metabelian, supersoluble, monomial
Aliases: C62.146D6, C33⋊20(C4⋊C4), (C6×C12).29S3, C33⋊5C4⋊5C4, (C3×C6).30Dic6, (C32×C6).78D4, (C32×C6).14Q8, C2.1(C33⋊8Q8), C6.7(C32⋊4Q8), C3⋊3(C6.Dic6), C6.22(C32⋊7D4), C2.1(C33⋊15D4), (C3×C62).52C22, C32⋊15(Dic3⋊C4), (C3×C6×C12).3C2, C6.15(C4×C3⋊S3), (C3×C6).83(C4×S3), (C2×C12).6(C3⋊S3), C2.4(C4×C33⋊C2), (C32×C6).62(C2×C4), (C2×C33⋊5C4).5C2, (C3×C6).113(C3⋊D4), (C2×C4).1(C33⋊C2), C22.4(C2×C33⋊C2), (C2×C6).41(C2×C3⋊S3), SmallGroup(432,504)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.146D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c5 >
Subgroups: 1512 in 364 conjugacy classes, 173 normal (15 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C33, C3⋊Dic3, C3×C12, C62, Dic3⋊C4, C32×C6, C2×C3⋊Dic3, C6×C12, C33⋊5C4, C33⋊5C4, C32×C12, C3×C62, C6.Dic6, C2×C33⋊5C4, C3×C6×C12, C62.146D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, C3⋊D4, C2×C3⋊S3, Dic3⋊C4, C33⋊C2, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C2×C33⋊C2, C6.Dic6, C33⋊8Q8, C4×C33⋊C2, C33⋊15D4, C62.146D6
►Regular action on 432 points
Generators in S432
(1 194 64 7 200 70)(2 195 65 8 201 71)(3 196 66 9 202 72)(4 197 67 10 203 61)(5 198 68 11 204 62)(6 199 69 12 193 63)(13 248 33 19 242 27)(14 249 34 20 243 28)(15 250 35 21 244 29)(16 251 36 22 245 30)(17 252 25 23 246 31)(18 241 26 24 247 32)(37 78 223 43 84 217)(38 79 224 44 73 218)(39 80 225 45 74 219)(40 81 226 46 75 220)(41 82 227 47 76 221)(42 83 228 48 77 222)(49 383 297 55 377 291)(50 384 298 56 378 292)(51 373 299 57 379 293)(52 374 300 58 380 294)(53 375 289 59 381 295)(54 376 290 60 382 296)(85 135 205 91 141 211)(86 136 206 92 142 212)(87 137 207 93 143 213)(88 138 208 94 144 214)(89 139 209 95 133 215)(90 140 210 96 134 216)(97 168 176 103 162 170)(98 157 177 104 163 171)(99 158 178 105 164 172)(100 159 179 106 165 173)(101 160 180 107 166 174)(102 161 169 108 167 175)(109 337 318 115 343 324)(110 338 319 116 344 313)(111 339 320 117 345 314)(112 340 321 118 346 315)(113 341 322 119 347 316)(114 342 323 120 348 317)(121 154 416 127 148 410)(122 155 417 128 149 411)(123 156 418 129 150 412)(124 145 419 130 151 413)(125 146 420 131 152 414)(126 147 409 132 153 415)(181 371 310 187 365 304)(182 372 311 188 366 305)(183 361 312 189 367 306)(184 362 301 190 368 307)(185 363 302 191 369 308)(186 364 303 192 370 309)(229 400 271 235 406 265)(230 401 272 236 407 266)(231 402 273 237 408 267)(232 403 274 238 397 268)(233 404 275 239 398 269)(234 405 276 240 399 270)(253 280 391 259 286 385)(254 281 392 260 287 386)(255 282 393 261 288 387)(256 283 394 262 277 388)(257 284 395 263 278 389)(258 285 396 264 279 390)(325 427 349 331 421 355)(326 428 350 332 422 356)(327 429 351 333 423 357)(328 430 352 334 424 358)(329 431 353 335 425 359)(330 432 354 336 426 360)
(1 25 294 311 137 169)(2 26 295 312 138 170)(3 27 296 301 139 171)(4 28 297 302 140 172)(5 29 298 303 141 173)(6 30 299 304 142 174)(7 31 300 305 143 175)(8 32 289 306 144 176)(9 33 290 307 133 177)(10 34 291 308 134 178)(11 35 292 309 135 179)(12 36 293 310 136 180)(13 54 190 209 98 196)(14 55 191 210 99 197)(15 56 192 211 100 198)(16 57 181 212 101 199)(17 58 182 213 102 200)(18 59 183 214 103 201)(19 60 184 215 104 202)(20 49 185 216 105 203)(21 50 186 205 106 204)(22 51 187 206 107 193)(23 52 188 207 108 194)(24 53 189 208 97 195)(37 153 235 388 344 427)(38 154 236 389 345 428)(39 155 237 390 346 429)(40 156 238 391 347 430)(41 145 239 392 348 431)(42 146 240 393 337 432)(43 147 229 394 338 421)(44 148 230 395 339 422)(45 149 231 396 340 423)(46 150 232 385 341 424)(47 151 233 386 342 425)(48 152 234 387 343 426)(61 243 383 363 90 164)(62 244 384 364 91 165)(63 245 373 365 92 166)(64 246 374 366 93 167)(65 247 375 367 94 168)(66 248 376 368 95 157)(67 249 377 369 96 158)(68 250 378 370 85 159)(69 251 379 371 86 160)(70 252 380 372 87 161)(71 241 381 361 88 162)(72 242 382 362 89 163)(73 410 401 263 320 356)(74 411 402 264 321 357)(75 412 403 253 322 358)(76 413 404 254 323 359)(77 414 405 255 324 360)(78 415 406 256 313 349)(79 416 407 257 314 350)(80 417 408 258 315 351)(81 418 397 259 316 352)(82 419 398 260 317 353)(83 420 399 261 318 354)(84 409 400 262 319 355)(109 330 222 125 276 282)(110 331 223 126 265 283)(111 332 224 127 266 284)(112 333 225 128 267 285)(113 334 226 129 268 286)(114 335 227 130 269 287)(115 336 228 131 270 288)(116 325 217 132 271 277)(117 326 218 121 272 278)(118 327 219 122 273 279)(119 328 220 123 274 280)(120 329 221 124 275 281)
(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)
(1 229 7 235)(2 426 8 432)(3 239 9 233)(4 424 10 430)(5 237 11 231)(6 422 12 428)(13 124 19 130)(14 113 20 119)(15 122 21 128)(16 111 22 117)(17 132 23 126)(18 109 24 115)(25 147 31 153)(26 343 32 337)(27 145 33 151)(28 341 34 347)(29 155 35 149)(30 339 36 345)(37 294 43 300)(38 174 44 180)(39 292 45 298)(40 172 46 178)(41 290 47 296)(42 170 48 176)(49 280 55 286)(50 225 56 219)(51 278 57 284)(52 223 58 217)(53 288 59 282)(54 221 60 227)(61 358 67 352)(62 408 68 402)(63 356 69 350)(64 406 70 400)(65 354 71 360)(66 404 72 398)(73 160 79 166)(74 384 80 378)(75 158 81 164)(76 382 82 376)(77 168 83 162)(78 380 84 374)(85 321 91 315)(86 416 92 410)(87 319 93 313)(88 414 94 420)(89 317 95 323)(90 412 96 418)(97 228 103 222)(98 281 104 287)(99 226 105 220)(100 279 106 285)(101 224 107 218)(102 277 108 283)(110 213 116 207)(112 211 118 205)(114 209 120 215)(121 212 127 206)(123 210 129 216)(125 208 131 214)(133 342 139 348)(134 156 140 150)(135 340 141 346)(136 154 142 148)(137 338 143 344)(138 152 144 146)(157 254 163 260)(159 264 165 258)(161 262 167 256)(169 394 175 388)(171 392 177 386)(173 390 179 396)(181 266 187 272)(182 325 188 331)(183 276 189 270)(184 335 190 329)(185 274 191 268)(186 333 192 327)(193 326 199 332)(194 265 200 271)(195 336 201 330)(196 275 202 269)(197 334 203 328)(198 273 204 267)(230 310 236 304)(232 308 238 302)(234 306 240 312)(241 324 247 318)(242 419 248 413)(243 322 249 316)(244 417 250 411)(245 320 251 314)(246 415 252 409)(253 377 259 383)(255 375 261 381)(257 373 263 379)(289 393 295 387)(291 391 297 385)(293 389 299 395)(301 431 307 425)(303 429 309 423)(305 427 311 421)(349 372 355 366)(351 370 357 364)(353 368 359 362)(361 405 367 399)(363 403 369 397)(365 401 371 407)
G:=sub<Sym(432)| (1,194,64,7,200,70)(2,195,65,8,201,71)(3,196,66,9,202,72)(4,197,67,10,203,61)(5,198,68,11,204,62)(6,199,69,12,193,63)(13,248,33,19,242,27)(14,249,34,20,243,28)(15,250,35,21,244,29)(16,251,36,22,245,30)(17,252,25,23,246,31)(18,241,26,24,247,32)(37,78,223,43,84,217)(38,79,224,44,73,218)(39,80,225,45,74,219)(40,81,226,46,75,220)(41,82,227,47,76,221)(42,83,228,48,77,222)(49,383,297,55,377,291)(50,384,298,56,378,292)(51,373,299,57,379,293)(52,374,300,58,380,294)(53,375,289,59,381,295)(54,376,290,60,382,296)(85,135,205,91,141,211)(86,136,206,92,142,212)(87,137,207,93,143,213)(88,138,208,94,144,214)(89,139,209,95,133,215)(90,140,210,96,134,216)(97,168,176,103,162,170)(98,157,177,104,163,171)(99,158,178,105,164,172)(100,159,179,106,165,173)(101,160,180,107,166,174)(102,161,169,108,167,175)(109,337,318,115,343,324)(110,338,319,116,344,313)(111,339,320,117,345,314)(112,340,321,118,346,315)(113,341,322,119,347,316)(114,342,323,120,348,317)(121,154,416,127,148,410)(122,155,417,128,149,411)(123,156,418,129,150,412)(124,145,419,130,151,413)(125,146,420,131,152,414)(126,147,409,132,153,415)(181,371,310,187,365,304)(182,372,311,188,366,305)(183,361,312,189,367,306)(184,362,301,190,368,307)(185,363,302,191,369,308)(186,364,303,192,370,309)(229,400,271,235,406,265)(230,401,272,236,407,266)(231,402,273,237,408,267)(232,403,274,238,397,268)(233,404,275,239,398,269)(234,405,276,240,399,270)(253,280,391,259,286,385)(254,281,392,260,287,386)(255,282,393,261,288,387)(256,283,394,262,277,388)(257,284,395,263,278,389)(258,285,396,264,279,390)(325,427,349,331,421,355)(326,428,350,332,422,356)(327,429,351,333,423,357)(328,430,352,334,424,358)(329,431,353,335,425,359)(330,432,354,336,426,360), (1,25,294,311,137,169)(2,26,295,312,138,170)(3,27,296,301,139,171)(4,28,297,302,140,172)(5,29,298,303,141,173)(6,30,299,304,142,174)(7,31,300,305,143,175)(8,32,289,306,144,176)(9,33,290,307,133,177)(10,34,291,308,134,178)(11,35,292,309,135,179)(12,36,293,310,136,180)(13,54,190,209,98,196)(14,55,191,210,99,197)(15,56,192,211,100,198)(16,57,181,212,101,199)(17,58,182,213,102,200)(18,59,183,214,103,201)(19,60,184,215,104,202)(20,49,185,216,105,203)(21,50,186,205,106,204)(22,51,187,206,107,193)(23,52,188,207,108,194)(24,53,189,208,97,195)(37,153,235,388,344,427)(38,154,236,389,345,428)(39,155,237,390,346,429)(40,156,238,391,347,430)(41,145,239,392,348,431)(42,146,240,393,337,432)(43,147,229,394,338,421)(44,148,230,395,339,422)(45,149,231,396,340,423)(46,150,232,385,341,424)(47,151,233,386,342,425)(48,152,234,387,343,426)(61,243,383,363,90,164)(62,244,384,364,91,165)(63,245,373,365,92,166)(64,246,374,366,93,167)(65,247,375,367,94,168)(66,248,376,368,95,157)(67,249,377,369,96,158)(68,250,378,370,85,159)(69,251,379,371,86,160)(70,252,380,372,87,161)(71,241,381,361,88,162)(72,242,382,362,89,163)(73,410,401,263,320,356)(74,411,402,264,321,357)(75,412,403,253,322,358)(76,413,404,254,323,359)(77,414,405,255,324,360)(78,415,406,256,313,349)(79,416,407,257,314,350)(80,417,408,258,315,351)(81,418,397,259,316,352)(82,419,398,260,317,353)(83,420,399,261,318,354)(84,409,400,262,319,355)(109,330,222,125,276,282)(110,331,223,126,265,283)(111,332,224,127,266,284)(112,333,225,128,267,285)(113,334,226,129,268,286)(114,335,227,130,269,287)(115,336,228,131,270,288)(116,325,217,132,271,277)(117,326,218,121,272,278)(118,327,219,122,273,279)(119,328,220,123,274,280)(120,329,221,124,275,281), (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), (1,229,7,235)(2,426,8,432)(3,239,9,233)(4,424,10,430)(5,237,11,231)(6,422,12,428)(13,124,19,130)(14,113,20,119)(15,122,21,128)(16,111,22,117)(17,132,23,126)(18,109,24,115)(25,147,31,153)(26,343,32,337)(27,145,33,151)(28,341,34,347)(29,155,35,149)(30,339,36,345)(37,294,43,300)(38,174,44,180)(39,292,45,298)(40,172,46,178)(41,290,47,296)(42,170,48,176)(49,280,55,286)(50,225,56,219)(51,278,57,284)(52,223,58,217)(53,288,59,282)(54,221,60,227)(61,358,67,352)(62,408,68,402)(63,356,69,350)(64,406,70,400)(65,354,71,360)(66,404,72,398)(73,160,79,166)(74,384,80,378)(75,158,81,164)(76,382,82,376)(77,168,83,162)(78,380,84,374)(85,321,91,315)(86,416,92,410)(87,319,93,313)(88,414,94,420)(89,317,95,323)(90,412,96,418)(97,228,103,222)(98,281,104,287)(99,226,105,220)(100,279,106,285)(101,224,107,218)(102,277,108,283)(110,213,116,207)(112,211,118,205)(114,209,120,215)(121,212,127,206)(123,210,129,216)(125,208,131,214)(133,342,139,348)(134,156,140,150)(135,340,141,346)(136,154,142,148)(137,338,143,344)(138,152,144,146)(157,254,163,260)(159,264,165,258)(161,262,167,256)(169,394,175,388)(171,392,177,386)(173,390,179,396)(181,266,187,272)(182,325,188,331)(183,276,189,270)(184,335,190,329)(185,274,191,268)(186,333,192,327)(193,326,199,332)(194,265,200,271)(195,336,201,330)(196,275,202,269)(197,334,203,328)(198,273,204,267)(230,310,236,304)(232,308,238,302)(234,306,240,312)(241,324,247,318)(242,419,248,413)(243,322,249,316)(244,417,250,411)(245,320,251,314)(246,415,252,409)(253,377,259,383)(255,375,261,381)(257,373,263,379)(289,393,295,387)(291,391,297,385)(293,389,299,395)(301,431,307,425)(303,429,309,423)(305,427,311,421)(349,372,355,366)(351,370,357,364)(353,368,359,362)(361,405,367,399)(363,403,369,397)(365,401,371,407)>;
G:=Group( (1,194,64,7,200,70)(2,195,65,8,201,71)(3,196,66,9,202,72)(4,197,67,10,203,61)(5,198,68,11,204,62)(6,199,69,12,193,63)(13,248,33,19,242,27)(14,249,34,20,243,28)(15,250,35,21,244,29)(16,251,36,22,245,30)(17,252,25,23,246,31)(18,241,26,24,247,32)(37,78,223,43,84,217)(38,79,224,44,73,218)(39,80,225,45,74,219)(40,81,226,46,75,220)(41,82,227,47,76,221)(42,83,228,48,77,222)(49,383,297,55,377,291)(50,384,298,56,378,292)(51,373,299,57,379,293)(52,374,300,58,380,294)(53,375,289,59,381,295)(54,376,290,60,382,296)(85,135,205,91,141,211)(86,136,206,92,142,212)(87,137,207,93,143,213)(88,138,208,94,144,214)(89,139,209,95,133,215)(90,140,210,96,134,216)(97,168,176,103,162,170)(98,157,177,104,163,171)(99,158,178,105,164,172)(100,159,179,106,165,173)(101,160,180,107,166,174)(102,161,169,108,167,175)(109,337,318,115,343,324)(110,338,319,116,344,313)(111,339,320,117,345,314)(112,340,321,118,346,315)(113,341,322,119,347,316)(114,342,323,120,348,317)(121,154,416,127,148,410)(122,155,417,128,149,411)(123,156,418,129,150,412)(124,145,419,130,151,413)(125,146,420,131,152,414)(126,147,409,132,153,415)(181,371,310,187,365,304)(182,372,311,188,366,305)(183,361,312,189,367,306)(184,362,301,190,368,307)(185,363,302,191,369,308)(186,364,303,192,370,309)(229,400,271,235,406,265)(230,401,272,236,407,266)(231,402,273,237,408,267)(232,403,274,238,397,268)(233,404,275,239,398,269)(234,405,276,240,399,270)(253,280,391,259,286,385)(254,281,392,260,287,386)(255,282,393,261,288,387)(256,283,394,262,277,388)(257,284,395,263,278,389)(258,285,396,264,279,390)(325,427,349,331,421,355)(326,428,350,332,422,356)(327,429,351,333,423,357)(328,430,352,334,424,358)(329,431,353,335,425,359)(330,432,354,336,426,360), (1,25,294,311,137,169)(2,26,295,312,138,170)(3,27,296,301,139,171)(4,28,297,302,140,172)(5,29,298,303,141,173)(6,30,299,304,142,174)(7,31,300,305,143,175)(8,32,289,306,144,176)(9,33,290,307,133,177)(10,34,291,308,134,178)(11,35,292,309,135,179)(12,36,293,310,136,180)(13,54,190,209,98,196)(14,55,191,210,99,197)(15,56,192,211,100,198)(16,57,181,212,101,199)(17,58,182,213,102,200)(18,59,183,214,103,201)(19,60,184,215,104,202)(20,49,185,216,105,203)(21,50,186,205,106,204)(22,51,187,206,107,193)(23,52,188,207,108,194)(24,53,189,208,97,195)(37,153,235,388,344,427)(38,154,236,389,345,428)(39,155,237,390,346,429)(40,156,238,391,347,430)(41,145,239,392,348,431)(42,146,240,393,337,432)(43,147,229,394,338,421)(44,148,230,395,339,422)(45,149,231,396,340,423)(46,150,232,385,341,424)(47,151,233,386,342,425)(48,152,234,387,343,426)(61,243,383,363,90,164)(62,244,384,364,91,165)(63,245,373,365,92,166)(64,246,374,366,93,167)(65,247,375,367,94,168)(66,248,376,368,95,157)(67,249,377,369,96,158)(68,250,378,370,85,159)(69,251,379,371,86,160)(70,252,380,372,87,161)(71,241,381,361,88,162)(72,242,382,362,89,163)(73,410,401,263,320,356)(74,411,402,264,321,357)(75,412,403,253,322,358)(76,413,404,254,323,359)(77,414,405,255,324,360)(78,415,406,256,313,349)(79,416,407,257,314,350)(80,417,408,258,315,351)(81,418,397,259,316,352)(82,419,398,260,317,353)(83,420,399,261,318,354)(84,409,400,262,319,355)(109,330,222,125,276,282)(110,331,223,126,265,283)(111,332,224,127,266,284)(112,333,225,128,267,285)(113,334,226,129,268,286)(114,335,227,130,269,287)(115,336,228,131,270,288)(116,325,217,132,271,277)(117,326,218,121,272,278)(118,327,219,122,273,279)(119,328,220,123,274,280)(120,329,221,124,275,281), (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), (1,229,7,235)(2,426,8,432)(3,239,9,233)(4,424,10,430)(5,237,11,231)(6,422,12,428)(13,124,19,130)(14,113,20,119)(15,122,21,128)(16,111,22,117)(17,132,23,126)(18,109,24,115)(25,147,31,153)(26,343,32,337)(27,145,33,151)(28,341,34,347)(29,155,35,149)(30,339,36,345)(37,294,43,300)(38,174,44,180)(39,292,45,298)(40,172,46,178)(41,290,47,296)(42,170,48,176)(49,280,55,286)(50,225,56,219)(51,278,57,284)(52,223,58,217)(53,288,59,282)(54,221,60,227)(61,358,67,352)(62,408,68,402)(63,356,69,350)(64,406,70,400)(65,354,71,360)(66,404,72,398)(73,160,79,166)(74,384,80,378)(75,158,81,164)(76,382,82,376)(77,168,83,162)(78,380,84,374)(85,321,91,315)(86,416,92,410)(87,319,93,313)(88,414,94,420)(89,317,95,323)(90,412,96,418)(97,228,103,222)(98,281,104,287)(99,226,105,220)(100,279,106,285)(101,224,107,218)(102,277,108,283)(110,213,116,207)(112,211,118,205)(114,209,120,215)(121,212,127,206)(123,210,129,216)(125,208,131,214)(133,342,139,348)(134,156,140,150)(135,340,141,346)(136,154,142,148)(137,338,143,344)(138,152,144,146)(157,254,163,260)(159,264,165,258)(161,262,167,256)(169,394,175,388)(171,392,177,386)(173,390,179,396)(181,266,187,272)(182,325,188,331)(183,276,189,270)(184,335,190,329)(185,274,191,268)(186,333,192,327)(193,326,199,332)(194,265,200,271)(195,336,201,330)(196,275,202,269)(197,334,203,328)(198,273,204,267)(230,310,236,304)(232,308,238,302)(234,306,240,312)(241,324,247,318)(242,419,248,413)(243,322,249,316)(244,417,250,411)(245,320,251,314)(246,415,252,409)(253,377,259,383)(255,375,261,381)(257,373,263,379)(289,393,295,387)(291,391,297,385)(293,389,299,395)(301,431,307,425)(303,429,309,423)(305,427,311,421)(349,372,355,366)(351,370,357,364)(353,368,359,362)(361,405,367,399)(363,403,369,397)(365,401,371,407) );
G=PermutationGroup([[(1,194,64,7,200,70),(2,195,65,8,201,71),(3,196,66,9,202,72),(4,197,67,10,203,61),(5,198,68,11,204,62),(6,199,69,12,193,63),(13,248,33,19,242,27),(14,249,34,20,243,28),(15,250,35,21,244,29),(16,251,36,22,245,30),(17,252,25,23,246,31),(18,241,26,24,247,32),(37,78,223,43,84,217),(38,79,224,44,73,218),(39,80,225,45,74,219),(40,81,226,46,75,220),(41,82,227,47,76,221),(42,83,228,48,77,222),(49,383,297,55,377,291),(50,384,298,56,378,292),(51,373,299,57,379,293),(52,374,300,58,380,294),(53,375,289,59,381,295),(54,376,290,60,382,296),(85,135,205,91,141,211),(86,136,206,92,142,212),(87,137,207,93,143,213),(88,138,208,94,144,214),(89,139,209,95,133,215),(90,140,210,96,134,216),(97,168,176,103,162,170),(98,157,177,104,163,171),(99,158,178,105,164,172),(100,159,179,106,165,173),(101,160,180,107,166,174),(102,161,169,108,167,175),(109,337,318,115,343,324),(110,338,319,116,344,313),(111,339,320,117,345,314),(112,340,321,118,346,315),(113,341,322,119,347,316),(114,342,323,120,348,317),(121,154,416,127,148,410),(122,155,417,128,149,411),(123,156,418,129,150,412),(124,145,419,130,151,413),(125,146,420,131,152,414),(126,147,409,132,153,415),(181,371,310,187,365,304),(182,372,311,188,366,305),(183,361,312,189,367,306),(184,362,301,190,368,307),(185,363,302,191,369,308),(186,364,303,192,370,309),(229,400,271,235,406,265),(230,401,272,236,407,266),(231,402,273,237,408,267),(232,403,274,238,397,268),(233,404,275,239,398,269),(234,405,276,240,399,270),(253,280,391,259,286,385),(254,281,392,260,287,386),(255,282,393,261,288,387),(256,283,394,262,277,388),(257,284,395,263,278,389),(258,285,396,264,279,390),(325,427,349,331,421,355),(326,428,350,332,422,356),(327,429,351,333,423,357),(328,430,352,334,424,358),(329,431,353,335,425,359),(330,432,354,336,426,360)], [(1,25,294,311,137,169),(2,26,295,312,138,170),(3,27,296,301,139,171),(4,28,297,302,140,172),(5,29,298,303,141,173),(6,30,299,304,142,174),(7,31,300,305,143,175),(8,32,289,306,144,176),(9,33,290,307,133,177),(10,34,291,308,134,178),(11,35,292,309,135,179),(12,36,293,310,136,180),(13,54,190,209,98,196),(14,55,191,210,99,197),(15,56,192,211,100,198),(16,57,181,212,101,199),(17,58,182,213,102,200),(18,59,183,214,103,201),(19,60,184,215,104,202),(20,49,185,216,105,203),(21,50,186,205,106,204),(22,51,187,206,107,193),(23,52,188,207,108,194),(24,53,189,208,97,195),(37,153,235,388,344,427),(38,154,236,389,345,428),(39,155,237,390,346,429),(40,156,238,391,347,430),(41,145,239,392,348,431),(42,146,240,393,337,432),(43,147,229,394,338,421),(44,148,230,395,339,422),(45,149,231,396,340,423),(46,150,232,385,341,424),(47,151,233,386,342,425),(48,152,234,387,343,426),(61,243,383,363,90,164),(62,244,384,364,91,165),(63,245,373,365,92,166),(64,246,374,366,93,167),(65,247,375,367,94,168),(66,248,376,368,95,157),(67,249,377,369,96,158),(68,250,378,370,85,159),(69,251,379,371,86,160),(70,252,380,372,87,161),(71,241,381,361,88,162),(72,242,382,362,89,163),(73,410,401,263,320,356),(74,411,402,264,321,357),(75,412,403,253,322,358),(76,413,404,254,323,359),(77,414,405,255,324,360),(78,415,406,256,313,349),(79,416,407,257,314,350),(80,417,408,258,315,351),(81,418,397,259,316,352),(82,419,398,260,317,353),(83,420,399,261,318,354),(84,409,400,262,319,355),(109,330,222,125,276,282),(110,331,223,126,265,283),(111,332,224,127,266,284),(112,333,225,128,267,285),(113,334,226,129,268,286),(114,335,227,130,269,287),(115,336,228,131,270,288),(116,325,217,132,271,277),(117,326,218,121,272,278),(118,327,219,122,273,279),(119,328,220,123,274,280),(120,329,221,124,275,281)], [(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)], [(1,229,7,235),(2,426,8,432),(3,239,9,233),(4,424,10,430),(5,237,11,231),(6,422,12,428),(13,124,19,130),(14,113,20,119),(15,122,21,128),(16,111,22,117),(17,132,23,126),(18,109,24,115),(25,147,31,153),(26,343,32,337),(27,145,33,151),(28,341,34,347),(29,155,35,149),(30,339,36,345),(37,294,43,300),(38,174,44,180),(39,292,45,298),(40,172,46,178),(41,290,47,296),(42,170,48,176),(49,280,55,286),(50,225,56,219),(51,278,57,284),(52,223,58,217),(53,288,59,282),(54,221,60,227),(61,358,67,352),(62,408,68,402),(63,356,69,350),(64,406,70,400),(65,354,71,360),(66,404,72,398),(73,160,79,166),(74,384,80,378),(75,158,81,164),(76,382,82,376),(77,168,83,162),(78,380,84,374),(85,321,91,315),(86,416,92,410),(87,319,93,313),(88,414,94,420),(89,317,95,323),(90,412,96,418),(97,228,103,222),(98,281,104,287),(99,226,105,220),(100,279,106,285),(101,224,107,218),(102,277,108,283),(110,213,116,207),(112,211,118,205),(114,209,120,215),(121,212,127,206),(123,210,129,216),(125,208,131,214),(133,342,139,348),(134,156,140,150),(135,340,141,346),(136,154,142,148),(137,338,143,344),(138,152,144,146),(157,254,163,260),(159,264,165,258),(161,262,167,256),(169,394,175,388),(171,392,177,386),(173,390,179,396),(181,266,187,272),(182,325,188,331),(183,276,189,270),(184,335,190,329),(185,274,191,268),(186,333,192,327),(193,326,199,332),(194,265,200,271),(195,336,201,330),(196,275,202,269),(197,334,203,328),(198,273,204,267),(230,310,236,304),(232,308,238,302),(234,306,240,312),(241,324,247,318),(242,419,248,413),(243,322,249,316),(244,417,250,411),(245,320,251,314),(246,415,252,409),(253,377,259,383),(255,375,261,381),(257,373,263,379),(289,393,295,387),(291,391,297,385),(293,389,299,395),(301,431,307,425),(303,429,309,423),(305,427,311,421),(349,372,355,366),(351,370,357,364),(353,368,359,362),(361,405,367,399),(363,403,369,397),(365,401,371,407)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3M | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6AM | 12A | ··· | 12AZ |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 54 | 54 | 54 | 54 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | Dic6 | C4×S3 | C3⋊D4 |
| kernel | C62.146D6 | C2×C33⋊5C4 | C3×C6×C12 | C33⋊5C4 | C6×C12 | C32×C6 | C32×C6 | C62 | C3×C6 | C3×C6 | C3×C6 |
| # reps | 1 | 2 | 1 | 4 | 13 | 1 | 1 | 13 | 26 | 26 | 26 |
Matrix representation of C62.146D6 ►in GL6(𝔽13)
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 2 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 10 | 3 |
G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[11,2,0,0,0,0,11,9,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[10,3,0,0,0,0,6,3,0,0,0,0,0,0,11,2,0,0,0,0,4,2,0,0,0,0,0,0,10,10,0,0,0,0,7,3] >;
C62.146D6 in GAP, Magma, Sage, TeX
C_6^2._{146}D_6 % in TeX
G:=Group("C6^2.146D6"); // GroupNames label
G:=SmallGroup(432,504);
// by ID
G=gap.SmallGroup(432,504);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,1124,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
// generators/relations