direct product, abelian, monomial
Aliases: C62×C12, SmallGroup(432,730)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C62×C12 |
| C1 — C62×C12 |
| C1 — C62×C12 |
Generators and relations for C62×C12
G = < a,b,c | a6=b6=c12=1, ab=ba, ac=ca, bc=cb >
Subgroups: 756, all normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C22×C4, C3×C6, C2×C12, C22×C6, C33, C3×C12, C62, C22×C12, C32×C6, C32×C6, C6×C12, C2×C62, C32×C12, C3×C62, C2×C6×C12, C3×C6×C12, C63, C62×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C22×C4, C3×C6, C2×C12, C22×C6, C33, C3×C12, C62, C22×C12, C32×C6, C6×C12, C2×C62, C32×C12, C3×C62, C2×C6×C12, C3×C6×C12, C63, C62×C12
►Regular action on 432 points
Generators in S432
(1 395 245 281 129 87)(2 396 246 282 130 88)(3 385 247 283 131 89)(4 386 248 284 132 90)(5 387 249 285 121 91)(6 388 250 286 122 92)(7 389 251 287 123 93)(8 390 252 288 124 94)(9 391 241 277 125 95)(10 392 242 278 126 96)(11 393 243 279 127 85)(12 394 244 280 128 86)(13 263 207 69 105 365)(14 264 208 70 106 366)(15 253 209 71 107 367)(16 254 210 72 108 368)(17 255 211 61 97 369)(18 256 212 62 98 370)(19 257 213 63 99 371)(20 258 214 64 100 372)(21 259 215 65 101 361)(22 260 216 66 102 362)(23 261 205 67 103 363)(24 262 206 68 104 364)(25 405 333 291 139 175)(26 406 334 292 140 176)(27 407 335 293 141 177)(28 408 336 294 142 178)(29 397 325 295 143 179)(30 398 326 296 144 180)(31 399 327 297 133 169)(32 400 328 298 134 170)(33 401 329 299 135 171)(34 402 330 300 136 172)(35 403 331 289 137 173)(36 404 332 290 138 174)(37 73 345 303 231 187)(38 74 346 304 232 188)(39 75 347 305 233 189)(40 76 348 306 234 190)(41 77 337 307 235 191)(42 78 338 308 236 192)(43 79 339 309 237 181)(44 80 340 310 238 182)(45 81 341 311 239 183)(46 82 342 312 240 184)(47 83 343 301 229 185)(48 84 344 302 230 186)(49 429 357 315 163 199)(50 430 358 316 164 200)(51 431 359 317 165 201)(52 432 360 318 166 202)(53 421 349 319 167 203)(54 422 350 320 168 204)(55 423 351 321 157 193)(56 424 352 322 158 194)(57 425 353 323 159 195)(58 426 354 324 160 196)(59 427 355 313 161 197)(60 428 356 314 162 198)(109 381 417 267 219 151)(110 382 418 268 220 152)(111 383 419 269 221 153)(112 384 420 270 222 154)(113 373 409 271 223 155)(114 374 410 272 224 156)(115 375 411 273 225 145)(116 376 412 274 226 146)(117 377 413 275 227 147)(118 378 414 276 228 148)(119 379 415 265 217 149)(120 380 416 266 218 150)
(1 257 233 359 335 153)(2 258 234 360 336 154)(3 259 235 349 325 155)(4 260 236 350 326 156)(5 261 237 351 327 145)(6 262 238 352 328 146)(7 263 239 353 329 147)(8 264 240 354 330 148)(9 253 229 355 331 149)(10 254 230 356 332 150)(11 255 231 357 333 151)(12 256 232 358 334 152)(13 311 425 401 227 93)(14 312 426 402 228 94)(15 301 427 403 217 95)(16 302 428 404 218 96)(17 303 429 405 219 85)(18 304 430 406 220 86)(19 305 431 407 221 87)(20 306 432 408 222 88)(21 307 421 397 223 89)(22 308 422 398 224 90)(23 309 423 399 225 91)(24 310 424 400 226 92)(25 267 127 369 345 49)(26 268 128 370 346 50)(27 269 129 371 347 51)(28 270 130 372 348 52)(29 271 131 361 337 53)(30 272 132 362 338 54)(31 273 121 363 339 55)(32 274 122 364 340 56)(33 275 123 365 341 57)(34 276 124 366 342 58)(35 265 125 367 343 59)(36 266 126 368 344 60)(37 163 139 381 243 61)(38 164 140 382 244 62)(39 165 141 383 245 63)(40 166 142 384 246 64)(41 167 143 373 247 65)(42 168 144 374 248 66)(43 157 133 375 249 67)(44 158 134 376 250 68)(45 159 135 377 251 69)(46 160 136 378 252 70)(47 161 137 379 241 71)(48 162 138 380 242 72)(73 199 175 417 279 97)(74 200 176 418 280 98)(75 201 177 419 281 99)(76 202 178 420 282 100)(77 203 179 409 283 101)(78 204 180 410 284 102)(79 193 169 411 285 103)(80 194 170 412 286 104)(81 195 171 413 287 105)(82 196 172 414 288 106)(83 197 173 415 277 107)(84 198 174 416 278 108)(109 393 211 187 315 291)(110 394 212 188 316 292)(111 395 213 189 317 293)(112 396 214 190 318 294)(113 385 215 191 319 295)(114 386 216 192 320 296)(115 387 205 181 321 297)(116 388 206 182 322 298)(117 389 207 183 323 299)(118 390 208 184 324 300)(119 391 209 185 313 289)(120 392 210 186 314 290)
(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)
G:=sub<Sym(432)| (1,395,245,281,129,87)(2,396,246,282,130,88)(3,385,247,283,131,89)(4,386,248,284,132,90)(5,387,249,285,121,91)(6,388,250,286,122,92)(7,389,251,287,123,93)(8,390,252,288,124,94)(9,391,241,277,125,95)(10,392,242,278,126,96)(11,393,243,279,127,85)(12,394,244,280,128,86)(13,263,207,69,105,365)(14,264,208,70,106,366)(15,253,209,71,107,367)(16,254,210,72,108,368)(17,255,211,61,97,369)(18,256,212,62,98,370)(19,257,213,63,99,371)(20,258,214,64,100,372)(21,259,215,65,101,361)(22,260,216,66,102,362)(23,261,205,67,103,363)(24,262,206,68,104,364)(25,405,333,291,139,175)(26,406,334,292,140,176)(27,407,335,293,141,177)(28,408,336,294,142,178)(29,397,325,295,143,179)(30,398,326,296,144,180)(31,399,327,297,133,169)(32,400,328,298,134,170)(33,401,329,299,135,171)(34,402,330,300,136,172)(35,403,331,289,137,173)(36,404,332,290,138,174)(37,73,345,303,231,187)(38,74,346,304,232,188)(39,75,347,305,233,189)(40,76,348,306,234,190)(41,77,337,307,235,191)(42,78,338,308,236,192)(43,79,339,309,237,181)(44,80,340,310,238,182)(45,81,341,311,239,183)(46,82,342,312,240,184)(47,83,343,301,229,185)(48,84,344,302,230,186)(49,429,357,315,163,199)(50,430,358,316,164,200)(51,431,359,317,165,201)(52,432,360,318,166,202)(53,421,349,319,167,203)(54,422,350,320,168,204)(55,423,351,321,157,193)(56,424,352,322,158,194)(57,425,353,323,159,195)(58,426,354,324,160,196)(59,427,355,313,161,197)(60,428,356,314,162,198)(109,381,417,267,219,151)(110,382,418,268,220,152)(111,383,419,269,221,153)(112,384,420,270,222,154)(113,373,409,271,223,155)(114,374,410,272,224,156)(115,375,411,273,225,145)(116,376,412,274,226,146)(117,377,413,275,227,147)(118,378,414,276,228,148)(119,379,415,265,217,149)(120,380,416,266,218,150), (1,257,233,359,335,153)(2,258,234,360,336,154)(3,259,235,349,325,155)(4,260,236,350,326,156)(5,261,237,351,327,145)(6,262,238,352,328,146)(7,263,239,353,329,147)(8,264,240,354,330,148)(9,253,229,355,331,149)(10,254,230,356,332,150)(11,255,231,357,333,151)(12,256,232,358,334,152)(13,311,425,401,227,93)(14,312,426,402,228,94)(15,301,427,403,217,95)(16,302,428,404,218,96)(17,303,429,405,219,85)(18,304,430,406,220,86)(19,305,431,407,221,87)(20,306,432,408,222,88)(21,307,421,397,223,89)(22,308,422,398,224,90)(23,309,423,399,225,91)(24,310,424,400,226,92)(25,267,127,369,345,49)(26,268,128,370,346,50)(27,269,129,371,347,51)(28,270,130,372,348,52)(29,271,131,361,337,53)(30,272,132,362,338,54)(31,273,121,363,339,55)(32,274,122,364,340,56)(33,275,123,365,341,57)(34,276,124,366,342,58)(35,265,125,367,343,59)(36,266,126,368,344,60)(37,163,139,381,243,61)(38,164,140,382,244,62)(39,165,141,383,245,63)(40,166,142,384,246,64)(41,167,143,373,247,65)(42,168,144,374,248,66)(43,157,133,375,249,67)(44,158,134,376,250,68)(45,159,135,377,251,69)(46,160,136,378,252,70)(47,161,137,379,241,71)(48,162,138,380,242,72)(73,199,175,417,279,97)(74,200,176,418,280,98)(75,201,177,419,281,99)(76,202,178,420,282,100)(77,203,179,409,283,101)(78,204,180,410,284,102)(79,193,169,411,285,103)(80,194,170,412,286,104)(81,195,171,413,287,105)(82,196,172,414,288,106)(83,197,173,415,277,107)(84,198,174,416,278,108)(109,393,211,187,315,291)(110,394,212,188,316,292)(111,395,213,189,317,293)(112,396,214,190,318,294)(113,385,215,191,319,295)(114,386,216,192,320,296)(115,387,205,181,321,297)(116,388,206,182,322,298)(117,389,207,183,323,299)(118,390,208,184,324,300)(119,391,209,185,313,289)(120,392,210,186,314,290), (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)>;
G:=Group( (1,395,245,281,129,87)(2,396,246,282,130,88)(3,385,247,283,131,89)(4,386,248,284,132,90)(5,387,249,285,121,91)(6,388,250,286,122,92)(7,389,251,287,123,93)(8,390,252,288,124,94)(9,391,241,277,125,95)(10,392,242,278,126,96)(11,393,243,279,127,85)(12,394,244,280,128,86)(13,263,207,69,105,365)(14,264,208,70,106,366)(15,253,209,71,107,367)(16,254,210,72,108,368)(17,255,211,61,97,369)(18,256,212,62,98,370)(19,257,213,63,99,371)(20,258,214,64,100,372)(21,259,215,65,101,361)(22,260,216,66,102,362)(23,261,205,67,103,363)(24,262,206,68,104,364)(25,405,333,291,139,175)(26,406,334,292,140,176)(27,407,335,293,141,177)(28,408,336,294,142,178)(29,397,325,295,143,179)(30,398,326,296,144,180)(31,399,327,297,133,169)(32,400,328,298,134,170)(33,401,329,299,135,171)(34,402,330,300,136,172)(35,403,331,289,137,173)(36,404,332,290,138,174)(37,73,345,303,231,187)(38,74,346,304,232,188)(39,75,347,305,233,189)(40,76,348,306,234,190)(41,77,337,307,235,191)(42,78,338,308,236,192)(43,79,339,309,237,181)(44,80,340,310,238,182)(45,81,341,311,239,183)(46,82,342,312,240,184)(47,83,343,301,229,185)(48,84,344,302,230,186)(49,429,357,315,163,199)(50,430,358,316,164,200)(51,431,359,317,165,201)(52,432,360,318,166,202)(53,421,349,319,167,203)(54,422,350,320,168,204)(55,423,351,321,157,193)(56,424,352,322,158,194)(57,425,353,323,159,195)(58,426,354,324,160,196)(59,427,355,313,161,197)(60,428,356,314,162,198)(109,381,417,267,219,151)(110,382,418,268,220,152)(111,383,419,269,221,153)(112,384,420,270,222,154)(113,373,409,271,223,155)(114,374,410,272,224,156)(115,375,411,273,225,145)(116,376,412,274,226,146)(117,377,413,275,227,147)(118,378,414,276,228,148)(119,379,415,265,217,149)(120,380,416,266,218,150), (1,257,233,359,335,153)(2,258,234,360,336,154)(3,259,235,349,325,155)(4,260,236,350,326,156)(5,261,237,351,327,145)(6,262,238,352,328,146)(7,263,239,353,329,147)(8,264,240,354,330,148)(9,253,229,355,331,149)(10,254,230,356,332,150)(11,255,231,357,333,151)(12,256,232,358,334,152)(13,311,425,401,227,93)(14,312,426,402,228,94)(15,301,427,403,217,95)(16,302,428,404,218,96)(17,303,429,405,219,85)(18,304,430,406,220,86)(19,305,431,407,221,87)(20,306,432,408,222,88)(21,307,421,397,223,89)(22,308,422,398,224,90)(23,309,423,399,225,91)(24,310,424,400,226,92)(25,267,127,369,345,49)(26,268,128,370,346,50)(27,269,129,371,347,51)(28,270,130,372,348,52)(29,271,131,361,337,53)(30,272,132,362,338,54)(31,273,121,363,339,55)(32,274,122,364,340,56)(33,275,123,365,341,57)(34,276,124,366,342,58)(35,265,125,367,343,59)(36,266,126,368,344,60)(37,163,139,381,243,61)(38,164,140,382,244,62)(39,165,141,383,245,63)(40,166,142,384,246,64)(41,167,143,373,247,65)(42,168,144,374,248,66)(43,157,133,375,249,67)(44,158,134,376,250,68)(45,159,135,377,251,69)(46,160,136,378,252,70)(47,161,137,379,241,71)(48,162,138,380,242,72)(73,199,175,417,279,97)(74,200,176,418,280,98)(75,201,177,419,281,99)(76,202,178,420,282,100)(77,203,179,409,283,101)(78,204,180,410,284,102)(79,193,169,411,285,103)(80,194,170,412,286,104)(81,195,171,413,287,105)(82,196,172,414,288,106)(83,197,173,415,277,107)(84,198,174,416,278,108)(109,393,211,187,315,291)(110,394,212,188,316,292)(111,395,213,189,317,293)(112,396,214,190,318,294)(113,385,215,191,319,295)(114,386,216,192,320,296)(115,387,205,181,321,297)(116,388,206,182,322,298)(117,389,207,183,323,299)(118,390,208,184,324,300)(119,391,209,185,313,289)(120,392,210,186,314,290), (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) );
G=PermutationGroup([[(1,395,245,281,129,87),(2,396,246,282,130,88),(3,385,247,283,131,89),(4,386,248,284,132,90),(5,387,249,285,121,91),(6,388,250,286,122,92),(7,389,251,287,123,93),(8,390,252,288,124,94),(9,391,241,277,125,95),(10,392,242,278,126,96),(11,393,243,279,127,85),(12,394,244,280,128,86),(13,263,207,69,105,365),(14,264,208,70,106,366),(15,253,209,71,107,367),(16,254,210,72,108,368),(17,255,211,61,97,369),(18,256,212,62,98,370),(19,257,213,63,99,371),(20,258,214,64,100,372),(21,259,215,65,101,361),(22,260,216,66,102,362),(23,261,205,67,103,363),(24,262,206,68,104,364),(25,405,333,291,139,175),(26,406,334,292,140,176),(27,407,335,293,141,177),(28,408,336,294,142,178),(29,397,325,295,143,179),(30,398,326,296,144,180),(31,399,327,297,133,169),(32,400,328,298,134,170),(33,401,329,299,135,171),(34,402,330,300,136,172),(35,403,331,289,137,173),(36,404,332,290,138,174),(37,73,345,303,231,187),(38,74,346,304,232,188),(39,75,347,305,233,189),(40,76,348,306,234,190),(41,77,337,307,235,191),(42,78,338,308,236,192),(43,79,339,309,237,181),(44,80,340,310,238,182),(45,81,341,311,239,183),(46,82,342,312,240,184),(47,83,343,301,229,185),(48,84,344,302,230,186),(49,429,357,315,163,199),(50,430,358,316,164,200),(51,431,359,317,165,201),(52,432,360,318,166,202),(53,421,349,319,167,203),(54,422,350,320,168,204),(55,423,351,321,157,193),(56,424,352,322,158,194),(57,425,353,323,159,195),(58,426,354,324,160,196),(59,427,355,313,161,197),(60,428,356,314,162,198),(109,381,417,267,219,151),(110,382,418,268,220,152),(111,383,419,269,221,153),(112,384,420,270,222,154),(113,373,409,271,223,155),(114,374,410,272,224,156),(115,375,411,273,225,145),(116,376,412,274,226,146),(117,377,413,275,227,147),(118,378,414,276,228,148),(119,379,415,265,217,149),(120,380,416,266,218,150)], [(1,257,233,359,335,153),(2,258,234,360,336,154),(3,259,235,349,325,155),(4,260,236,350,326,156),(5,261,237,351,327,145),(6,262,238,352,328,146),(7,263,239,353,329,147),(8,264,240,354,330,148),(9,253,229,355,331,149),(10,254,230,356,332,150),(11,255,231,357,333,151),(12,256,232,358,334,152),(13,311,425,401,227,93),(14,312,426,402,228,94),(15,301,427,403,217,95),(16,302,428,404,218,96),(17,303,429,405,219,85),(18,304,430,406,220,86),(19,305,431,407,221,87),(20,306,432,408,222,88),(21,307,421,397,223,89),(22,308,422,398,224,90),(23,309,423,399,225,91),(24,310,424,400,226,92),(25,267,127,369,345,49),(26,268,128,370,346,50),(27,269,129,371,347,51),(28,270,130,372,348,52),(29,271,131,361,337,53),(30,272,132,362,338,54),(31,273,121,363,339,55),(32,274,122,364,340,56),(33,275,123,365,341,57),(34,276,124,366,342,58),(35,265,125,367,343,59),(36,266,126,368,344,60),(37,163,139,381,243,61),(38,164,140,382,244,62),(39,165,141,383,245,63),(40,166,142,384,246,64),(41,167,143,373,247,65),(42,168,144,374,248,66),(43,157,133,375,249,67),(44,158,134,376,250,68),(45,159,135,377,251,69),(46,160,136,378,252,70),(47,161,137,379,241,71),(48,162,138,380,242,72),(73,199,175,417,279,97),(74,200,176,418,280,98),(75,201,177,419,281,99),(76,202,178,420,282,100),(77,203,179,409,283,101),(78,204,180,410,284,102),(79,193,169,411,285,103),(80,194,170,412,286,104),(81,195,171,413,287,105),(82,196,172,414,288,106),(83,197,173,415,277,107),(84,198,174,416,278,108),(109,393,211,187,315,291),(110,394,212,188,316,292),(111,395,213,189,317,293),(112,396,214,190,318,294),(113,385,215,191,319,295),(114,386,216,192,320,296),(115,387,205,181,321,297),(116,388,206,182,322,298),(117,389,207,183,323,299),(118,390,208,184,324,300),(119,391,209,185,313,289),(120,392,210,186,314,290)], [(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)]])
432 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | ··· | 3Z | 4A | ··· | 4H | 6A | ··· | 6FZ | 12A | ··· | 12GZ |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
432 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 |
| kernel | C62×C12 | C3×C6×C12 | C63 | C2×C6×C12 | C3×C62 | C6×C12 | C2×C62 | C62 |
| # reps | 1 | 6 | 1 | 26 | 8 | 156 | 26 | 208 |
Matrix representation of C62×C12 ►in GL3(𝔽13) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 4 |
| 11 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(13))| [4,0,0,0,4,0,0,0,9],[12,0,0,0,1,0,0,0,4],[11,0,0,0,9,0,0,0,3] >;
C62×C12 in GAP, Magma, Sage, TeX
C_6^2\times C_{12} % in TeX
G:=Group("C6^2xC12"); // GroupNames label
G:=SmallGroup(432,730);
// by ID
G=gap.SmallGroup(432,730);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-3,-2,1512]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations