metabelian, supersoluble, monomial
Aliases: C12⋊1Dic9, C36⋊1Dic3, C6.10D36, C18.10D12, C6.8Dic18, C18.8Dic6, C62.125D6, (C3×C36)⋊3C4, C4⋊(C9⋊Dic3), (C6×C36).8C2, (C2×C36).8S3, (C2×C12).8D9, (C3×C18).9Q8, (C6×C12).30S3, C3⋊2(C4⋊Dic9), C9⋊2(C4⋊Dic3), (C3×C6).57D12, (C2×C18).37D6, (C3×C18).32D4, (C2×C6).37D18, C6.15(C2×Dic9), (C3×C6).25Dic6, C2.1(C36⋊S3), C6.4(C12⋊S3), (C6×C18).39C22, C12.2(C3⋊Dic3), C3.(C12⋊Dic3), (C3×C12).16Dic3, C18.15(C2×Dic3), C6.5(C32⋊4Q8), C2.2(C12.D9), C32.4(C4⋊Dic3), (C3×C9)⋊9(C4⋊C4), (C2×C4).3(C9⋊S3), C22.5(C2×C9⋊S3), C6.9(C2×C3⋊Dic3), C2.4(C2×C9⋊Dic3), (C2×C12).7(C3⋊S3), (C3×C18).39(C2×C4), (C2×C9⋊Dic3).6C2, (C3×C6).62(C2×Dic3), (C2×C6).31(C2×C3⋊S3), SmallGroup(432,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊Dic9
G = < a,b,c | a12=b18=1, c2=b9, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 612 in 130 conjugacy classes, 83 normal (25 characteristic)
C1, C2 [×3], C3, C3 [×3], C4 [×2], C4 [×2], C22, C6 [×3], C6 [×9], C2×C4, C2×C4 [×2], C9 [×3], C32, Dic3 [×8], C12 [×8], C2×C6, C2×C6 [×3], C4⋊C4, C18 [×9], C3×C6 [×3], C2×Dic3 [×8], C2×C12, C2×C12 [×3], C3×C9, Dic9 [×6], C36 [×6], C2×C18 [×3], C3⋊Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3 [×4], C3×C18 [×3], C2×Dic9 [×6], C2×C36 [×3], C2×C3⋊Dic3 [×2], C6×C12, C9⋊Dic3 [×2], C3×C36 [×2], C6×C18, C4⋊Dic9 [×3], C12⋊Dic3, C2×C9⋊Dic3 [×2], C6×C36, C12⋊Dic9
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, D9 [×3], C3⋊S3, Dic6 [×4], D12 [×4], C2×Dic3 [×4], Dic9 [×6], D18 [×3], C3⋊Dic3 [×2], C2×C3⋊S3, C4⋊Dic3 [×4], C9⋊S3, Dic18 [×3], D36 [×3], C2×Dic9 [×3], C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C9⋊Dic3 [×2], C2×C9⋊S3, C4⋊Dic9 [×3], C12⋊Dic3, C12.D9, C36⋊S3, C2×C9⋊Dic3, C12⋊Dic9
►Regular action on 432 points
Generators in S432
(1 182 344 417 78 293 242 114 386 255 337 363)(2 183 345 418 79 294 243 115 387 256 338 364)(3 184 346 419 80 295 244 116 388 257 339 365)(4 185 347 420 81 296 245 117 389 258 340 366)(5 186 348 421 82 297 246 118 390 259 341 367)(6 187 349 422 83 298 247 119 391 260 342 368)(7 188 350 423 84 299 248 120 392 261 325 369)(8 189 351 424 85 300 249 121 393 262 326 370)(9 190 352 425 86 301 250 122 394 263 327 371)(10 191 353 426 87 302 251 123 395 264 328 372)(11 192 354 427 88 303 252 124 396 265 329 373)(12 193 355 428 89 304 235 125 379 266 330 374)(13 194 356 429 90 305 236 126 380 267 331 375)(14 195 357 430 73 306 237 109 381 268 332 376)(15 196 358 431 74 289 238 110 382 269 333 377)(16 197 359 432 75 290 239 111 383 270 334 378)(17 198 360 415 76 291 240 112 384 253 335 361)(18 181 343 416 77 292 241 113 385 254 336 362)(19 144 43 215 410 153 272 313 58 172 97 223)(20 127 44 216 411 154 273 314 59 173 98 224)(21 128 45 199 412 155 274 315 60 174 99 225)(22 129 46 200 413 156 275 316 61 175 100 226)(23 130 47 201 414 157 276 317 62 176 101 227)(24 131 48 202 397 158 277 318 63 177 102 228)(25 132 49 203 398 159 278 319 64 178 103 229)(26 133 50 204 399 160 279 320 65 179 104 230)(27 134 51 205 400 161 280 321 66 180 105 231)(28 135 52 206 401 162 281 322 67 163 106 232)(29 136 53 207 402 145 282 323 68 164 107 233)(30 137 54 208 403 146 283 324 69 165 108 234)(31 138 37 209 404 147 284 307 70 166 91 217)(32 139 38 210 405 148 285 308 71 167 92 218)(33 140 39 211 406 149 286 309 72 168 93 219)(34 141 40 212 407 150 287 310 55 169 94 220)(35 142 41 213 408 151 288 311 56 170 95 221)(36 143 42 214 409 152 271 312 57 171 96 222)
(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 29 10 20)(2 28 11 19)(3 27 12 36)(4 26 13 35)(5 25 14 34)(6 24 15 33)(7 23 16 32)(8 22 17 31)(9 21 18 30)(37 326 46 335)(38 325 47 334)(39 342 48 333)(40 341 49 332)(41 340 50 331)(42 339 51 330)(43 338 52 329)(44 337 53 328)(45 336 54 327)(55 82 64 73)(56 81 65 90)(57 80 66 89)(58 79 67 88)(59 78 68 87)(60 77 69 86)(61 76 70 85)(62 75 71 84)(63 74 72 83)(91 351 100 360)(92 350 101 359)(93 349 102 358)(94 348 103 357)(95 347 104 356)(96 346 105 355)(97 345 106 354)(98 344 107 353)(99 343 108 352)(109 150 118 159)(110 149 119 158)(111 148 120 157)(112 147 121 156)(113 146 122 155)(114 145 123 154)(115 162 124 153)(116 161 125 152)(117 160 126 151)(127 363 136 372)(128 362 137 371)(129 361 138 370)(130 378 139 369)(131 377 140 368)(132 376 141 367)(133 375 142 366)(134 374 143 365)(135 373 144 364)(163 427 172 418)(164 426 173 417)(165 425 174 416)(166 424 175 415)(167 423 176 432)(168 422 177 431)(169 421 178 430)(170 420 179 429)(171 419 180 428)(181 234 190 225)(182 233 191 224)(183 232 192 223)(184 231 193 222)(185 230 194 221)(186 229 195 220)(187 228 196 219)(188 227 197 218)(189 226 198 217)(199 254 208 263)(200 253 209 262)(201 270 210 261)(202 269 211 260)(203 268 212 259)(204 267 213 258)(205 266 214 257)(206 265 215 256)(207 264 216 255)(235 271 244 280)(236 288 245 279)(237 287 246 278)(238 286 247 277)(239 285 248 276)(240 284 249 275)(241 283 250 274)(242 282 251 273)(243 281 252 272)(289 309 298 318)(290 308 299 317)(291 307 300 316)(292 324 301 315)(293 323 302 314)(294 322 303 313)(295 321 304 312)(296 320 305 311)(297 319 306 310)(379 409 388 400)(380 408 389 399)(381 407 390 398)(382 406 391 397)(383 405 392 414)(384 404 393 413)(385 403 394 412)(386 402 395 411)(387 401 396 410)
G:=sub<Sym(432)| (1,182,344,417,78,293,242,114,386,255,337,363)(2,183,345,418,79,294,243,115,387,256,338,364)(3,184,346,419,80,295,244,116,388,257,339,365)(4,185,347,420,81,296,245,117,389,258,340,366)(5,186,348,421,82,297,246,118,390,259,341,367)(6,187,349,422,83,298,247,119,391,260,342,368)(7,188,350,423,84,299,248,120,392,261,325,369)(8,189,351,424,85,300,249,121,393,262,326,370)(9,190,352,425,86,301,250,122,394,263,327,371)(10,191,353,426,87,302,251,123,395,264,328,372)(11,192,354,427,88,303,252,124,396,265,329,373)(12,193,355,428,89,304,235,125,379,266,330,374)(13,194,356,429,90,305,236,126,380,267,331,375)(14,195,357,430,73,306,237,109,381,268,332,376)(15,196,358,431,74,289,238,110,382,269,333,377)(16,197,359,432,75,290,239,111,383,270,334,378)(17,198,360,415,76,291,240,112,384,253,335,361)(18,181,343,416,77,292,241,113,385,254,336,362)(19,144,43,215,410,153,272,313,58,172,97,223)(20,127,44,216,411,154,273,314,59,173,98,224)(21,128,45,199,412,155,274,315,60,174,99,225)(22,129,46,200,413,156,275,316,61,175,100,226)(23,130,47,201,414,157,276,317,62,176,101,227)(24,131,48,202,397,158,277,318,63,177,102,228)(25,132,49,203,398,159,278,319,64,178,103,229)(26,133,50,204,399,160,279,320,65,179,104,230)(27,134,51,205,400,161,280,321,66,180,105,231)(28,135,52,206,401,162,281,322,67,163,106,232)(29,136,53,207,402,145,282,323,68,164,107,233)(30,137,54,208,403,146,283,324,69,165,108,234)(31,138,37,209,404,147,284,307,70,166,91,217)(32,139,38,210,405,148,285,308,71,167,92,218)(33,140,39,211,406,149,286,309,72,168,93,219)(34,141,40,212,407,150,287,310,55,169,94,220)(35,142,41,213,408,151,288,311,56,170,95,221)(36,143,42,214,409,152,271,312,57,171,96,222), (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), (1,29,10,20)(2,28,11,19)(3,27,12,36)(4,26,13,35)(5,25,14,34)(6,24,15,33)(7,23,16,32)(8,22,17,31)(9,21,18,30)(37,326,46,335)(38,325,47,334)(39,342,48,333)(40,341,49,332)(41,340,50,331)(42,339,51,330)(43,338,52,329)(44,337,53,328)(45,336,54,327)(55,82,64,73)(56,81,65,90)(57,80,66,89)(58,79,67,88)(59,78,68,87)(60,77,69,86)(61,76,70,85)(62,75,71,84)(63,74,72,83)(91,351,100,360)(92,350,101,359)(93,349,102,358)(94,348,103,357)(95,347,104,356)(96,346,105,355)(97,345,106,354)(98,344,107,353)(99,343,108,352)(109,150,118,159)(110,149,119,158)(111,148,120,157)(112,147,121,156)(113,146,122,155)(114,145,123,154)(115,162,124,153)(116,161,125,152)(117,160,126,151)(127,363,136,372)(128,362,137,371)(129,361,138,370)(130,378,139,369)(131,377,140,368)(132,376,141,367)(133,375,142,366)(134,374,143,365)(135,373,144,364)(163,427,172,418)(164,426,173,417)(165,425,174,416)(166,424,175,415)(167,423,176,432)(168,422,177,431)(169,421,178,430)(170,420,179,429)(171,419,180,428)(181,234,190,225)(182,233,191,224)(183,232,192,223)(184,231,193,222)(185,230,194,221)(186,229,195,220)(187,228,196,219)(188,227,197,218)(189,226,198,217)(199,254,208,263)(200,253,209,262)(201,270,210,261)(202,269,211,260)(203,268,212,259)(204,267,213,258)(205,266,214,257)(206,265,215,256)(207,264,216,255)(235,271,244,280)(236,288,245,279)(237,287,246,278)(238,286,247,277)(239,285,248,276)(240,284,249,275)(241,283,250,274)(242,282,251,273)(243,281,252,272)(289,309,298,318)(290,308,299,317)(291,307,300,316)(292,324,301,315)(293,323,302,314)(294,322,303,313)(295,321,304,312)(296,320,305,311)(297,319,306,310)(379,409,388,400)(380,408,389,399)(381,407,390,398)(382,406,391,397)(383,405,392,414)(384,404,393,413)(385,403,394,412)(386,402,395,411)(387,401,396,410)>;
G:=Group( (1,182,344,417,78,293,242,114,386,255,337,363)(2,183,345,418,79,294,243,115,387,256,338,364)(3,184,346,419,80,295,244,116,388,257,339,365)(4,185,347,420,81,296,245,117,389,258,340,366)(5,186,348,421,82,297,246,118,390,259,341,367)(6,187,349,422,83,298,247,119,391,260,342,368)(7,188,350,423,84,299,248,120,392,261,325,369)(8,189,351,424,85,300,249,121,393,262,326,370)(9,190,352,425,86,301,250,122,394,263,327,371)(10,191,353,426,87,302,251,123,395,264,328,372)(11,192,354,427,88,303,252,124,396,265,329,373)(12,193,355,428,89,304,235,125,379,266,330,374)(13,194,356,429,90,305,236,126,380,267,331,375)(14,195,357,430,73,306,237,109,381,268,332,376)(15,196,358,431,74,289,238,110,382,269,333,377)(16,197,359,432,75,290,239,111,383,270,334,378)(17,198,360,415,76,291,240,112,384,253,335,361)(18,181,343,416,77,292,241,113,385,254,336,362)(19,144,43,215,410,153,272,313,58,172,97,223)(20,127,44,216,411,154,273,314,59,173,98,224)(21,128,45,199,412,155,274,315,60,174,99,225)(22,129,46,200,413,156,275,316,61,175,100,226)(23,130,47,201,414,157,276,317,62,176,101,227)(24,131,48,202,397,158,277,318,63,177,102,228)(25,132,49,203,398,159,278,319,64,178,103,229)(26,133,50,204,399,160,279,320,65,179,104,230)(27,134,51,205,400,161,280,321,66,180,105,231)(28,135,52,206,401,162,281,322,67,163,106,232)(29,136,53,207,402,145,282,323,68,164,107,233)(30,137,54,208,403,146,283,324,69,165,108,234)(31,138,37,209,404,147,284,307,70,166,91,217)(32,139,38,210,405,148,285,308,71,167,92,218)(33,140,39,211,406,149,286,309,72,168,93,219)(34,141,40,212,407,150,287,310,55,169,94,220)(35,142,41,213,408,151,288,311,56,170,95,221)(36,143,42,214,409,152,271,312,57,171,96,222), (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), (1,29,10,20)(2,28,11,19)(3,27,12,36)(4,26,13,35)(5,25,14,34)(6,24,15,33)(7,23,16,32)(8,22,17,31)(9,21,18,30)(37,326,46,335)(38,325,47,334)(39,342,48,333)(40,341,49,332)(41,340,50,331)(42,339,51,330)(43,338,52,329)(44,337,53,328)(45,336,54,327)(55,82,64,73)(56,81,65,90)(57,80,66,89)(58,79,67,88)(59,78,68,87)(60,77,69,86)(61,76,70,85)(62,75,71,84)(63,74,72,83)(91,351,100,360)(92,350,101,359)(93,349,102,358)(94,348,103,357)(95,347,104,356)(96,346,105,355)(97,345,106,354)(98,344,107,353)(99,343,108,352)(109,150,118,159)(110,149,119,158)(111,148,120,157)(112,147,121,156)(113,146,122,155)(114,145,123,154)(115,162,124,153)(116,161,125,152)(117,160,126,151)(127,363,136,372)(128,362,137,371)(129,361,138,370)(130,378,139,369)(131,377,140,368)(132,376,141,367)(133,375,142,366)(134,374,143,365)(135,373,144,364)(163,427,172,418)(164,426,173,417)(165,425,174,416)(166,424,175,415)(167,423,176,432)(168,422,177,431)(169,421,178,430)(170,420,179,429)(171,419,180,428)(181,234,190,225)(182,233,191,224)(183,232,192,223)(184,231,193,222)(185,230,194,221)(186,229,195,220)(187,228,196,219)(188,227,197,218)(189,226,198,217)(199,254,208,263)(200,253,209,262)(201,270,210,261)(202,269,211,260)(203,268,212,259)(204,267,213,258)(205,266,214,257)(206,265,215,256)(207,264,216,255)(235,271,244,280)(236,288,245,279)(237,287,246,278)(238,286,247,277)(239,285,248,276)(240,284,249,275)(241,283,250,274)(242,282,251,273)(243,281,252,272)(289,309,298,318)(290,308,299,317)(291,307,300,316)(292,324,301,315)(293,323,302,314)(294,322,303,313)(295,321,304,312)(296,320,305,311)(297,319,306,310)(379,409,388,400)(380,408,389,399)(381,407,390,398)(382,406,391,397)(383,405,392,414)(384,404,393,413)(385,403,394,412)(386,402,395,411)(387,401,396,410) );
G=PermutationGroup([(1,182,344,417,78,293,242,114,386,255,337,363),(2,183,345,418,79,294,243,115,387,256,338,364),(3,184,346,419,80,295,244,116,388,257,339,365),(4,185,347,420,81,296,245,117,389,258,340,366),(5,186,348,421,82,297,246,118,390,259,341,367),(6,187,349,422,83,298,247,119,391,260,342,368),(7,188,350,423,84,299,248,120,392,261,325,369),(8,189,351,424,85,300,249,121,393,262,326,370),(9,190,352,425,86,301,250,122,394,263,327,371),(10,191,353,426,87,302,251,123,395,264,328,372),(11,192,354,427,88,303,252,124,396,265,329,373),(12,193,355,428,89,304,235,125,379,266,330,374),(13,194,356,429,90,305,236,126,380,267,331,375),(14,195,357,430,73,306,237,109,381,268,332,376),(15,196,358,431,74,289,238,110,382,269,333,377),(16,197,359,432,75,290,239,111,383,270,334,378),(17,198,360,415,76,291,240,112,384,253,335,361),(18,181,343,416,77,292,241,113,385,254,336,362),(19,144,43,215,410,153,272,313,58,172,97,223),(20,127,44,216,411,154,273,314,59,173,98,224),(21,128,45,199,412,155,274,315,60,174,99,225),(22,129,46,200,413,156,275,316,61,175,100,226),(23,130,47,201,414,157,276,317,62,176,101,227),(24,131,48,202,397,158,277,318,63,177,102,228),(25,132,49,203,398,159,278,319,64,178,103,229),(26,133,50,204,399,160,279,320,65,179,104,230),(27,134,51,205,400,161,280,321,66,180,105,231),(28,135,52,206,401,162,281,322,67,163,106,232),(29,136,53,207,402,145,282,323,68,164,107,233),(30,137,54,208,403,146,283,324,69,165,108,234),(31,138,37,209,404,147,284,307,70,166,91,217),(32,139,38,210,405,148,285,308,71,167,92,218),(33,140,39,211,406,149,286,309,72,168,93,219),(34,141,40,212,407,150,287,310,55,169,94,220),(35,142,41,213,408,151,288,311,56,170,95,221),(36,143,42,214,409,152,271,312,57,171,96,222)], [(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)], [(1,29,10,20),(2,28,11,19),(3,27,12,36),(4,26,13,35),(5,25,14,34),(6,24,15,33),(7,23,16,32),(8,22,17,31),(9,21,18,30),(37,326,46,335),(38,325,47,334),(39,342,48,333),(40,341,49,332),(41,340,50,331),(42,339,51,330),(43,338,52,329),(44,337,53,328),(45,336,54,327),(55,82,64,73),(56,81,65,90),(57,80,66,89),(58,79,67,88),(59,78,68,87),(60,77,69,86),(61,76,70,85),(62,75,71,84),(63,74,72,83),(91,351,100,360),(92,350,101,359),(93,349,102,358),(94,348,103,357),(95,347,104,356),(96,346,105,355),(97,345,106,354),(98,344,107,353),(99,343,108,352),(109,150,118,159),(110,149,119,158),(111,148,120,157),(112,147,121,156),(113,146,122,155),(114,145,123,154),(115,162,124,153),(116,161,125,152),(117,160,126,151),(127,363,136,372),(128,362,137,371),(129,361,138,370),(130,378,139,369),(131,377,140,368),(132,376,141,367),(133,375,142,366),(134,374,143,365),(135,373,144,364),(163,427,172,418),(164,426,173,417),(165,425,174,416),(166,424,175,415),(167,423,176,432),(168,422,177,431),(169,421,178,430),(170,420,179,429),(171,419,180,428),(181,234,190,225),(182,233,191,224),(183,232,192,223),(184,231,193,222),(185,230,194,221),(186,229,195,220),(187,228,196,219),(188,227,197,218),(189,226,198,217),(199,254,208,263),(200,253,209,262),(201,270,210,261),(202,269,211,260),(203,268,212,259),(204,267,213,258),(205,266,214,257),(206,265,215,256),(207,264,216,255),(235,271,244,280),(236,288,245,279),(237,287,246,278),(238,286,247,277),(239,285,248,276),(240,284,249,275),(241,283,250,274),(242,282,251,273),(243,281,252,272),(289,309,298,318),(290,308,299,317),(291,307,300,316),(292,324,301,315),(293,323,302,314),(294,322,303,313),(295,321,304,312),(296,320,305,311),(297,319,306,310),(379,409,388,400),(380,408,389,399),(381,407,390,398),(382,406,391,397),(383,405,392,414),(384,404,393,413),(385,403,394,412),(386,402,395,411),(387,401,396,410)])
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | + | - | + | + | - | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | S3 | D4 | Q8 | Dic3 | D6 | Dic3 | D6 | D9 | Dic6 | D12 | Dic6 | D12 | Dic9 | D18 | Dic18 | D36 |
| kernel | C12⋊Dic9 | C2×C9⋊Dic3 | C6×C36 | C3×C36 | C2×C36 | C6×C12 | C3×C18 | C3×C18 | C36 | C2×C18 | C3×C12 | C62 | C2×C12 | C18 | C18 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 4 | 3 | 1 | 1 | 1 | 6 | 3 | 2 | 1 | 9 | 6 | 6 | 2 | 2 | 18 | 9 | 18 | 18 |
Matrix representation of C12⋊Dic9 ►in GL4(𝔽37) generated by
| 27 | 32 | 0 | 0 |
| 5 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 36 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 17 | 31 |
| 0 | 0 | 6 | 11 |
| 29 | 1 | 0 | 0 |
| 9 | 8 | 0 | 0 |
| 0 | 0 | 0 | 31 |
| 0 | 0 | 31 | 0 |
G:=sub<GL(4,GF(37))| [27,5,0,0,32,32,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,36,1,0,0,0,0,17,6,0,0,31,11],[29,9,0,0,1,8,0,0,0,0,0,31,0,0,31,0] >;
C12⋊Dic9 in GAP, Magma, Sage, TeX
C_{12}\rtimes {\rm Dic}_9 % in TeX
G:=Group("C12:Dic9"); // GroupNames label
G:=SmallGroup(432,182);
// by ID
G=gap.SmallGroup(432,182);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^12=b^18=1,c^2=b^9,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations