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## G = C32×C51order 459 = 33·17

### Abelian group of type [3,3,51]

Aliases: C32×C51, SmallGroup(459,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C51
 Chief series C1 — C17 — C51 — C3×C51 — C32×C51
 Lower central C1 — C32×C51
 Upper central 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, C32, C17, C33, C51, C3×C51, C32×C51
Quotients: C1, C3, C32, C17, C33, C51, C3×C51, C32×C51

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

459 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C17 C51 kernel C32×C51 C3×C51 C33 C32 # reps 1 26 16 416

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

 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
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

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