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## G = C3×C122order 432 = 24·33

### Abelian group of type [3,12,12]

Aliases: C3×C122, SmallGroup(432,512)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C122
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C3×C62 — C3×C6×C12 — C3×C122
 Lower central C1 — C3×C122
 Upper central C1 — C3×C122

Generators and relations for C3×C122
G = < a,b,c | a3=b12=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 420, all normal (6 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, C3×C6, C2×C12, C33, C3×C12, C62, C4×C12, C32×C6, C6×C12, C32×C12, C3×C62, C122, C3×C6×C12, C3×C122
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, C3×C6, C2×C12, C33, C3×C12, C62, C4×C12, C32×C6, C6×C12, C32×C12, C3×C62, C122, C3×C6×C12, C3×C122

Smallest permutation representation of C3×C122
Regular action on 432 points
Generators in S432
(1 383 252)(2 384 241)(3 373 242)(4 374 243)(5 375 244)(6 376 245)(7 377 246)(8 378 247)(9 379 248)(10 380 249)(11 381 250)(12 382 251)(13 212 170)(14 213 171)(15 214 172)(16 215 173)(17 216 174)(18 205 175)(19 206 176)(20 207 177)(21 208 178)(22 209 179)(23 210 180)(24 211 169)(25 74 412)(26 75 413)(27 76 414)(28 77 415)(29 78 416)(30 79 417)(31 80 418)(32 81 419)(33 82 420)(34 83 409)(35 84 410)(36 73 411)(37 60 64)(38 49 65)(39 50 66)(40 51 67)(41 52 68)(42 53 69)(43 54 70)(44 55 71)(45 56 72)(46 57 61)(47 58 62)(48 59 63)(85 137 332)(86 138 333)(87 139 334)(88 140 335)(89 141 336)(90 142 325)(91 143 326)(92 144 327)(93 133 328)(94 134 329)(95 135 330)(96 136 331)(97 204 240)(98 193 229)(99 194 230)(100 195 231)(101 196 232)(102 197 233)(103 198 234)(104 199 235)(105 200 236)(106 201 237)(107 202 238)(108 203 239)(109 145 257)(110 146 258)(111 147 259)(112 148 260)(113 149 261)(114 150 262)(115 151 263)(116 152 264)(117 153 253)(118 154 254)(119 155 255)(120 156 256)(121 432 162)(122 421 163)(123 422 164)(124 423 165)(125 424 166)(126 425 167)(127 426 168)(128 427 157)(129 428 158)(130 429 159)(131 430 160)(132 431 161)(181 356 368)(182 357 369)(183 358 370)(184 359 371)(185 360 372)(186 349 361)(187 350 362)(188 351 363)(189 352 364)(190 353 365)(191 354 366)(192 355 367)(217 305 317)(218 306 318)(219 307 319)(220 308 320)(221 309 321)(222 310 322)(223 311 323)(224 312 324)(225 301 313)(226 302 314)(227 303 315)(228 304 316)(265 391 343)(266 392 344)(267 393 345)(268 394 346)(269 395 347)(270 396 348)(271 385 337)(272 386 338)(273 387 339)(274 388 340)(275 389 341)(276 390 342)(277 403 291)(278 404 292)(279 405 293)(280 406 294)(281 407 295)(282 408 296)(283 397 297)(284 398 298)(285 399 299)(286 400 300)(287 401 289)(288 402 290)
(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)
(1 362 208 86 80 262 230 293 225 51 386 158)(2 363 209 87 81 263 231 294 226 52 387 159)(3 364 210 88 82 264 232 295 227 53 388 160)(4 365 211 89 83 253 233 296 228 54 389 161)(5 366 212 90 84 254 234 297 217 55 390 162)(6 367 213 91 73 255 235 298 218 56 391 163)(7 368 214 92 74 256 236 299 219 57 392 164)(8 369 215 93 75 257 237 300 220 58 393 165)(9 370 216 94 76 258 238 289 221 59 394 166)(10 371 205 95 77 259 239 290 222 60 395 167)(11 372 206 96 78 260 240 291 223 49 396 168)(12 361 207 85 79 261 229 292 224 50 385 157)(13 325 35 154 198 397 317 44 276 432 244 354)(14 326 36 155 199 398 318 45 265 421 245 355)(15 327 25 156 200 399 319 46 266 422 246 356)(16 328 26 145 201 400 320 47 267 423 247 357)(17 329 27 146 202 401 321 48 268 424 248 358)(18 330 28 147 203 402 322 37 269 425 249 359)(19 331 29 148 204 403 323 38 270 426 250 360)(20 332 30 149 193 404 324 39 271 427 251 349)(21 333 31 150 194 405 313 40 272 428 252 350)(22 334 32 151 195 406 314 41 273 429 241 351)(23 335 33 152 196 407 315 42 274 430 242 352)(24 336 34 153 197 408 316 43 275 431 243 353)(61 344 123 377 181 172 144 412 120 105 285 307)(62 345 124 378 182 173 133 413 109 106 286 308)(63 346 125 379 183 174 134 414 110 107 287 309)(64 347 126 380 184 175 135 415 111 108 288 310)(65 348 127 381 185 176 136 416 112 97 277 311)(66 337 128 382 186 177 137 417 113 98 278 312)(67 338 129 383 187 178 138 418 114 99 279 301)(68 339 130 384 188 179 139 419 115 100 280 302)(69 340 131 373 189 180 140 420 116 101 281 303)(70 341 132 374 190 169 141 409 117 102 282 304)(71 342 121 375 191 170 142 410 118 103 283 305)(72 343 122 376 192 171 143 411 119 104 284 306)

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

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

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

432 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3Z 4A ··· 4L 6A ··· 6BZ 12A ··· 12KZ order 1 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

432 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C12 kernel C3×C122 C3×C6×C12 C122 C32×C12 C6×C12 C3×C12 # reps 1 3 26 12 78 312

Matrix representation of C3×C122 in GL3(𝔽13) generated by

 3 0 0 0 3 0 0 0 9
,
 9 0 0 0 5 0 0 0 8
,
 10 0 0 0 8 0 0 0 4
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,9],[9,0,0,0,5,0,0,0,8],[10,0,0,0,8,0,0,0,4] >;

C3×C122 in GAP, Magma, Sage, TeX

C_3\times C_{12}^2
% in TeX

G:=Group("C3xC12^2");
// GroupNames label

G:=SmallGroup(432,512);
// by ID

G=gap.SmallGroup(432,512);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,-2,756,1520]);
// Polycyclic

G:=Group<a,b,c|a^3=b^12=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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