direct product, metabelian, supersoluble, monomial
Aliases: C2×C12.D9, C18⋊2Dic6, C6⋊2Dic18, C36.59D6, C12.59D18, C62.134D6, (C3×C18)⋊4Q8, (C2×C36).9S3, C9⋊3(C2×Dic6), (C2×C12).9D9, (C6×C12).31S3, (C6×C36).10C2, C3⋊3(C2×Dic18), (C2×C18).41D6, (C2×C6).41D18, (C3×C12).194D6, (C3×C6).27Dic6, C6.39(C22×D9), C18.39(C22×S3), (C3×C36).63C22, (C6×C18).47C22, (C3×C18).48C23, C32.5(C2×Dic6), C6.6(C32⋊4Q8), C9⋊Dic3.13C22, (C3×C9)⋊7(C2×Q8), C4.11(C2×C9⋊S3), C12.62(C2×C3⋊S3), (C2×C4).4(C9⋊S3), C3.(C2×C32⋊4Q8), C22.8(C2×C9⋊S3), C2.3(C22×C9⋊S3), (C2×C12).8(C3⋊S3), C6.28(C22×C3⋊S3), (C2×C9⋊Dic3).8C2, (C3×C6).162(C22×S3), (C2×C6).34(C2×C3⋊S3), SmallGroup(432,380)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.D9
G = < a,b,c,d | a2=b12=c9=1, d2=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 916 in 190 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C9, C32, Dic3, C12, C2×C6, C2×C6, C2×Q8, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×C9, Dic9, C36, C2×C18, C3⋊Dic3, C3×C12, C62, C2×Dic6, C3×C18, C3×C18, Dic18, C2×Dic9, C2×C36, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C9⋊Dic3, C3×C36, C6×C18, C2×Dic18, C2×C32⋊4Q8, C12.D9, C2×C9⋊Dic3, C6×C36, C2×C12.D9
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, C3⋊S3, Dic6, C22×S3, D18, C2×C3⋊S3, C2×Dic6, C9⋊S3, Dic18, C22×D9, C32⋊4Q8, C22×C3⋊S3, C2×C9⋊S3, C2×Dic18, C2×C32⋊4Q8, C12.D9, C22×C9⋊S3, C2×C12.D9
►Regular action on 432 points
Generators in S432
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 97)(11 98)(12 99)(13 303)(14 304)(15 305)(16 306)(17 307)(18 308)(19 309)(20 310)(21 311)(22 312)(23 301)(24 302)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 49)(35 50)(36 51)(37 282)(38 283)(39 284)(40 285)(41 286)(42 287)(43 288)(44 277)(45 278)(46 279)(47 280)(48 281)(61 424)(62 425)(63 426)(64 427)(65 428)(66 429)(67 430)(68 431)(69 432)(70 421)(71 422)(72 423)(73 220)(74 221)(75 222)(76 223)(77 224)(78 225)(79 226)(80 227)(81 228)(82 217)(83 218)(84 219)(85 368)(86 369)(87 370)(88 371)(89 372)(90 361)(91 362)(92 363)(93 364)(94 365)(95 366)(96 367)(109 376)(110 377)(111 378)(112 379)(113 380)(114 381)(115 382)(116 383)(117 384)(118 373)(119 374)(120 375)(121 415)(122 416)(123 417)(124 418)(125 419)(126 420)(127 409)(128 410)(129 411)(130 412)(131 413)(132 414)(133 385)(134 386)(135 387)(136 388)(137 389)(138 390)(139 391)(140 392)(141 393)(142 394)(143 395)(144 396)(145 240)(146 229)(147 230)(148 231)(149 232)(150 233)(151 234)(152 235)(153 236)(154 237)(155 238)(156 239)(157 327)(158 328)(159 329)(160 330)(161 331)(162 332)(163 333)(164 334)(165 335)(166 336)(167 325)(168 326)(169 337)(170 338)(171 339)(172 340)(173 341)(174 342)(175 343)(176 344)(177 345)(178 346)(179 347)(180 348)(181 202)(182 203)(183 204)(184 193)(185 194)(186 195)(187 196)(188 197)(189 198)(190 199)(191 200)(192 201)(205 276)(206 265)(207 266)(208 267)(209 268)(210 269)(211 270)(212 271)(213 272)(214 273)(215 274)(216 275)(241 296)(242 297)(243 298)(244 299)(245 300)(246 289)(247 290)(248 291)(249 292)(250 293)(251 294)(252 295)(253 349)(254 350)(255 351)(256 352)(257 353)(258 354)(259 355)(260 356)(261 357)(262 358)(263 359)(264 360)(313 407)(314 408)(315 397)(316 398)(317 399)(318 400)(319 401)(320 402)(321 403)(322 404)(323 405)(324 406)
(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)
(1 183 130 179 229 214 51 285 395)(2 184 131 180 230 215 52 286 396)(3 185 132 169 231 216 53 287 385)(4 186 121 170 232 205 54 288 386)(5 187 122 171 233 206 55 277 387)(6 188 123 172 234 207 56 278 388)(7 189 124 173 235 208 57 279 389)(8 190 125 174 236 209 58 280 390)(9 191 126 175 237 210 59 281 391)(10 192 127 176 238 211 60 282 392)(11 181 128 177 239 212 49 283 393)(12 182 129 178 240 213 50 284 394)(13 72 318 162 87 299 356 221 379)(14 61 319 163 88 300 357 222 380)(15 62 320 164 89 289 358 223 381)(16 63 321 165 90 290 359 224 382)(17 64 322 166 91 291 360 225 383)(18 65 323 167 92 292 349 226 384)(19 66 324 168 93 293 350 227 373)(20 67 313 157 94 294 351 228 374)(21 68 314 158 95 295 352 217 375)(22 69 315 159 96 296 353 218 376)(23 70 316 160 85 297 354 219 377)(24 71 317 161 86 298 355 220 378)(25 41 144 101 193 413 348 147 274)(26 42 133 102 194 414 337 148 275)(27 43 134 103 195 415 338 149 276)(28 44 135 104 196 416 339 150 265)(29 45 136 105 197 417 340 151 266)(30 46 137 106 198 418 341 152 267)(31 47 138 107 199 419 342 153 268)(32 48 139 108 200 420 343 154 269)(33 37 140 97 201 409 344 155 270)(34 38 141 98 202 410 345 156 271)(35 39 142 99 203 411 346 145 272)(36 40 143 100 204 412 347 146 273)(73 111 302 422 399 331 369 243 259)(74 112 303 423 400 332 370 244 260)(75 113 304 424 401 333 371 245 261)(76 114 305 425 402 334 372 246 262)(77 115 306 426 403 335 361 247 263)(78 116 307 427 404 336 362 248 264)(79 117 308 428 405 325 363 249 253)(80 118 309 429 406 326 364 250 254)(81 119 310 430 407 327 365 251 255)(82 120 311 431 408 328 366 252 256)(83 109 312 432 397 329 367 241 257)(84 110 301 421 398 330 368 242 258)
(1 117 7 111)(2 116 8 110)(3 115 9 109)(4 114 10 120)(5 113 11 119)(6 112 12 118)(13 142 19 136)(14 141 20 135)(15 140 21 134)(16 139 22 133)(17 138 23 144)(18 137 24 143)(25 322 31 316)(26 321 32 315)(27 320 33 314)(28 319 34 313)(29 318 35 324)(30 317 36 323)(37 68 43 62)(38 67 44 61)(39 66 45 72)(40 65 46 71)(41 64 47 70)(42 63 48 69)(49 407 55 401)(50 406 56 400)(51 405 57 399)(52 404 58 398)(53 403 59 397)(54 402 60 408)(73 183 79 189)(74 182 80 188)(75 181 81 187)(76 192 82 186)(77 191 83 185)(78 190 84 184)(85 147 91 153)(86 146 92 152)(87 145 93 151)(88 156 94 150)(89 155 95 149)(90 154 96 148)(97 375 103 381)(98 374 104 380)(99 373 105 379)(100 384 106 378)(101 383 107 377)(102 382 108 376)(121 262 127 256)(122 261 128 255)(123 260 129 254)(124 259 130 253)(125 258 131 264)(126 257 132 263)(157 265 163 271)(158 276 164 270)(159 275 165 269)(160 274 166 268)(161 273 167 267)(162 272 168 266)(169 247 175 241)(170 246 176 252)(171 245 177 251)(172 244 178 250)(173 243 179 249)(174 242 180 248)(193 225 199 219)(194 224 200 218)(195 223 201 217)(196 222 202 228)(197 221 203 227)(198 220 204 226)(205 334 211 328)(206 333 212 327)(207 332 213 326)(208 331 214 325)(209 330 215 336)(210 329 216 335)(229 363 235 369)(230 362 236 368)(231 361 237 367)(232 372 238 366)(233 371 239 365)(234 370 240 364)(277 424 283 430)(278 423 284 429)(279 422 285 428)(280 421 286 427)(281 432 287 426)(282 431 288 425)(289 344 295 338)(290 343 296 337)(291 342 297 348)(292 341 298 347)(293 340 299 346)(294 339 300 345)(301 396 307 390)(302 395 308 389)(303 394 309 388)(304 393 310 387)(305 392 311 386)(306 391 312 385)(349 418 355 412)(350 417 356 411)(351 416 357 410)(352 415 358 409)(353 414 359 420)(354 413 360 419)
G:=sub<Sym(432)| (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,97)(11,98)(12,99)(13,303)(14,304)(15,305)(16,306)(17,307)(18,308)(19,309)(20,310)(21,311)(22,312)(23,301)(24,302)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,282)(38,283)(39,284)(40,285)(41,286)(42,287)(43,288)(44,277)(45,278)(46,279)(47,280)(48,281)(61,424)(62,425)(63,426)(64,427)(65,428)(66,429)(67,430)(68,431)(69,432)(70,421)(71,422)(72,423)(73,220)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,228)(82,217)(83,218)(84,219)(85,368)(86,369)(87,370)(88,371)(89,372)(90,361)(91,362)(92,363)(93,364)(94,365)(95,366)(96,367)(109,376)(110,377)(111,378)(112,379)(113,380)(114,381)(115,382)(116,383)(117,384)(118,373)(119,374)(120,375)(121,415)(122,416)(123,417)(124,418)(125,419)(126,420)(127,409)(128,410)(129,411)(130,412)(131,413)(132,414)(133,385)(134,386)(135,387)(136,388)(137,389)(138,390)(139,391)(140,392)(141,393)(142,394)(143,395)(144,396)(145,240)(146,229)(147,230)(148,231)(149,232)(150,233)(151,234)(152,235)(153,236)(154,237)(155,238)(156,239)(157,327)(158,328)(159,329)(160,330)(161,331)(162,332)(163,333)(164,334)(165,335)(166,336)(167,325)(168,326)(169,337)(170,338)(171,339)(172,340)(173,341)(174,342)(175,343)(176,344)(177,345)(178,346)(179,347)(180,348)(181,202)(182,203)(183,204)(184,193)(185,194)(186,195)(187,196)(188,197)(189,198)(190,199)(191,200)(192,201)(205,276)(206,265)(207,266)(208,267)(209,268)(210,269)(211,270)(212,271)(213,272)(214,273)(215,274)(216,275)(241,296)(242,297)(243,298)(244,299)(245,300)(246,289)(247,290)(248,291)(249,292)(250,293)(251,294)(252,295)(253,349)(254,350)(255,351)(256,352)(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)(264,360)(313,407)(314,408)(315,397)(316,398)(317,399)(318,400)(319,401)(320,402)(321,403)(322,404)(323,405)(324,406), (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), (1,183,130,179,229,214,51,285,395)(2,184,131,180,230,215,52,286,396)(3,185,132,169,231,216,53,287,385)(4,186,121,170,232,205,54,288,386)(5,187,122,171,233,206,55,277,387)(6,188,123,172,234,207,56,278,388)(7,189,124,173,235,208,57,279,389)(8,190,125,174,236,209,58,280,390)(9,191,126,175,237,210,59,281,391)(10,192,127,176,238,211,60,282,392)(11,181,128,177,239,212,49,283,393)(12,182,129,178,240,213,50,284,394)(13,72,318,162,87,299,356,221,379)(14,61,319,163,88,300,357,222,380)(15,62,320,164,89,289,358,223,381)(16,63,321,165,90,290,359,224,382)(17,64,322,166,91,291,360,225,383)(18,65,323,167,92,292,349,226,384)(19,66,324,168,93,293,350,227,373)(20,67,313,157,94,294,351,228,374)(21,68,314,158,95,295,352,217,375)(22,69,315,159,96,296,353,218,376)(23,70,316,160,85,297,354,219,377)(24,71,317,161,86,298,355,220,378)(25,41,144,101,193,413,348,147,274)(26,42,133,102,194,414,337,148,275)(27,43,134,103,195,415,338,149,276)(28,44,135,104,196,416,339,150,265)(29,45,136,105,197,417,340,151,266)(30,46,137,106,198,418,341,152,267)(31,47,138,107,199,419,342,153,268)(32,48,139,108,200,420,343,154,269)(33,37,140,97,201,409,344,155,270)(34,38,141,98,202,410,345,156,271)(35,39,142,99,203,411,346,145,272)(36,40,143,100,204,412,347,146,273)(73,111,302,422,399,331,369,243,259)(74,112,303,423,400,332,370,244,260)(75,113,304,424,401,333,371,245,261)(76,114,305,425,402,334,372,246,262)(77,115,306,426,403,335,361,247,263)(78,116,307,427,404,336,362,248,264)(79,117,308,428,405,325,363,249,253)(80,118,309,429,406,326,364,250,254)(81,119,310,430,407,327,365,251,255)(82,120,311,431,408,328,366,252,256)(83,109,312,432,397,329,367,241,257)(84,110,301,421,398,330,368,242,258), (1,117,7,111)(2,116,8,110)(3,115,9,109)(4,114,10,120)(5,113,11,119)(6,112,12,118)(13,142,19,136)(14,141,20,135)(15,140,21,134)(16,139,22,133)(17,138,23,144)(18,137,24,143)(25,322,31,316)(26,321,32,315)(27,320,33,314)(28,319,34,313)(29,318,35,324)(30,317,36,323)(37,68,43,62)(38,67,44,61)(39,66,45,72)(40,65,46,71)(41,64,47,70)(42,63,48,69)(49,407,55,401)(50,406,56,400)(51,405,57,399)(52,404,58,398)(53,403,59,397)(54,402,60,408)(73,183,79,189)(74,182,80,188)(75,181,81,187)(76,192,82,186)(77,191,83,185)(78,190,84,184)(85,147,91,153)(86,146,92,152)(87,145,93,151)(88,156,94,150)(89,155,95,149)(90,154,96,148)(97,375,103,381)(98,374,104,380)(99,373,105,379)(100,384,106,378)(101,383,107,377)(102,382,108,376)(121,262,127,256)(122,261,128,255)(123,260,129,254)(124,259,130,253)(125,258,131,264)(126,257,132,263)(157,265,163,271)(158,276,164,270)(159,275,165,269)(160,274,166,268)(161,273,167,267)(162,272,168,266)(169,247,175,241)(170,246,176,252)(171,245,177,251)(172,244,178,250)(173,243,179,249)(174,242,180,248)(193,225,199,219)(194,224,200,218)(195,223,201,217)(196,222,202,228)(197,221,203,227)(198,220,204,226)(205,334,211,328)(206,333,212,327)(207,332,213,326)(208,331,214,325)(209,330,215,336)(210,329,216,335)(229,363,235,369)(230,362,236,368)(231,361,237,367)(232,372,238,366)(233,371,239,365)(234,370,240,364)(277,424,283,430)(278,423,284,429)(279,422,285,428)(280,421,286,427)(281,432,287,426)(282,431,288,425)(289,344,295,338)(290,343,296,337)(291,342,297,348)(292,341,298,347)(293,340,299,346)(294,339,300,345)(301,396,307,390)(302,395,308,389)(303,394,309,388)(304,393,310,387)(305,392,311,386)(306,391,312,385)(349,418,355,412)(350,417,356,411)(351,416,357,410)(352,415,358,409)(353,414,359,420)(354,413,360,419)>;
G:=Group( (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,97)(11,98)(12,99)(13,303)(14,304)(15,305)(16,306)(17,307)(18,308)(19,309)(20,310)(21,311)(22,312)(23,301)(24,302)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,282)(38,283)(39,284)(40,285)(41,286)(42,287)(43,288)(44,277)(45,278)(46,279)(47,280)(48,281)(61,424)(62,425)(63,426)(64,427)(65,428)(66,429)(67,430)(68,431)(69,432)(70,421)(71,422)(72,423)(73,220)(74,221)(75,222)(76,223)(77,224)(78,225)(79,226)(80,227)(81,228)(82,217)(83,218)(84,219)(85,368)(86,369)(87,370)(88,371)(89,372)(90,361)(91,362)(92,363)(93,364)(94,365)(95,366)(96,367)(109,376)(110,377)(111,378)(112,379)(113,380)(114,381)(115,382)(116,383)(117,384)(118,373)(119,374)(120,375)(121,415)(122,416)(123,417)(124,418)(125,419)(126,420)(127,409)(128,410)(129,411)(130,412)(131,413)(132,414)(133,385)(134,386)(135,387)(136,388)(137,389)(138,390)(139,391)(140,392)(141,393)(142,394)(143,395)(144,396)(145,240)(146,229)(147,230)(148,231)(149,232)(150,233)(151,234)(152,235)(153,236)(154,237)(155,238)(156,239)(157,327)(158,328)(159,329)(160,330)(161,331)(162,332)(163,333)(164,334)(165,335)(166,336)(167,325)(168,326)(169,337)(170,338)(171,339)(172,340)(173,341)(174,342)(175,343)(176,344)(177,345)(178,346)(179,347)(180,348)(181,202)(182,203)(183,204)(184,193)(185,194)(186,195)(187,196)(188,197)(189,198)(190,199)(191,200)(192,201)(205,276)(206,265)(207,266)(208,267)(209,268)(210,269)(211,270)(212,271)(213,272)(214,273)(215,274)(216,275)(241,296)(242,297)(243,298)(244,299)(245,300)(246,289)(247,290)(248,291)(249,292)(250,293)(251,294)(252,295)(253,349)(254,350)(255,351)(256,352)(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)(264,360)(313,407)(314,408)(315,397)(316,398)(317,399)(318,400)(319,401)(320,402)(321,403)(322,404)(323,405)(324,406), (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), (1,183,130,179,229,214,51,285,395)(2,184,131,180,230,215,52,286,396)(3,185,132,169,231,216,53,287,385)(4,186,121,170,232,205,54,288,386)(5,187,122,171,233,206,55,277,387)(6,188,123,172,234,207,56,278,388)(7,189,124,173,235,208,57,279,389)(8,190,125,174,236,209,58,280,390)(9,191,126,175,237,210,59,281,391)(10,192,127,176,238,211,60,282,392)(11,181,128,177,239,212,49,283,393)(12,182,129,178,240,213,50,284,394)(13,72,318,162,87,299,356,221,379)(14,61,319,163,88,300,357,222,380)(15,62,320,164,89,289,358,223,381)(16,63,321,165,90,290,359,224,382)(17,64,322,166,91,291,360,225,383)(18,65,323,167,92,292,349,226,384)(19,66,324,168,93,293,350,227,373)(20,67,313,157,94,294,351,228,374)(21,68,314,158,95,295,352,217,375)(22,69,315,159,96,296,353,218,376)(23,70,316,160,85,297,354,219,377)(24,71,317,161,86,298,355,220,378)(25,41,144,101,193,413,348,147,274)(26,42,133,102,194,414,337,148,275)(27,43,134,103,195,415,338,149,276)(28,44,135,104,196,416,339,150,265)(29,45,136,105,197,417,340,151,266)(30,46,137,106,198,418,341,152,267)(31,47,138,107,199,419,342,153,268)(32,48,139,108,200,420,343,154,269)(33,37,140,97,201,409,344,155,270)(34,38,141,98,202,410,345,156,271)(35,39,142,99,203,411,346,145,272)(36,40,143,100,204,412,347,146,273)(73,111,302,422,399,331,369,243,259)(74,112,303,423,400,332,370,244,260)(75,113,304,424,401,333,371,245,261)(76,114,305,425,402,334,372,246,262)(77,115,306,426,403,335,361,247,263)(78,116,307,427,404,336,362,248,264)(79,117,308,428,405,325,363,249,253)(80,118,309,429,406,326,364,250,254)(81,119,310,430,407,327,365,251,255)(82,120,311,431,408,328,366,252,256)(83,109,312,432,397,329,367,241,257)(84,110,301,421,398,330,368,242,258), (1,117,7,111)(2,116,8,110)(3,115,9,109)(4,114,10,120)(5,113,11,119)(6,112,12,118)(13,142,19,136)(14,141,20,135)(15,140,21,134)(16,139,22,133)(17,138,23,144)(18,137,24,143)(25,322,31,316)(26,321,32,315)(27,320,33,314)(28,319,34,313)(29,318,35,324)(30,317,36,323)(37,68,43,62)(38,67,44,61)(39,66,45,72)(40,65,46,71)(41,64,47,70)(42,63,48,69)(49,407,55,401)(50,406,56,400)(51,405,57,399)(52,404,58,398)(53,403,59,397)(54,402,60,408)(73,183,79,189)(74,182,80,188)(75,181,81,187)(76,192,82,186)(77,191,83,185)(78,190,84,184)(85,147,91,153)(86,146,92,152)(87,145,93,151)(88,156,94,150)(89,155,95,149)(90,154,96,148)(97,375,103,381)(98,374,104,380)(99,373,105,379)(100,384,106,378)(101,383,107,377)(102,382,108,376)(121,262,127,256)(122,261,128,255)(123,260,129,254)(124,259,130,253)(125,258,131,264)(126,257,132,263)(157,265,163,271)(158,276,164,270)(159,275,165,269)(160,274,166,268)(161,273,167,267)(162,272,168,266)(169,247,175,241)(170,246,176,252)(171,245,177,251)(172,244,178,250)(173,243,179,249)(174,242,180,248)(193,225,199,219)(194,224,200,218)(195,223,201,217)(196,222,202,228)(197,221,203,227)(198,220,204,226)(205,334,211,328)(206,333,212,327)(207,332,213,326)(208,331,214,325)(209,330,215,336)(210,329,216,335)(229,363,235,369)(230,362,236,368)(231,361,237,367)(232,372,238,366)(233,371,239,365)(234,370,240,364)(277,424,283,430)(278,423,284,429)(279,422,285,428)(280,421,286,427)(281,432,287,426)(282,431,288,425)(289,344,295,338)(290,343,296,337)(291,342,297,348)(292,341,298,347)(293,340,299,346)(294,339,300,345)(301,396,307,390)(302,395,308,389)(303,394,309,388)(304,393,310,387)(305,392,311,386)(306,391,312,385)(349,418,355,412)(350,417,356,411)(351,416,357,410)(352,415,358,409)(353,414,359,420)(354,413,360,419) );
G=PermutationGroup([[(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,97),(11,98),(12,99),(13,303),(14,304),(15,305),(16,306),(17,307),(18,308),(19,309),(20,310),(21,311),(22,312),(23,301),(24,302),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,49),(35,50),(36,51),(37,282),(38,283),(39,284),(40,285),(41,286),(42,287),(43,288),(44,277),(45,278),(46,279),(47,280),(48,281),(61,424),(62,425),(63,426),(64,427),(65,428),(66,429),(67,430),(68,431),(69,432),(70,421),(71,422),(72,423),(73,220),(74,221),(75,222),(76,223),(77,224),(78,225),(79,226),(80,227),(81,228),(82,217),(83,218),(84,219),(85,368),(86,369),(87,370),(88,371),(89,372),(90,361),(91,362),(92,363),(93,364),(94,365),(95,366),(96,367),(109,376),(110,377),(111,378),(112,379),(113,380),(114,381),(115,382),(116,383),(117,384),(118,373),(119,374),(120,375),(121,415),(122,416),(123,417),(124,418),(125,419),(126,420),(127,409),(128,410),(129,411),(130,412),(131,413),(132,414),(133,385),(134,386),(135,387),(136,388),(137,389),(138,390),(139,391),(140,392),(141,393),(142,394),(143,395),(144,396),(145,240),(146,229),(147,230),(148,231),(149,232),(150,233),(151,234),(152,235),(153,236),(154,237),(155,238),(156,239),(157,327),(158,328),(159,329),(160,330),(161,331),(162,332),(163,333),(164,334),(165,335),(166,336),(167,325),(168,326),(169,337),(170,338),(171,339),(172,340),(173,341),(174,342),(175,343),(176,344),(177,345),(178,346),(179,347),(180,348),(181,202),(182,203),(183,204),(184,193),(185,194),(186,195),(187,196),(188,197),(189,198),(190,199),(191,200),(192,201),(205,276),(206,265),(207,266),(208,267),(209,268),(210,269),(211,270),(212,271),(213,272),(214,273),(215,274),(216,275),(241,296),(242,297),(243,298),(244,299),(245,300),(246,289),(247,290),(248,291),(249,292),(250,293),(251,294),(252,295),(253,349),(254,350),(255,351),(256,352),(257,353),(258,354),(259,355),(260,356),(261,357),(262,358),(263,359),(264,360),(313,407),(314,408),(315,397),(316,398),(317,399),(318,400),(319,401),(320,402),(321,403),(322,404),(323,405),(324,406)], [(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)], [(1,183,130,179,229,214,51,285,395),(2,184,131,180,230,215,52,286,396),(3,185,132,169,231,216,53,287,385),(4,186,121,170,232,205,54,288,386),(5,187,122,171,233,206,55,277,387),(6,188,123,172,234,207,56,278,388),(7,189,124,173,235,208,57,279,389),(8,190,125,174,236,209,58,280,390),(9,191,126,175,237,210,59,281,391),(10,192,127,176,238,211,60,282,392),(11,181,128,177,239,212,49,283,393),(12,182,129,178,240,213,50,284,394),(13,72,318,162,87,299,356,221,379),(14,61,319,163,88,300,357,222,380),(15,62,320,164,89,289,358,223,381),(16,63,321,165,90,290,359,224,382),(17,64,322,166,91,291,360,225,383),(18,65,323,167,92,292,349,226,384),(19,66,324,168,93,293,350,227,373),(20,67,313,157,94,294,351,228,374),(21,68,314,158,95,295,352,217,375),(22,69,315,159,96,296,353,218,376),(23,70,316,160,85,297,354,219,377),(24,71,317,161,86,298,355,220,378),(25,41,144,101,193,413,348,147,274),(26,42,133,102,194,414,337,148,275),(27,43,134,103,195,415,338,149,276),(28,44,135,104,196,416,339,150,265),(29,45,136,105,197,417,340,151,266),(30,46,137,106,198,418,341,152,267),(31,47,138,107,199,419,342,153,268),(32,48,139,108,200,420,343,154,269),(33,37,140,97,201,409,344,155,270),(34,38,141,98,202,410,345,156,271),(35,39,142,99,203,411,346,145,272),(36,40,143,100,204,412,347,146,273),(73,111,302,422,399,331,369,243,259),(74,112,303,423,400,332,370,244,260),(75,113,304,424,401,333,371,245,261),(76,114,305,425,402,334,372,246,262),(77,115,306,426,403,335,361,247,263),(78,116,307,427,404,336,362,248,264),(79,117,308,428,405,325,363,249,253),(80,118,309,429,406,326,364,250,254),(81,119,310,430,407,327,365,251,255),(82,120,311,431,408,328,366,252,256),(83,109,312,432,397,329,367,241,257),(84,110,301,421,398,330,368,242,258)], [(1,117,7,111),(2,116,8,110),(3,115,9,109),(4,114,10,120),(5,113,11,119),(6,112,12,118),(13,142,19,136),(14,141,20,135),(15,140,21,134),(16,139,22,133),(17,138,23,144),(18,137,24,143),(25,322,31,316),(26,321,32,315),(27,320,33,314),(28,319,34,313),(29,318,35,324),(30,317,36,323),(37,68,43,62),(38,67,44,61),(39,66,45,72),(40,65,46,71),(41,64,47,70),(42,63,48,69),(49,407,55,401),(50,406,56,400),(51,405,57,399),(52,404,58,398),(53,403,59,397),(54,402,60,408),(73,183,79,189),(74,182,80,188),(75,181,81,187),(76,192,82,186),(77,191,83,185),(78,190,84,184),(85,147,91,153),(86,146,92,152),(87,145,93,151),(88,156,94,150),(89,155,95,149),(90,154,96,148),(97,375,103,381),(98,374,104,380),(99,373,105,379),(100,384,106,378),(101,383,107,377),(102,382,108,376),(121,262,127,256),(122,261,128,255),(123,260,129,254),(124,259,130,253),(125,258,131,264),(126,257,132,263),(157,265,163,271),(158,276,164,270),(159,275,165,269),(160,274,166,268),(161,273,167,267),(162,272,168,266),(169,247,175,241),(170,246,176,252),(171,245,177,251),(172,244,178,250),(173,243,179,249),(174,242,180,248),(193,225,199,219),(194,224,200,218),(195,223,201,217),(196,222,202,228),(197,221,203,227),(198,220,204,226),(205,334,211,328),(206,333,212,327),(207,332,213,326),(208,331,214,325),(209,330,215,336),(210,329,216,335),(229,363,235,369),(230,362,236,368),(231,361,237,367),(232,372,238,366),(233,371,239,365),(234,370,240,364),(277,424,283,430),(278,423,284,429),(279,422,285,428),(280,421,286,427),(281,432,287,426),(282,431,288,425),(289,344,295,338),(290,343,296,337),(291,342,297,348),(292,341,298,347),(293,340,299,346),(294,339,300,345),(301,396,307,390),(302,395,308,389),(303,394,309,388),(304,393,310,387),(305,392,311,386),(306,391,312,385),(349,418,355,412),(350,417,356,411),(351,416,357,410),(352,415,358,409),(353,414,359,420),(354,413,360,419)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 9A | ··· | 9I | 12A | ··· | 12P | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 54 | 54 | 54 | 54 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D6 | D6 | D9 | Dic6 | Dic6 | D18 | D18 | Dic18 |
| kernel | C2×C12.D9 | C12.D9 | C2×C9⋊Dic3 | C6×C36 | C2×C36 | C6×C12 | C3×C18 | C36 | C2×C18 | C3×C12 | C62 | C2×C12 | C18 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 2 | 1 | 3 | 1 | 2 | 6 | 3 | 2 | 1 | 9 | 12 | 4 | 18 | 9 | 36 |
Matrix representation of C2×C12.D9 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 1 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 5 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 36 | 1 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 0 | 26 | 17 |
| 32 | 32 | 0 | 0 | 0 | 0 |
| 27 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 16 | 0 | 0 |
| 0 | 0 | 3 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 24 |
| 0 | 0 | 0 | 0 | 26 | 35 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,35,36,0,0,0,0,0,0,27,32,0,0,0,0,5,32],[36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,26,0,0,0,0,11,17],[32,27,0,0,0,0,32,5,0,0,0,0,0,0,32,3,0,0,0,0,16,5,0,0,0,0,0,0,2,26,0,0,0,0,24,35] >;
C2×C12.D9 in GAP, Magma, Sage, TeX
C_2\times C_{12}.D_9 % in TeX
G:=Group("C2xC12.D9"); // GroupNames label
G:=SmallGroup(432,380);
// by ID
G=gap.SmallGroup(432,380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^9=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations