254 276)(144 255 271)(157 261 372)(158 262 367)(159 263 368)(160 264 369)(161 259 370)(162 260 371)(163 267 378)(164 268 373)(165 269 374)(166 270 375)(167 265 376)(168 266 377)(169 438 302)(170 433 303)(171 434 304)(172 435 305)(173 436 306)(174 437 301)(193 297 408)(194 298 403)(195 299 404)(196 300 405)(197 295 406)(198 296 407)(199 316 414)(200 317 409)(201 318 410)(202 313 411)(203 314 412)(204 315 413)(229 346 362)(230 347 363)(231 348 364)(232 343 365)(233 344 366)(234 345 361)(235 339 450)(236 340 445)(237 341 446)(238 342 447)(239 337 448)(240 338 449)(331 356 467)(332 357 468)(333 358 463)(334 359 464)(335 360 465)(336 355 466)
(1 134 11)(2 135 12)(3 136 7)(4 137 8)(5 138 9)(6 133 10)(13 369 253)(14 370 254)(15 371 255)(16 372 256)(17 367 257)(18 368 258)(19 26 382)(20 27 383)(21 28 384)(22 29 379)(23 30 380)(24 25 381)(31 269 147)(32 270 148)(33 265 149)(34 266 150)(35 267 145)(36 268 146)(37 393 166)(38 394 167)(39 395 168)(40 396 163)(41 391 164)(42 392 165)(43 294 172)(44 289 173)(45 290 174)(46 291 169)(47 292 170)(48 293 171)(49 56 412)(50 57 413)(51 58 414)(52 59 409)(53 60 410)(54 55 411)(61 404 177)(62 405 178)(63 406 179)(64 407 180)(65 408 175)(66 403 176)(67 423 300)(68 424 295)(69 425 296)(70 426 297)(71 421 298)(72 422 299)(73 324 202)(74 319 203)(75 320 204)(76 321 199)(77 322 200)(78 323 201)(79 197 442)(80 198 443)(81 193 444)(82 194 439)(83 195 440)(84 196 441)(85 343 214)(86 344 215)(87 345 216)(88 346 211)(89 347 212)(90 348 213)(91 434 311)(92 435 312)(93 436 307)(94 437 308)(95 438 309)(96 433 310)(97 114 226)(98 109 227)(99 110 228)(100 111 223)(101 112 224)(102 113 225)(103 459 232)(104 460 233)(105 461 234)(106 462 229)(107 457 230)(108 458 231)(115 471 483)(116 472 484)(117 473 485)(118 474 486)(119 469 481)(120 470 482)(121 464 341)(122 465 342)(123 466 337)(124 467 338)(125 468 339)(126 463 340)(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)(151 402 286)(152 397 287)(153 398 288)(154 399 283)(155 400 284)(156 401 285)(157 272 279)(158 273 280)(159 274 281)(160 275 282)(161 276 277)(162 271 278)(181 432 316)(182 427 317)(183 428 318)(184 429 313)(185 430 314)(186 431 315)(187 302 420)(188 303 415)(189 304 416)(190 305 417)(191 306 418)(192 301 419)(205 456 327)(206 451 328)(207 452 329)(208 453 330)(209 454 325)(210 455 326)(217 332 450)(218 333 445)(219 334 446)(220 335 447)(221 336 448)(222 331 449)(235 245 357)(236 246 358)(237 241 359)(238 242 360)(239 243 355)(240 244 356)(349 366 478)(350 361 479)(351 362 480)(352 363 475)(353 364 476)(354 365 477)
(1 128 257)(2 129 258)(3 130 253)(4 131 254)(5 132 255)(6 127 256)(7 252 369)(8 247 370)(9 248 371)(10 249 372)(11 250 367)(12 251 368)(13 136 375)(14 137 376)(15 138 377)(16 133 378)(17 134 373)(18 135 374)(19 33 143)(20 34 144)(21 35 139)(22 36 140)(23 31 141)(24 32 142)(25 270 387)(26 265 388)(27 266 389)(28 267 390)(29 268 385)(30 269 386)(37 160 283)(38 161 284)(39 162 285)(40 157 286)(41 158 287)(42 159 288)(43 54 405)(44 49 406)(45 50 407)(46 51 408)(47 52 403)(48 53 404)(55 178 294)(56 179 289)(57 180 290)(58 175 291)(59 176 292)(60 177 293)(61 171 410)(62 172 411)(63 173 412)(64 174 413)(65 169 414)(66 170 409)(67 190 429)(68 191 430)(69 192 431)(70 187 432)(71 188 427)(72 189 428)(73 196 312)(74 197 307)(75 198 308)(76 193 309)(77 194 310)(78 195 311)(79 436 203)(80 437 204)(81 438 199)(82 433 200)(83 434 201)(84 435 202)(85 208 447)(86 209 448)(87 210 449)(88 205 450)(89 206 445)(90 207 446)(91 323 440)(92 324 441)(93 319 442)(94 320 443)(95 321 444)(96 322 439)(97 342 459)(98 337 460)(99 338 461)(100 339 462)(101 340 457)(102 341 458)(103 226 465)(104 227 466)(105 228 467)(106 223 468)(107 224 463)(108 225 464)(109 123 233)(110 124 234)(111 125 229)(112 126 230)(113 121 231)(114 122 232)(115 360 477)(116 355 478)(117 356 479)(118 357 480)(119 358 475)(120 359 476)(145 261 384)(146 262 379)(147 263 380)(148 264 381)(149 259 382)(150 260 383)(151 396 272)(152 391 273)(153 392 274)(154 393 275)(155 394 276)(156 395 271)(163 279 402)(164 280 397)(165 281 398)(166 282 399)(167 277 400)(168 278 401)(181 297 420)(182 298 415)(183 299 416)(184 300 417)(185 295 418)(186 296 419)(211 327 332)(212 328 333)(213 329 334)(214 330 335)(215 325 336)(216 326 331)(217 346 456)(218 347 451)(219 348 452)(220 343 453)(221 344 454)(222 345 455)(235 351 474)(236 352 469)(237 353 470)(238 354 471)(239 349 472)(240 350 473)(241 364 482)(242 365 483)(243 366 484)(244 361 485)(245 362 486)(246 363 481)(301 315 425)(302 316 426)(303 317 421)(304 318 422)(305 313 423)(306 314 424)
(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)(7,330,172)(8,325,173)(9,326,174)(10,327,169)(11,328,170)(12,329,171)(13,343,178)(14,344,179)(15,345,180)(16,346,175)(17,347,176)(18,348,177)(19,98,418)(20,99,419)(21,100,420)(22,101,415)(23,102,416)(24,97,417)(25,114,190)(26,109,191)(27,110,192)(28,111,187)(29,112,188)(30,113,189)(31,341,183)(32,342,184)(33,337,185)(34,338,186)(35,339,181)(36,340,182)(37,360,202)(38,355,203)(39,356,204)(40,357,199)(41,358,200)(42,359,201)(49,131,448)(50,132,449)(51,127,450)(52,128,445)(53,129,446)(54,130,447)(55,375,220)(56,376,221)(57,377,222)(58,378,217)(59,373,218)(60,374,219)(61,368,213)(62,369,214)(63,370,215)(64,371,216)(65,372,211)(66,367,212)(67,387,232)(68,388,233)(69,389,234)(70,390,229)(71,385,230)(72,386,231)(73,393,238)(74,394,239)(75,395,240)(76,396,235)(77,391,236)(78,392,237)(79,161,478)(80,162,479)(81,157,480)(82,158,475)(83,159,476)(84,160,477)(85,405,253)(86,406,254)(87,407,255)(88,408,256)(89,403,257)(90,404,258)(91,398,482)(92,399,483)(93,400,484)(94,401,485)(95,402,486)(96,397,481)(103,423,264)(104,424,259)(105,425,260)(106,426,261)(107,421,262)(108,422,263)(115,435,283)(116,436,284)(117,437,285)(118,438,286)(119,433,287)(120,434,288)(121,428,269)(122,429,270)(123,430,265)(124,431,266)(125,432,267)(126,427,268)(133,456,291)(134,451,292)(135,452,293)(136,453,294)(137,454,289)(138,455,290)(139,462,297)(140,457,298)(141,458,299)(142,459,300)(143,460,295)(144,461,296)(145,468,316)(146,463,317)(147,464,318)(148,465,313)(149,466,314)(150,467,315)(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)(193,272,351)(194,273,352)(195,274,353)(196,275,354)(197,276,349)(198,271,350)(223,302,384)(224,303,379)(225,304,380)(226,305,381)(227,306,382)(228,301,383)(247,336,412)(248,331,413)(249,332,414)(250,333,409)(251,334,410)(252,335,411)(277,366,442)(278,361,443)(279,362,444)(280,363,439)(281,364,440)(282,365,441), (1,152,22)(2,153,23)(3,154,24)(4,155,19)(5,156,20)(6,151,21)(7,283,381)(8,284,382)(9,285,383)(10,286,384)(11,287,379)(12,288,380)(13,282,387)(14,277,388)(15,278,389)(16,279,390)(17,280,385)(18,281,386)(25,136,399)(26,137,400)(27,138,401)(28,133,402)(29,134,397)(30,135,398)(31,129,392)(32,130,393)(33,131,394)(34,132,395)(35,127,396)(36,128,391)(37,148,252)(38,149,247)(39,150,248)(40,145,249)(41,146,250)(42,147,251)(43,312,417)(44,307,418)(45,308,419)(46,309,420)(47,310,415)(48,311,416)(49,74,185)(50,75,186)(51,76,181)(52,77,182)(53,78,183)(54,73,184)(55,324,429)(56,319,430)(57,320,431)(58,321,432)(59,322,427)(60,323,428)(61,83,422)(62,84,423)(63,79,424)(64,80,425)(65,81,426)(66,82,421)(67,178,441)(68,179,442)(69,180,443)(70,175,444)(71,176,439)(72,177,440)(85,354,459)(86,349,460)(87,350,461)(88,351,462)(89,352,457)(90,353,458)(91,189,293)(92,190,294)(93,191,289)(94,192,290)(95,187,291)(96,188,292)(97,208,471)(98,209,472)(99,210,473)(100,205,474)(101,206,469)(102,207,470)(103,214,477)(104,215,478)(105,216,479)(106,211,480)(107,212,475)(108,213,476)(109,454,484)(110,455,485)(111,456,486)(112,451,481)(113,452,482)(114,453,483)(115,226,330)(116,227,325)(117,228,326)(118,223,327)(119,224,328)(120,225,329)(121,219,241)(122,220,242)(123,221,243)(124,222,244)(125,217,245)(126,218,246)(139,256,272)(140,257,273)(141,258,274)(142,253,275)(143,254,276)(144,255,271)(157,261,372)(158,262,367)(159,263,368)(160,264,369)(161,259,370)(162,260,371)(163,267,378)(164,268,373)(165,269,374)(166,270,375)(167,265,376)(168,266,377)(169,438,302)(170,433,303)(171,434,304)(172,435,305)(173,436,306)(174,437,301)(193,297,408)(194,298,403)(195,299,404)(196,300,405)(197,295,406)(198,296,407)(199,316,414)(200,317,409)(201,318,410)(202,313,411)(203,314,412)(204,315,413)(229,346,362)(230,347,363)(231,348,364)(232,343,365)(233,344,366)(234,345,361)(235,339,450)(236,340,445)(237,341,446)(238,342,447)(239,337,448)(240,338,449)(331,356,467)(332,357,468)(333,358,463)(334,359,464)(335,360,465)(336,355,466), (1,134,11)(2,135,12)(3,136,7)(4,137,8)(5,138,9)(6,133,10)(13,369,253)(14,370,254)(15,371,255)(16,372,256)(17,367,257)(18,368,258)(19,26,382)(20,27,383)(21,28,384)(22,29,379)(23,30,380)(24,25,381)(31,269,147)(32,270,148)(33,265,149)(34,266,150)(35,267,145)(36,268,146)(37,393,166)(38,394,167)(39,395,168)(40,396,163)(41,391,164)(42,392,165)(43,294,172)(44,289,173)(45,290,174)(46,291,169)(47,292,170)(48,293,171)(49,56,412)(50,57,413)(51,58,414)(52,59,409)(53,60,410)(54,55,411)(61,404,177)(62,405,178)(63,406,179)(64,407,180)(65,408,175)(66,403,176)(67,423,300)(68,424,295)(69,425,296)(70,426,297)(71,421,298)(72,422,299)(73,324,202)(74,319,203)(75,320,204)(76,321,199)(77,322,200)(78,323,201)(79,197,442)(80,198,443)(81,193,444)(82,194,439)(83,195,440)(84,196,441)(85,343,214)(86,344,215)(87,345,216)(88,346,211)(89,347,212)(90,348,213)(91,434,311)(92,435,312)(93,436,307)(94,437,308)(95,438,309)(96,433,310)(97,114,226)(98,109,227)(99,110,228)(100,111,223)(101,112,224)(102,113,225)(103,459,232)(104,460,233)(105,461,234)(106,462,229)(107,457,230)(108,458,231)(115,471,483)(116,472,484)(117,473,485)(118,474,486)(119,469,481)(120,470,482)(121,464,341)(122,465,342)(123,466,337)(124,467,338)(125,468,339)(126,463,340)(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)(151,402,286)(152,397,287)(153,398,288)(154,399,283)(155,400,284)(156,401,285)(157,272,279)(158,273,280)(159,274,281)(160,275,282)(161,276,277)(162,271,278)(181,432,316)(182,427,317)(183,428,318)(184,429,313)(185,430,314)(186,431,315)(187,302,420)(188,303,415)(189,304,416)(190,305,417)(191,306,418)(192,301,419)(205,456,327)(206,451,328)(207,452,329)(208,453,330)(209,454,325)(210,455,326)(217,332,450)(218,333,445)(219,334,446)(220,335,447)(221,336,448)(222,331,449)(235,245,357)(236,246,358)(237,241,359)(238,242,360)(239,243,355)(240,244,356)(349,366,478)(350,361,479)(351,362,480)(352,363,475)(353,364,476)(354,365,477), (1,128,257)(2,129,258)(3,130,253)(4,131,254)(5,132,255)(6,127,256)(7,252,369)(8,247,370)(9,248,371)(10,249,372)(11,250,367)(12,251,368)(13,136,375)(14,137,376)(15,138,377)(16,133,378)(17,134,373)(18,135,374)(19,33,143)(20,34,144)(21,35,139)(22,36,140)(23,31,141)(24,32,142)(25,270,387)(26,265,388)(27,266,389)(28,267,390)(29,268,385)(30,269,386)(37,160,283)(38,161,284)(39,162,285)(40,157,286)(41,158,287)(42,159,288)(43,54,405)(44,49,406)(45,50,407)(46,51,408)(47,52,403)(48,53,404)(55,178,294)(56,179,289)(57,180,290)(58,175,291)(59,176,292)(60,177,293)(61,171,410)(62,172,411)(63,173,412)(64,174,413)(65,169,414)(66,170,409)(67,190,429)(68,191,430)(69,192,431)(70,187,432)(71,188,427)(72,189,428)(73,196,312)(74,197,307)(75,198,308)(76,193,309)(77,194,310)(78,195,311)(79,436,203)(80,437,204)(81,438,199)(82,433,200)(83,434,201)(84,435,202)(85,208,447)(86,209,448)(87,210,449)(88,205,450)(89,206,445)(90,207,446)(91,323,440)(92,324,441)(93,319,442)(94,320,443)(95,321,444)(96,322,439)(97,342,459)(98,337,460)(99,338,461)(100,339,462)(101,340,457)(102,341,458)(103,226,465)(104,227,466)(105,228,467)(106,223,468)(107,224,463)(108,225,464)(109,123,233)(110,124,234)(111,125,229)(112,126,230)(113,121,231)(114,122,232)(115,360,477)(116,355,478)(117,356,479)(118,357,480)(119,358,475)(120,359,476)(145,261,384)(146,262,379)(147,263,380)(148,264,381)(149,259,382)(150,260,383)(151,396,272)(152,391,273)(153,392,274)(154,393,275)(155,394,276)(156,395,271)(163,279,402)(164,280,397)(165,281,398)(166,282,399)(167,277,400)(168,278,401)(181,297,420)(182,298,415)(183,299,416)(184,300,417)(185,295,418)(186,296,419)(211,327,332)(212,328,333)(213,329,334)(214,330,335)(215,325,336)(216,326,331)(217,346,456)(218,347,451)(219,348,452)(220,343,453)(221,344,454)(222,345,455)(235,351,474)(236,352,469)(237,353,470)(238,354,471)(239,349,472)(240,350,473)(241,364,482)(242,365,483)(243,366,484)(244,361,485)(245,362,486)(246,363,481)(301,315,425)(302,316,426)(303,317,421)(304,318,422)(305,313,423)(306,314,424), 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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),(7,330,172),(8,325,173),(9,326,174),(10,327,169),(11,328,170),(12,329,171),(13,343,178),(14,344,179),(15,345,180),(16,346,175),(17,347,176),(18,348,177),(19,98,418),(20,99,419),(21,100,420),(22,101,415),(23,102,416),(24,97,417),(25,114,190),(26,109,191),(27,110,192),(28,111,187),(29,112,188),(30,113,189),(31,341,183),(32,342,184),(33,337,185),(34,338,186),(35,339,181),(36,340,182),(37,360,202),(38,355,203),(39,356,204),(40,357,199),(41,358,200),(42,359,201),(49,131,448),(50,132,449),(51,127,450),(52,128,445),(53,129,446),(54,130,447),(55,375,220),(56,376,221),(57,377,222),(58,378,217),(59,373,218),(60,374,219),(61,368,213),(62,369,214),(63,370,215),(64,371,216),(65,372,211),(66,367,212),(67,387,232),(68,388,233),(69,389,234),(70,390,229),(71,385,230),(72,386,231),(73,393,238),(74,394,239),(75,395,240),(76,396,235),(77,391,236),(78,392,237),(79,161,478),(80,162,479),(81,157,480),(82,158,475),(83,159,476),(84,160,477),(85,405,253),(86,406,254),(87,407,255),(88,408,256),(89,403,257),(90,404,258),(91,398,482),(92,399,483),(93,400,484),(94,401,485),(95,402,486),(96,397,481),(103,423,264),(104,424,259),(105,425,260),(106,426,261),(107,421,262),(108,422,263),(115,435,283),(116,436,284),(117,437,285),(118,438,286),(119,433,287),(120,434,288),(121,428,269),(122,429,270),(123,430,265),(124,431,266),(125,432,267),(126,427,268),(133,456,291),(134,451,292),(135,452,293),(136,453,294),(137,454,289),(138,455,290),(139,462,297),(140,457,298),(141,458,299),(142,459,300),(143,460,295),(144,461,296),(145,468,316),(146,463,317),(147,464,318),(148,465,313),(149,466,314),(150,467,315),(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),(193,272,351),(194,273,352),(195,274,353),(196,275,354),(197,276,349),(198,271,350),(223,302,384),(224,303,379),(225,304,380),(226,305,381),(227,306,382),(228,301,383),(247,336,412),(248,331,413),(249,332,414),(250,333,409),(251,334,410),(252,335,411),(277,366,442),(278,361,443),(279,362,444),(280,363,439),(281,364,440),(282,365,441)], [(1,152,22),(2,153,23),(3,154,24),(4,155,19),(5,156,20),(6,151,21),(7,283,381),(8,284,382),(9,285,383),(10,286,384),(11,287,379),(12,288,380),(13,282,387),(14,277,388),(15,278,389),(16,279,390),(17,280,385),(18,281,386),(25,136,399),(26,137,400),(27,138,401),(28,133,402),(29,134,397),(30,135,398),(31,129,392),(32,130,393),(33,131,394),(34,132,395),(35,127,396),(36,128,391),(37,148,252),(38,149,247),(39,150,248),(40,145,249),(41,146,250),(42,147,251),(43,312,417),(44,307,418),(45,308,419),(46,309,420),(47,310,415),(48,311,416),(49,74,185),(50,75,186),(51,76,181),(52,77,182),(53,78,183),(54,73,184),(55,324,429),(56,319,430),(57,320,431),(58,321,432),(59,322,427),(60,323,428),(61,83,422),(62,84,423),(63,79,424),(64,80,425),(65,81,426),(66,82,421),(67,178,441),(68,179,442),(69,180,443),(70,175,444),(71,176,439),(72,177,440),(85,354,459),(86,349,460),(87,350,461),(88,351,462),(89,352,457),(90,353,458),(91,189,293),(92,190,294),(93,191,289),(94,192,290),(95,187,291),(96,188,292),(97,208,471),(98,209,472),(99,210,473),(100,205,474),(101,206,469),(102,207,470),(103,214,477),(104,215,478),(105,216,479),(106,211,480),(107,212,475),(108,213,476),(109,454,484),(110,455,485),(111,456,486),(112,451,481),(113,452,482),(114,453,483),(115,226,330),(116,227,325),(117,228,326),(118,223,327),(119,224,328),(120,225,329),(121,219,241),(122,220,242),(123,221,243),(124,222,244),(125,217,245),(126,218,246),(139,256,272),(140,257,273),(141,258,274),(142,253,275),(143,254,276),(144,255,271),(157,261,372),(158,262,367),(159,263,368),(160,264,369),(161,259,370),(162,260,371),(163,267,378),(164,268,373),(165,269,374),(166,270,375),(167,265,376),(168,266,377),(169,438,302),(170,433,303),(171,434,304),(172,435,305),(173,436,306),(174,437,301),(193,297,408),(194,298,403),(195,299,404),(196,300,405),(197,295,406),(198,296,407),(199,316,414),(200,317,409),(201,318,410),(202,313,411),(203,314,412),(204,315,413),(229,346,362),(230,347,363),(231,348,364),(232,343,365),(233,344,366),(234,345,361),(235,339,450),(236,340,445),(237,341,446),(238,342,447),(239,337,448),(240,338,449),(331,356,467),(332,357,468),(333,358,463),(334,359,464),(335,360,465),(336,355,466)], [(1,134,11),(2,135,12),(3,136,7),(4,137,8),(5,138,9),(6,133,10),(13,369,253),(14,370,254),(15,371,255),(16,372,256),(17,367,257),(18,368,258),(19,26,382),(20,27,383),(21,28,384),(22,29,379),(23,30,380),(24,25,381),(31,269,147),(32,270,148),(33,265,149),(34,266,150),(35,267,145),(36,268,146),(37,393,166),(38,394,167),(39,395,168),(40,396,163),(41,391,164),(42,392,165),(43,294,172),(44,289,173),(45,290,174),(46,291,169),(47,292,170),(48,293,171),(49,56,412),(50,57,413),(51,58,414),(52,59,409),(53,60,410),(54,55,411),(61,404,177),(62,405,178),(63,406,179),(64,407,180),(65,408,175),(66,403,176),(67,423,300),(68,424,295),(69,425,296),(70,426,297),(71,421,298),(72,422,299),(73,324,202),(74,319,203),(75,320,204),(76,321,199),(77,322,200),(78,323,201),(79,197,442),(80,198,443),(81,193,444),(82,194,439),(83,195,440),(84,196,441),(85,343,214),(86,344,215),(87,345,216),(88,346,211),(89,347,212),(90,348,213),(91,434,311),(92,435,312),(93,436,307),(94,437,308),(95,438,309),(96,433,310),(97,114,226),(98,109,227),(99,110,228),(100,111,223),(101,112,224),(102,113,225),(103,459,232),(104,460,233),(105,461,234),(106,462,229),(107,457,230),(108,458,231),(115,471,483),(116,472,484),(117,473,485),(118,474,486),(119,469,481),(120,470,482),(121,464,341),(122,465,342),(123,466,337),(124,467,338),(125,468,339),(126,463,340),(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),(151,402,286),(152,397,287),(153,398,288),(154,399,283),(155,400,284),(156,401,285),(157,272,279),(158,273,280),(159,274,281),(160,275,282),(161,276,277),(162,271,278),(181,432,316),(182,427,317),(183,428,318),(184,429,313),(185,430,314),(186,431,315),(187,302,420),(188,303,415),(189,304,416),(190,305,417),(191,306,418),(192,301,419),(205,456,327),(206,451,328),(207,452,329),(208,453,330),(209,454,325),(210,455,326),(217,332,450),(218,333,445),(219,334,446),(220,335,447),(221,336,448),(222,331,449),(235,245,357),(236,246,358),(237,241,359),(238,242,360),(239,243,355),(240,244,356),(349,366,478),(350,361,479),(351,362,480),(352,363,475),(353,364,476),(354,365,477)], [(1,128,257),(2,129,258),(3,130,253),(4,131,254),(5,132,255),(6,127,256),(7,252,369),(8,247,370),(9,248,371),(10,249,372),(11,250,367),(12,251,368),(13,136,375),(14,137,376),(15,138,377),(16,133,378),(17,134,373),(18,135,374),(19,33,143),(20,34,144),(21,35,139),(22,36,140),(23,31,141),(24,32,142),(25,270,387),(26,265,388),(27,266,389),(28,267,390),(29,268,385),(30,269,386),(37,160,283),(38,161,284),(39,162,285),(40,157,286),(41,158,287),(42,159,288),(43,54,405),(44,49,406),(45,50,407),(46,51,408),(47,52,403),(48,53,404),(55,178,294),(56,179,289),(57,180,290),(58,175,291),(59,176,292),(60,177,293),(61,171,410),(62,172,411),(63,173,412),(64,174,413),(65,169,414),(66,170,409),(67,190,429),(68,191,430),(69,192,431),(70,187,432),(71,188,427),(72,189,428),(73,196,312),(74,197,307),(75,198,308),(76,193,309),(77,194,310),(78,195,311),(79,436,203),(80,437,204),(81,438,199),(82,433,200),(83,434,201),(84,435,202),(85,208,447),(86,209,448),(87,210,449),(88,205,450),(89,206,445),(90,207,446),(91,323,440),(92,324,441),(93,319,442),(94,320,443),(95,321,444),(96,322,439),(97,342,459),(98,337,460),(99,338,461),(100,339,462),(101,340,457),(102,341,458),(103,226,465),(104,227,466),(105,228,467),(106,223,468),(107,224,463),(108,225,464),(109,123,233),(110,124,234),(111,125,229),(112,126,230),(113,121,231),(114,122,232),(115,360,477),(116,355,478),(117,356,479),(118,357,480),(119,358,475),(120,359,476),(145,261,384),(146,262,379),(147,263,380),(148,264,381),(149,259,382),(150,260,383),(151,396,272),(152,391,273),(153,392,274),(154,393,275),(155,394,276),(156,395,271),(163,279,402),(164,280,397),(165,281,398),(166,282,399),(167,277,400),(168,278,401),(181,297,420),(182,298,415),(183,299,416),(184,300,417),(185,295,418),(186,296,419),(211,327,332),(212,328,333),(213,329,334),(214,330,335),(215,325,336),(216,326,331),(217,346,456),(218,347,451),(219,348,452),(220,343,453),(221,344,454),(222,345,455),(235,351,474),(236,352,469),(237,353,470),(238,354,471),(239,349,472),(240,350,473),(241,364,482),(242,365,483),(243,366,484),(244,361,485),(245,362,486),(246,363,481),(301,315,425),(302,316,426),(303,317,421),(304,318,422),(305,313,423),(306,314,424)], [(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

׿
×
𝔽