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G = C3×C122order 432 = 24·33

Abelian group of type [3,12,12]

direct product, abelian, monomial

Aliases: C3×C122, SmallGroup(432,512)

Series: Derived Chief Lower central Upper central

C1 — C3×C122
C1C2C22C2×C6C62C3×C62C3×C6×C12 — C3×C122
C1 — C3×C122
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 [×3], C3 [×13], C4 [×6], C22, C6 [×39], C2×C4 [×3], C32 [×13], C12 [×78], C2×C6 [×13], C42, C3×C6 [×39], C2×C12 [×39], C33, C3×C12 [×78], C62 [×13], C4×C12 [×13], C32×C6 [×3], C6×C12 [×39], C32×C12 [×6], C3×C62, C122 [×13], C3×C6×C12 [×3], C3×C122
Quotients: C1, C2 [×3], C3 [×13], C4 [×6], C22, C6 [×39], C2×C4 [×3], C32 [×13], C12 [×78], C2×C6 [×13], C42, C3×C6 [×39], C2×C12 [×39], C33, C3×C12 [×78], C62 [×13], C4×C12 [×13], C32×C6 [×3], C6×C12 [×39], C32×C12 [×6], C3×C62, C122 [×13], C3×C6×C12 [×3], C3×C122

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

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

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

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

432 conjugacy classes

class 1 2A2B2C3A···3Z4A···4L6A···6BZ12A···12KZ
order12223···34···46···612···12
size11111···11···11···11···1

432 irreducible representations

dim111111
type++
imageC1C2C3C4C6C12
kernelC3×C122C3×C6×C12C122C32×C12C6×C12C3×C12
# reps13261278312

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

300
030
009
,
900
050
008
,
1000
080
004
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

׿
×
𝔽