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G = C32×C51order 459 = 33·17

Abelian group of type [3,3,51]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C51, SmallGroup(459,5)

Series: Derived Chief Lower central Upper central

C1 — C32×C51
C1C17C51C3×C51 — C32×C51
C1 — C32×C51
C1 — C32×C51

Generators and relations for C32×C51
 G = < a,b,c | a3=b3=c51=1, ab=ba, ac=ca, bc=cb >

Subgroups: 56, all normal (4 characteristic)
C1, C3 [×13], C32 [×13], C17, C33, C51 [×13], C3×C51 [×13], C32×C51
Quotients: C1, C3 [×13], C32 [×13], C17, C33, C51 [×13], C3×C51 [×13], C32×C51

Smallest permutation representation of C32×C51
Regular action on 459 points
Generators in S459
(1 340 189)(2 341 190)(3 342 191)(4 343 192)(5 344 193)(6 345 194)(7 346 195)(8 347 196)(9 348 197)(10 349 198)(11 350 199)(12 351 200)(13 352 201)(14 353 202)(15 354 203)(16 355 204)(17 356 154)(18 357 155)(19 307 156)(20 308 157)(21 309 158)(22 310 159)(23 311 160)(24 312 161)(25 313 162)(26 314 163)(27 315 164)(28 316 165)(29 317 166)(30 318 167)(31 319 168)(32 320 169)(33 321 170)(34 322 171)(35 323 172)(36 324 173)(37 325 174)(38 326 175)(39 327 176)(40 328 177)(41 329 178)(42 330 179)(43 331 180)(44 332 181)(45 333 182)(46 334 183)(47 335 184)(48 336 185)(49 337 186)(50 338 187)(51 339 188)(52 362 227)(53 363 228)(54 364 229)(55 365 230)(56 366 231)(57 367 232)(58 368 233)(59 369 234)(60 370 235)(61 371 236)(62 372 237)(63 373 238)(64 374 239)(65 375 240)(66 376 241)(67 377 242)(68 378 243)(69 379 244)(70 380 245)(71 381 246)(72 382 247)(73 383 248)(74 384 249)(75 385 250)(76 386 251)(77 387 252)(78 388 253)(79 389 254)(80 390 255)(81 391 205)(82 392 206)(83 393 207)(84 394 208)(85 395 209)(86 396 210)(87 397 211)(88 398 212)(89 399 213)(90 400 214)(91 401 215)(92 402 216)(93 403 217)(94 404 218)(95 405 219)(96 406 220)(97 407 221)(98 408 222)(99 358 223)(100 359 224)(101 360 225)(102 361 226)(103 411 268)(104 412 269)(105 413 270)(106 414 271)(107 415 272)(108 416 273)(109 417 274)(110 418 275)(111 419 276)(112 420 277)(113 421 278)(114 422 279)(115 423 280)(116 424 281)(117 425 282)(118 426 283)(119 427 284)(120 428 285)(121 429 286)(122 430 287)(123 431 288)(124 432 289)(125 433 290)(126 434 291)(127 435 292)(128 436 293)(129 437 294)(130 438 295)(131 439 296)(132 440 297)(133 441 298)(134 442 299)(135 443 300)(136 444 301)(137 445 302)(138 446 303)(139 447 304)(140 448 305)(141 449 306)(142 450 256)(143 451 257)(144 452 258)(145 453 259)(146 454 260)(147 455 261)(148 456 262)(149 457 263)(150 458 264)(151 459 265)(152 409 266)(153 410 267)
(1 142 79)(2 143 80)(3 144 81)(4 145 82)(5 146 83)(6 147 84)(7 148 85)(8 149 86)(9 150 87)(10 151 88)(11 152 89)(12 153 90)(13 103 91)(14 104 92)(15 105 93)(16 106 94)(17 107 95)(18 108 96)(19 109 97)(20 110 98)(21 111 99)(22 112 100)(23 113 101)(24 114 102)(25 115 52)(26 116 53)(27 117 54)(28 118 55)(29 119 56)(30 120 57)(31 121 58)(32 122 59)(33 123 60)(34 124 61)(35 125 62)(36 126 63)(37 127 64)(38 128 65)(39 129 66)(40 130 67)(41 131 68)(42 132 69)(43 133 70)(44 134 71)(45 135 72)(46 136 73)(47 137 74)(48 138 75)(49 139 76)(50 140 77)(51 141 78)(154 272 219)(155 273 220)(156 274 221)(157 275 222)(158 276 223)(159 277 224)(160 278 225)(161 279 226)(162 280 227)(163 281 228)(164 282 229)(165 283 230)(166 284 231)(167 285 232)(168 286 233)(169 287 234)(170 288 235)(171 289 236)(172 290 237)(173 291 238)(174 292 239)(175 293 240)(176 294 241)(177 295 242)(178 296 243)(179 297 244)(180 298 245)(181 299 246)(182 300 247)(183 301 248)(184 302 249)(185 303 250)(186 304 251)(187 305 252)(188 306 253)(189 256 254)(190 257 255)(191 258 205)(192 259 206)(193 260 207)(194 261 208)(195 262 209)(196 263 210)(197 264 211)(198 265 212)(199 266 213)(200 267 214)(201 268 215)(202 269 216)(203 270 217)(204 271 218)(307 417 407)(308 418 408)(309 419 358)(310 420 359)(311 421 360)(312 422 361)(313 423 362)(314 424 363)(315 425 364)(316 426 365)(317 427 366)(318 428 367)(319 429 368)(320 430 369)(321 431 370)(322 432 371)(323 433 372)(324 434 373)(325 435 374)(326 436 375)(327 437 376)(328 438 377)(329 439 378)(330 440 379)(331 441 380)(332 442 381)(333 443 382)(334 444 383)(335 445 384)(336 446 385)(337 447 386)(338 448 387)(339 449 388)(340 450 389)(341 451 390)(342 452 391)(343 453 392)(344 454 393)(345 455 394)(346 456 395)(347 457 396)(348 458 397)(349 459 398)(350 409 399)(351 410 400)(352 411 401)(353 412 402)(354 413 403)(355 414 404)(356 415 405)(357 416 406)
(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)

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

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

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

459 conjugacy classes

class 1 3A···3Z17A···17P51A···51OZ
order13···317···1751···51
size11···11···11···1

459 irreducible representations

dim1111
type+
imageC1C3C17C51
kernelC32×C51C3×C51C33C32
# reps12616416

Matrix representation of C32×C51 in GL3(𝔽103) generated by

5600
010
0046
,
100
010
0046
,
2800
0280
0076
G:=sub<GL(3,GF(103))| [56,0,0,0,1,0,0,0,46],[1,0,0,0,1,0,0,0,46],[28,0,0,0,28,0,0,0,76] >;

C32×C51 in GAP, Magma, Sage, TeX

C_3^2\times C_{51}
% in TeX

G:=Group("C3^2xC51");
// GroupNames label

G:=SmallGroup(459,5);
// by ID

G=gap.SmallGroup(459,5);
# by ID

G:=PCGroup([4,-3,-3,-3,-17]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^51=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽