metabelian, supersoluble, monomial
Aliases: C62.147D6, C4⋊(C33⋊5C4), C33⋊21(C4⋊C4), (C6×C12).33S3, (C32×C12)⋊5C4, (C3×C12)⋊7Dic3, (C3×C6).69D12, C12⋊1(C3⋊Dic3), (C3×C6).31Dic6, (C32×C6).64D4, (C32×C6).15Q8, C6.10(C12⋊S3), C3⋊2(C12⋊Dic3), C6.8(C32⋊4Q8), C2.2(C33⋊8Q8), C32⋊12(C4⋊Dic3), C2.1(C33⋊12D4), (C3×C62).53C22, (C3×C6×C12).4C2, C6.15(C2×C3⋊Dic3), C2.4(C2×C33⋊5C4), (C2×C12).10(C3⋊S3), (C32×C6).74(C2×C4), (C2×C33⋊5C4).6C2, (C3×C6).74(C2×Dic3), (C2×C4).3(C33⋊C2), C22.5(C2×C33⋊C2), (C2×C6).42(C2×C3⋊S3), SmallGroup(432,505)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.147D6
G = < a,b,c,d | a6=b6=1, c6=b3, 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, 227 normal (13 characteristic)
C1, C2, C3, C4, 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, C4⋊Dic3, C32×C6, C2×C3⋊Dic3, C6×C12, C33⋊5C4, C32×C12, C3×C62, C12⋊Dic3, C2×C33⋊5C4, C3×C6×C12, C62.147D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C3⋊S3, Dic6, D12, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4⋊Dic3, C33⋊C2, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C33⋊5C4, C2×C33⋊C2, C12⋊Dic3, C33⋊8Q8, C33⋊12D4, C2×C33⋊5C4, C62.147D6
►Regular action on 432 points
Generators in S432
(1 90 191 150 304 286)(2 91 192 151 305 287)(3 92 181 152 306 288)(4 93 182 153 307 277)(5 94 183 154 308 278)(6 95 184 155 309 279)(7 96 185 156 310 280)(8 85 186 145 311 281)(9 86 187 146 312 282)(10 87 188 147 301 283)(11 88 189 148 302 284)(12 89 190 149 303 285)(13 354 432 256 330 27)(14 355 421 257 331 28)(15 356 422 258 332 29)(16 357 423 259 333 30)(17 358 424 260 334 31)(18 359 425 261 335 32)(19 360 426 262 336 33)(20 349 427 263 325 34)(21 350 428 264 326 35)(22 351 429 253 327 36)(23 352 430 254 328 25)(24 353 431 255 329 26)(37 324 157 68 82 137)(38 313 158 69 83 138)(39 314 159 70 84 139)(40 315 160 71 73 140)(41 316 161 72 74 141)(42 317 162 61 75 142)(43 318 163 62 76 143)(44 319 164 63 77 144)(45 320 165 64 78 133)(46 321 166 65 79 134)(47 322 167 66 80 135)(48 323 168 67 81 136)(49 209 176 361 201 385)(50 210 177 362 202 386)(51 211 178 363 203 387)(52 212 179 364 204 388)(53 213 180 365 193 389)(54 214 169 366 194 390)(55 215 170 367 195 391)(56 216 171 368 196 392)(57 205 172 369 197 393)(58 206 173 370 198 394)(59 207 174 371 199 395)(60 208 175 372 200 396)(97 338 227 273 374 399)(98 339 228 274 375 400)(99 340 217 275 376 401)(100 341 218 276 377 402)(101 342 219 265 378 403)(102 343 220 266 379 404)(103 344 221 267 380 405)(104 345 222 268 381 406)(105 346 223 269 382 407)(106 347 224 270 383 408)(107 348 225 271 384 397)(108 337 226 272 373 398)(109 237 414 249 291 126)(110 238 415 250 292 127)(111 239 416 251 293 128)(112 240 417 252 294 129)(113 229 418 241 295 130)(114 230 419 242 296 131)(115 231 420 243 297 132)(116 232 409 244 298 121)(117 233 410 245 299 122)(118 234 411 246 300 123)(119 235 412 247 289 124)(120 236 413 248 290 125)
(1 356 294 7 350 300)(2 357 295 8 351 289)(3 358 296 9 352 290)(4 359 297 10 353 291)(5 360 298 11 354 292)(6 349 299 12 355 293)(13 250 278 19 244 284)(14 251 279 20 245 285)(15 252 280 21 246 286)(16 241 281 22 247 287)(17 242 282 23 248 288)(18 243 283 24 249 277)(25 413 306 31 419 312)(26 414 307 32 420 301)(27 415 308 33 409 302)(28 416 309 34 410 303)(29 417 310 35 411 304)(30 418 311 36 412 305)(37 213 339 43 207 345)(38 214 340 44 208 346)(39 215 341 45 209 347)(40 216 342 46 210 348)(41 205 343 47 211 337)(42 206 344 48 212 338)(49 106 139 55 100 133)(50 107 140 56 101 134)(51 108 141 57 102 135)(52 97 142 58 103 136)(53 98 143 59 104 137)(54 99 144 60 105 138)(61 198 380 67 204 374)(62 199 381 68 193 375)(63 200 382 69 194 376)(64 201 383 70 195 377)(65 202 384 71 196 378)(66 203 373 72 197 379)(73 392 403 79 386 397)(74 393 404 80 387 398)(75 394 405 81 388 399)(76 395 406 82 389 400)(77 396 407 83 390 401)(78 385 408 84 391 402)(85 429 124 91 423 130)(86 430 125 92 424 131)(87 431 126 93 425 132)(88 432 127 94 426 121)(89 421 128 95 427 122)(90 422 129 96 428 123)(109 182 261 115 188 255)(110 183 262 116 189 256)(111 184 263 117 190 257)(112 185 264 118 191 258)(113 186 253 119 192 259)(114 187 254 120 181 260)(145 327 235 151 333 229)(146 328 236 152 334 230)(147 329 237 153 335 231)(148 330 238 154 336 232)(149 331 239 155 325 233)(150 332 240 156 326 234)(157 365 274 163 371 268)(158 366 275 164 372 269)(159 367 276 165 361 270)(160 368 265 166 362 271)(161 369 266 167 363 272)(162 370 267 168 364 273)(169 217 319 175 223 313)(170 218 320 176 224 314)(171 219 321 177 225 315)(172 220 322 178 226 316)(173 221 323 179 227 317)(174 222 324 180 228 318)
(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)
(1 269 150 105)(2 268 151 104)(3 267 152 103)(4 266 153 102)(5 265 154 101)(6 276 155 100)(7 275 156 99)(8 274 145 98)(9 273 146 97)(10 272 147 108)(11 271 148 107)(12 270 149 106)(13 202 256 210)(14 201 257 209)(15 200 258 208)(16 199 259 207)(17 198 260 206)(18 197 261 205)(19 196 262 216)(20 195 263 215)(21 194 264 214)(22 193 253 213)(23 204 254 212)(24 203 255 211)(25 388 430 179)(26 387 431 178)(27 386 432 177)(28 385 421 176)(29 396 422 175)(30 395 423 174)(31 394 424 173)(32 393 425 172)(33 392 426 171)(34 391 427 170)(35 390 428 169)(36 389 429 180)(37 247 68 119)(38 246 69 118)(39 245 70 117)(40 244 71 116)(41 243 72 115)(42 242 61 114)(43 241 62 113)(44 252 63 112)(45 251 64 111)(46 250 65 110)(47 249 66 109)(48 248 67 120)(49 355 361 331)(50 354 362 330)(51 353 363 329)(52 352 364 328)(53 351 365 327)(54 350 366 326)(55 349 367 325)(56 360 368 336)(57 359 369 335)(58 358 370 334)(59 357 371 333)(60 356 372 332)(73 121 315 409)(74 132 316 420)(75 131 317 419)(76 130 318 418)(77 129 319 417)(78 128 320 416)(79 127 321 415)(80 126 322 414)(81 125 323 413)(82 124 324 412)(83 123 313 411)(84 122 314 410)(85 228 311 400)(86 227 312 399)(87 226 301 398)(88 225 302 397)(89 224 303 408)(90 223 304 407)(91 222 305 406)(92 221 306 405)(93 220 307 404)(94 219 308 403)(95 218 309 402)(96 217 310 401)(133 293 165 239)(134 292 166 238)(135 291 167 237)(136 290 168 236)(137 289 157 235)(138 300 158 234)(139 299 159 233)(140 298 160 232)(141 297 161 231)(142 296 162 230)(143 295 163 229)(144 294 164 240)(181 344 288 380)(182 343 277 379)(183 342 278 378)(184 341 279 377)(185 340 280 376)(186 339 281 375)(187 338 282 374)(188 337 283 373)(189 348 284 384)(190 347 285 383)(191 346 286 382)(192 345 287 381)
G:=sub<Sym(432)| (1,90,191,150,304,286)(2,91,192,151,305,287)(3,92,181,152,306,288)(4,93,182,153,307,277)(5,94,183,154,308,278)(6,95,184,155,309,279)(7,96,185,156,310,280)(8,85,186,145,311,281)(9,86,187,146,312,282)(10,87,188,147,301,283)(11,88,189,148,302,284)(12,89,190,149,303,285)(13,354,432,256,330,27)(14,355,421,257,331,28)(15,356,422,258,332,29)(16,357,423,259,333,30)(17,358,424,260,334,31)(18,359,425,261,335,32)(19,360,426,262,336,33)(20,349,427,263,325,34)(21,350,428,264,326,35)(22,351,429,253,327,36)(23,352,430,254,328,25)(24,353,431,255,329,26)(37,324,157,68,82,137)(38,313,158,69,83,138)(39,314,159,70,84,139)(40,315,160,71,73,140)(41,316,161,72,74,141)(42,317,162,61,75,142)(43,318,163,62,76,143)(44,319,164,63,77,144)(45,320,165,64,78,133)(46,321,166,65,79,134)(47,322,167,66,80,135)(48,323,168,67,81,136)(49,209,176,361,201,385)(50,210,177,362,202,386)(51,211,178,363,203,387)(52,212,179,364,204,388)(53,213,180,365,193,389)(54,214,169,366,194,390)(55,215,170,367,195,391)(56,216,171,368,196,392)(57,205,172,369,197,393)(58,206,173,370,198,394)(59,207,174,371,199,395)(60,208,175,372,200,396)(97,338,227,273,374,399)(98,339,228,274,375,400)(99,340,217,275,376,401)(100,341,218,276,377,402)(101,342,219,265,378,403)(102,343,220,266,379,404)(103,344,221,267,380,405)(104,345,222,268,381,406)(105,346,223,269,382,407)(106,347,224,270,383,408)(107,348,225,271,384,397)(108,337,226,272,373,398)(109,237,414,249,291,126)(110,238,415,250,292,127)(111,239,416,251,293,128)(112,240,417,252,294,129)(113,229,418,241,295,130)(114,230,419,242,296,131)(115,231,420,243,297,132)(116,232,409,244,298,121)(117,233,410,245,299,122)(118,234,411,246,300,123)(119,235,412,247,289,124)(120,236,413,248,290,125), (1,356,294,7,350,300)(2,357,295,8,351,289)(3,358,296,9,352,290)(4,359,297,10,353,291)(5,360,298,11,354,292)(6,349,299,12,355,293)(13,250,278,19,244,284)(14,251,279,20,245,285)(15,252,280,21,246,286)(16,241,281,22,247,287)(17,242,282,23,248,288)(18,243,283,24,249,277)(25,413,306,31,419,312)(26,414,307,32,420,301)(27,415,308,33,409,302)(28,416,309,34,410,303)(29,417,310,35,411,304)(30,418,311,36,412,305)(37,213,339,43,207,345)(38,214,340,44,208,346)(39,215,341,45,209,347)(40,216,342,46,210,348)(41,205,343,47,211,337)(42,206,344,48,212,338)(49,106,139,55,100,133)(50,107,140,56,101,134)(51,108,141,57,102,135)(52,97,142,58,103,136)(53,98,143,59,104,137)(54,99,144,60,105,138)(61,198,380,67,204,374)(62,199,381,68,193,375)(63,200,382,69,194,376)(64,201,383,70,195,377)(65,202,384,71,196,378)(66,203,373,72,197,379)(73,392,403,79,386,397)(74,393,404,80,387,398)(75,394,405,81,388,399)(76,395,406,82,389,400)(77,396,407,83,390,401)(78,385,408,84,391,402)(85,429,124,91,423,130)(86,430,125,92,424,131)(87,431,126,93,425,132)(88,432,127,94,426,121)(89,421,128,95,427,122)(90,422,129,96,428,123)(109,182,261,115,188,255)(110,183,262,116,189,256)(111,184,263,117,190,257)(112,185,264,118,191,258)(113,186,253,119,192,259)(114,187,254,120,181,260)(145,327,235,151,333,229)(146,328,236,152,334,230)(147,329,237,153,335,231)(148,330,238,154,336,232)(149,331,239,155,325,233)(150,332,240,156,326,234)(157,365,274,163,371,268)(158,366,275,164,372,269)(159,367,276,165,361,270)(160,368,265,166,362,271)(161,369,266,167,363,272)(162,370,267,168,364,273)(169,217,319,175,223,313)(170,218,320,176,224,314)(171,219,321,177,225,315)(172,220,322,178,226,316)(173,221,323,179,227,317)(174,222,324,180,228,318), (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), (1,269,150,105)(2,268,151,104)(3,267,152,103)(4,266,153,102)(5,265,154,101)(6,276,155,100)(7,275,156,99)(8,274,145,98)(9,273,146,97)(10,272,147,108)(11,271,148,107)(12,270,149,106)(13,202,256,210)(14,201,257,209)(15,200,258,208)(16,199,259,207)(17,198,260,206)(18,197,261,205)(19,196,262,216)(20,195,263,215)(21,194,264,214)(22,193,253,213)(23,204,254,212)(24,203,255,211)(25,388,430,179)(26,387,431,178)(27,386,432,177)(28,385,421,176)(29,396,422,175)(30,395,423,174)(31,394,424,173)(32,393,425,172)(33,392,426,171)(34,391,427,170)(35,390,428,169)(36,389,429,180)(37,247,68,119)(38,246,69,118)(39,245,70,117)(40,244,71,116)(41,243,72,115)(42,242,61,114)(43,241,62,113)(44,252,63,112)(45,251,64,111)(46,250,65,110)(47,249,66,109)(48,248,67,120)(49,355,361,331)(50,354,362,330)(51,353,363,329)(52,352,364,328)(53,351,365,327)(54,350,366,326)(55,349,367,325)(56,360,368,336)(57,359,369,335)(58,358,370,334)(59,357,371,333)(60,356,372,332)(73,121,315,409)(74,132,316,420)(75,131,317,419)(76,130,318,418)(77,129,319,417)(78,128,320,416)(79,127,321,415)(80,126,322,414)(81,125,323,413)(82,124,324,412)(83,123,313,411)(84,122,314,410)(85,228,311,400)(86,227,312,399)(87,226,301,398)(88,225,302,397)(89,224,303,408)(90,223,304,407)(91,222,305,406)(92,221,306,405)(93,220,307,404)(94,219,308,403)(95,218,309,402)(96,217,310,401)(133,293,165,239)(134,292,166,238)(135,291,167,237)(136,290,168,236)(137,289,157,235)(138,300,158,234)(139,299,159,233)(140,298,160,232)(141,297,161,231)(142,296,162,230)(143,295,163,229)(144,294,164,240)(181,344,288,380)(182,343,277,379)(183,342,278,378)(184,341,279,377)(185,340,280,376)(186,339,281,375)(187,338,282,374)(188,337,283,373)(189,348,284,384)(190,347,285,383)(191,346,286,382)(192,345,287,381)>;
G:=Group( (1,90,191,150,304,286)(2,91,192,151,305,287)(3,92,181,152,306,288)(4,93,182,153,307,277)(5,94,183,154,308,278)(6,95,184,155,309,279)(7,96,185,156,310,280)(8,85,186,145,311,281)(9,86,187,146,312,282)(10,87,188,147,301,283)(11,88,189,148,302,284)(12,89,190,149,303,285)(13,354,432,256,330,27)(14,355,421,257,331,28)(15,356,422,258,332,29)(16,357,423,259,333,30)(17,358,424,260,334,31)(18,359,425,261,335,32)(19,360,426,262,336,33)(20,349,427,263,325,34)(21,350,428,264,326,35)(22,351,429,253,327,36)(23,352,430,254,328,25)(24,353,431,255,329,26)(37,324,157,68,82,137)(38,313,158,69,83,138)(39,314,159,70,84,139)(40,315,160,71,73,140)(41,316,161,72,74,141)(42,317,162,61,75,142)(43,318,163,62,76,143)(44,319,164,63,77,144)(45,320,165,64,78,133)(46,321,166,65,79,134)(47,322,167,66,80,135)(48,323,168,67,81,136)(49,209,176,361,201,385)(50,210,177,362,202,386)(51,211,178,363,203,387)(52,212,179,364,204,388)(53,213,180,365,193,389)(54,214,169,366,194,390)(55,215,170,367,195,391)(56,216,171,368,196,392)(57,205,172,369,197,393)(58,206,173,370,198,394)(59,207,174,371,199,395)(60,208,175,372,200,396)(97,338,227,273,374,399)(98,339,228,274,375,400)(99,340,217,275,376,401)(100,341,218,276,377,402)(101,342,219,265,378,403)(102,343,220,266,379,404)(103,344,221,267,380,405)(104,345,222,268,381,406)(105,346,223,269,382,407)(106,347,224,270,383,408)(107,348,225,271,384,397)(108,337,226,272,373,398)(109,237,414,249,291,126)(110,238,415,250,292,127)(111,239,416,251,293,128)(112,240,417,252,294,129)(113,229,418,241,295,130)(114,230,419,242,296,131)(115,231,420,243,297,132)(116,232,409,244,298,121)(117,233,410,245,299,122)(118,234,411,246,300,123)(119,235,412,247,289,124)(120,236,413,248,290,125), (1,356,294,7,350,300)(2,357,295,8,351,289)(3,358,296,9,352,290)(4,359,297,10,353,291)(5,360,298,11,354,292)(6,349,299,12,355,293)(13,250,278,19,244,284)(14,251,279,20,245,285)(15,252,280,21,246,286)(16,241,281,22,247,287)(17,242,282,23,248,288)(18,243,283,24,249,277)(25,413,306,31,419,312)(26,414,307,32,420,301)(27,415,308,33,409,302)(28,416,309,34,410,303)(29,417,310,35,411,304)(30,418,311,36,412,305)(37,213,339,43,207,345)(38,214,340,44,208,346)(39,215,341,45,209,347)(40,216,342,46,210,348)(41,205,343,47,211,337)(42,206,344,48,212,338)(49,106,139,55,100,133)(50,107,140,56,101,134)(51,108,141,57,102,135)(52,97,142,58,103,136)(53,98,143,59,104,137)(54,99,144,60,105,138)(61,198,380,67,204,374)(62,199,381,68,193,375)(63,200,382,69,194,376)(64,201,383,70,195,377)(65,202,384,71,196,378)(66,203,373,72,197,379)(73,392,403,79,386,397)(74,393,404,80,387,398)(75,394,405,81,388,399)(76,395,406,82,389,400)(77,396,407,83,390,401)(78,385,408,84,391,402)(85,429,124,91,423,130)(86,430,125,92,424,131)(87,431,126,93,425,132)(88,432,127,94,426,121)(89,421,128,95,427,122)(90,422,129,96,428,123)(109,182,261,115,188,255)(110,183,262,116,189,256)(111,184,263,117,190,257)(112,185,264,118,191,258)(113,186,253,119,192,259)(114,187,254,120,181,260)(145,327,235,151,333,229)(146,328,236,152,334,230)(147,329,237,153,335,231)(148,330,238,154,336,232)(149,331,239,155,325,233)(150,332,240,156,326,234)(157,365,274,163,371,268)(158,366,275,164,372,269)(159,367,276,165,361,270)(160,368,265,166,362,271)(161,369,266,167,363,272)(162,370,267,168,364,273)(169,217,319,175,223,313)(170,218,320,176,224,314)(171,219,321,177,225,315)(172,220,322,178,226,316)(173,221,323,179,227,317)(174,222,324,180,228,318), (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), (1,269,150,105)(2,268,151,104)(3,267,152,103)(4,266,153,102)(5,265,154,101)(6,276,155,100)(7,275,156,99)(8,274,145,98)(9,273,146,97)(10,272,147,108)(11,271,148,107)(12,270,149,106)(13,202,256,210)(14,201,257,209)(15,200,258,208)(16,199,259,207)(17,198,260,206)(18,197,261,205)(19,196,262,216)(20,195,263,215)(21,194,264,214)(22,193,253,213)(23,204,254,212)(24,203,255,211)(25,388,430,179)(26,387,431,178)(27,386,432,177)(28,385,421,176)(29,396,422,175)(30,395,423,174)(31,394,424,173)(32,393,425,172)(33,392,426,171)(34,391,427,170)(35,390,428,169)(36,389,429,180)(37,247,68,119)(38,246,69,118)(39,245,70,117)(40,244,71,116)(41,243,72,115)(42,242,61,114)(43,241,62,113)(44,252,63,112)(45,251,64,111)(46,250,65,110)(47,249,66,109)(48,248,67,120)(49,355,361,331)(50,354,362,330)(51,353,363,329)(52,352,364,328)(53,351,365,327)(54,350,366,326)(55,349,367,325)(56,360,368,336)(57,359,369,335)(58,358,370,334)(59,357,371,333)(60,356,372,332)(73,121,315,409)(74,132,316,420)(75,131,317,419)(76,130,318,418)(77,129,319,417)(78,128,320,416)(79,127,321,415)(80,126,322,414)(81,125,323,413)(82,124,324,412)(83,123,313,411)(84,122,314,410)(85,228,311,400)(86,227,312,399)(87,226,301,398)(88,225,302,397)(89,224,303,408)(90,223,304,407)(91,222,305,406)(92,221,306,405)(93,220,307,404)(94,219,308,403)(95,218,309,402)(96,217,310,401)(133,293,165,239)(134,292,166,238)(135,291,167,237)(136,290,168,236)(137,289,157,235)(138,300,158,234)(139,299,159,233)(140,298,160,232)(141,297,161,231)(142,296,162,230)(143,295,163,229)(144,294,164,240)(181,344,288,380)(182,343,277,379)(183,342,278,378)(184,341,279,377)(185,340,280,376)(186,339,281,375)(187,338,282,374)(188,337,283,373)(189,348,284,384)(190,347,285,383)(191,346,286,382)(192,345,287,381) );
G=PermutationGroup([[(1,90,191,150,304,286),(2,91,192,151,305,287),(3,92,181,152,306,288),(4,93,182,153,307,277),(5,94,183,154,308,278),(6,95,184,155,309,279),(7,96,185,156,310,280),(8,85,186,145,311,281),(9,86,187,146,312,282),(10,87,188,147,301,283),(11,88,189,148,302,284),(12,89,190,149,303,285),(13,354,432,256,330,27),(14,355,421,257,331,28),(15,356,422,258,332,29),(16,357,423,259,333,30),(17,358,424,260,334,31),(18,359,425,261,335,32),(19,360,426,262,336,33),(20,349,427,263,325,34),(21,350,428,264,326,35),(22,351,429,253,327,36),(23,352,430,254,328,25),(24,353,431,255,329,26),(37,324,157,68,82,137),(38,313,158,69,83,138),(39,314,159,70,84,139),(40,315,160,71,73,140),(41,316,161,72,74,141),(42,317,162,61,75,142),(43,318,163,62,76,143),(44,319,164,63,77,144),(45,320,165,64,78,133),(46,321,166,65,79,134),(47,322,167,66,80,135),(48,323,168,67,81,136),(49,209,176,361,201,385),(50,210,177,362,202,386),(51,211,178,363,203,387),(52,212,179,364,204,388),(53,213,180,365,193,389),(54,214,169,366,194,390),(55,215,170,367,195,391),(56,216,171,368,196,392),(57,205,172,369,197,393),(58,206,173,370,198,394),(59,207,174,371,199,395),(60,208,175,372,200,396),(97,338,227,273,374,399),(98,339,228,274,375,400),(99,340,217,275,376,401),(100,341,218,276,377,402),(101,342,219,265,378,403),(102,343,220,266,379,404),(103,344,221,267,380,405),(104,345,222,268,381,406),(105,346,223,269,382,407),(106,347,224,270,383,408),(107,348,225,271,384,397),(108,337,226,272,373,398),(109,237,414,249,291,126),(110,238,415,250,292,127),(111,239,416,251,293,128),(112,240,417,252,294,129),(113,229,418,241,295,130),(114,230,419,242,296,131),(115,231,420,243,297,132),(116,232,409,244,298,121),(117,233,410,245,299,122),(118,234,411,246,300,123),(119,235,412,247,289,124),(120,236,413,248,290,125)], [(1,356,294,7,350,300),(2,357,295,8,351,289),(3,358,296,9,352,290),(4,359,297,10,353,291),(5,360,298,11,354,292),(6,349,299,12,355,293),(13,250,278,19,244,284),(14,251,279,20,245,285),(15,252,280,21,246,286),(16,241,281,22,247,287),(17,242,282,23,248,288),(18,243,283,24,249,277),(25,413,306,31,419,312),(26,414,307,32,420,301),(27,415,308,33,409,302),(28,416,309,34,410,303),(29,417,310,35,411,304),(30,418,311,36,412,305),(37,213,339,43,207,345),(38,214,340,44,208,346),(39,215,341,45,209,347),(40,216,342,46,210,348),(41,205,343,47,211,337),(42,206,344,48,212,338),(49,106,139,55,100,133),(50,107,140,56,101,134),(51,108,141,57,102,135),(52,97,142,58,103,136),(53,98,143,59,104,137),(54,99,144,60,105,138),(61,198,380,67,204,374),(62,199,381,68,193,375),(63,200,382,69,194,376),(64,201,383,70,195,377),(65,202,384,71,196,378),(66,203,373,72,197,379),(73,392,403,79,386,397),(74,393,404,80,387,398),(75,394,405,81,388,399),(76,395,406,82,389,400),(77,396,407,83,390,401),(78,385,408,84,391,402),(85,429,124,91,423,130),(86,430,125,92,424,131),(87,431,126,93,425,132),(88,432,127,94,426,121),(89,421,128,95,427,122),(90,422,129,96,428,123),(109,182,261,115,188,255),(110,183,262,116,189,256),(111,184,263,117,190,257),(112,185,264,118,191,258),(113,186,253,119,192,259),(114,187,254,120,181,260),(145,327,235,151,333,229),(146,328,236,152,334,230),(147,329,237,153,335,231),(148,330,238,154,336,232),(149,331,239,155,325,233),(150,332,240,156,326,234),(157,365,274,163,371,268),(158,366,275,164,372,269),(159,367,276,165,361,270),(160,368,265,166,362,271),(161,369,266,167,363,272),(162,370,267,168,364,273),(169,217,319,175,223,313),(170,218,320,176,224,314),(171,219,321,177,225,315),(172,220,322,178,226,316),(173,221,323,179,227,317),(174,222,324,180,228,318)], [(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)], [(1,269,150,105),(2,268,151,104),(3,267,152,103),(4,266,153,102),(5,265,154,101),(6,276,155,100),(7,275,156,99),(8,274,145,98),(9,273,146,97),(10,272,147,108),(11,271,148,107),(12,270,149,106),(13,202,256,210),(14,201,257,209),(15,200,258,208),(16,199,259,207),(17,198,260,206),(18,197,261,205),(19,196,262,216),(20,195,263,215),(21,194,264,214),(22,193,253,213),(23,204,254,212),(24,203,255,211),(25,388,430,179),(26,387,431,178),(27,386,432,177),(28,385,421,176),(29,396,422,175),(30,395,423,174),(31,394,424,173),(32,393,425,172),(33,392,426,171),(34,391,427,170),(35,390,428,169),(36,389,429,180),(37,247,68,119),(38,246,69,118),(39,245,70,117),(40,244,71,116),(41,243,72,115),(42,242,61,114),(43,241,62,113),(44,252,63,112),(45,251,64,111),(46,250,65,110),(47,249,66,109),(48,248,67,120),(49,355,361,331),(50,354,362,330),(51,353,363,329),(52,352,364,328),(53,351,365,327),(54,350,366,326),(55,349,367,325),(56,360,368,336),(57,359,369,335),(58,358,370,334),(59,357,371,333),(60,356,372,332),(73,121,315,409),(74,132,316,420),(75,131,317,419),(76,130,318,418),(77,129,319,417),(78,128,320,416),(79,127,321,415),(80,126,322,414),(81,125,323,413),(82,124,324,412),(83,123,313,411),(84,122,314,410),(85,228,311,400),(86,227,312,399),(87,226,301,398),(88,225,302,397),(89,224,303,408),(90,223,304,407),(91,222,305,406),(92,221,306,405),(93,220,307,404),(94,219,308,403),(95,218,309,402),(96,217,310,401),(133,293,165,239),(134,292,166,238),(135,291,167,237),(136,290,168,236),(137,289,157,235),(138,300,158,234),(139,299,159,233),(140,298,160,232),(141,297,161,231),(142,296,162,230),(143,295,163,229),(144,294,164,240),(181,344,288,380),(182,343,277,379),(183,342,278,378),(184,341,279,377),(185,340,280,376),(186,339,281,375),(187,338,282,374),(188,337,283,373),(189,348,284,384),(190,347,285,383),(191,346,286,382),(192,345,287,381)]])
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 | Dic3 | D6 | Dic6 | D12 |
| kernel | C62.147D6 | C2×C33⋊5C4 | C3×C6×C12 | C32×C12 | C6×C12 | C32×C6 | C32×C6 | C3×C12 | C62 | C3×C6 | C3×C6 |
| # reps | 1 | 2 | 1 | 4 | 13 | 1 | 1 | 26 | 13 | 26 | 26 |
Matrix representation of C62.147D6 ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 0 | 10 | 7 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,3,7,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,6,10,0,0,0,0,0,3,7] >;
C62.147D6 in GAP, Magma, Sage, TeX
C_6^2._{147}D_6 % in TeX
G:=Group("C6^2.147D6"); // GroupNames label
G:=SmallGroup(432,505);
// by ID
G=gap.SmallGroup(432,505);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,1124,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^3,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