direct product, metabelian, supersoluble, monomial, A-group
Aliases: C4×C9⋊Dic3, C36⋊2Dic3, C12⋊2Dic9, C62.123D6, (C3×C36)⋊5C4, (C3×C9)⋊4C42, C9⋊2(C4×Dic3), C3⋊2(C4×Dic9), C6.13(C4×D9), C18.14(C4×S3), (C6×C12).49S3, (C6×C36).16C2, (C2×C36).17S3, (C2×C12).17D9, (C2×C18).35D6, (C2×C6).35D18, C6.14(C2×Dic9), (C6×C18).37C22, C12.6(C3⋊Dic3), (C3×C12).26Dic3, C18.14(C2×Dic3), C32.4(C4×Dic3), C6.8(C4×C3⋊S3), C2.2(C4×C9⋊S3), C3.(C4×C3⋊Dic3), (C2×C4).6(C9⋊S3), (C3×C6).70(C4×S3), C22.3(C2×C9⋊S3), C2.2(C2×C9⋊Dic3), C6.8(C2×C3⋊Dic3), (C3×C18).38(C2×C4), (C2×C9⋊Dic3).9C2, (C2×C12).16(C3⋊S3), (C3×C6).61(C2×Dic3), (C2×C6).29(C2×C3⋊S3), SmallGroup(432,180)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C4×C9⋊Dic3 |
Generators and relations for C4×C9⋊Dic3
G = < a,b,c,d | a4=b9=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 612 in 150 conjugacy classes, 87 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C32, Dic3, C12, C2×C6, C2×C6, C42, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, Dic9, C36, C2×C18, C3⋊Dic3, C3×C12, C62, C4×Dic3, C3×C18, C3×C18, C2×Dic9, C2×C36, C2×C3⋊Dic3, C6×C12, C9⋊Dic3, C3×C36, C6×C18, C4×Dic9, C4×C3⋊Dic3, C2×C9⋊Dic3, C6×C36, C4×C9⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, D9, C3⋊S3, C4×S3, C2×Dic3, Dic9, D18, C3⋊Dic3, C2×C3⋊S3, C4×Dic3, C9⋊S3, C4×D9, C2×Dic9, C4×C3⋊S3, C2×C3⋊Dic3, C9⋊Dic3, C2×C9⋊S3, C4×Dic9, C4×C3⋊Dic3, C4×C9⋊S3, C2×C9⋊Dic3, C4×C9⋊Dic3
►Regular action on 432 points
Generators in S432
(1 178 70 151)(2 179 71 152)(3 180 72 153)(4 172 64 145)(5 173 65 146)(6 174 66 147)(7 175 67 148)(8 176 68 149)(9 177 69 150)(10 341 422 314)(11 342 423 315)(12 334 415 307)(13 335 416 308)(14 336 417 309)(15 337 418 310)(16 338 419 311)(17 339 420 312)(18 340 421 313)(19 330 411 303)(20 331 412 304)(21 332 413 305)(22 333 414 306)(23 325 406 298)(24 326 407 299)(25 327 408 300)(26 328 409 301)(27 329 410 302)(28 163 55 136)(29 164 56 137)(30 165 57 138)(31 166 58 139)(32 167 59 140)(33 168 60 141)(34 169 61 142)(35 170 62 143)(36 171 63 144)(37 159 51 132)(38 160 52 133)(39 161 53 134)(40 162 54 135)(41 154 46 127)(42 155 47 128)(43 156 48 129)(44 157 49 130)(45 158 50 131)(73 208 100 181)(74 209 101 182)(75 210 102 183)(76 211 103 184)(77 212 104 185)(78 213 105 186)(79 214 106 187)(80 215 107 188)(81 216 108 189)(82 229 109 190)(83 230 110 191)(84 231 111 192)(85 232 112 193)(86 233 113 194)(87 234 114 195)(88 226 115 196)(89 227 116 197)(90 228 117 198)(91 224 118 199)(92 225 119 200)(93 217 120 201)(94 218 121 202)(95 219 122 203)(96 220 123 204)(97 221 124 205)(98 222 125 206)(99 223 126 207)(235 370 262 343)(236 371 263 344)(237 372 264 345)(238 373 265 346)(239 374 266 347)(240 375 267 348)(241 376 268 349)(242 377 269 350)(243 378 270 351)(244 379 271 352)(245 380 272 353)(246 381 273 354)(247 382 274 355)(248 383 275 356)(249 384 276 357)(250 385 277 358)(251 386 278 359)(252 387 279 360)(253 388 280 361)(254 389 281 362)(255 390 282 363)(256 391 283 364)(257 392 284 365)(258 393 285 366)(259 394 286 367)(260 395 287 368)(261 396 288 369)(289 424 316 397)(290 425 317 398)(291 426 318 399)(292 427 319 400)(293 428 320 401)(294 429 321 402)(295 430 322 403)(296 431 323 404)(297 432 324 405)
(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)
(1 102 35 124 43 116)(2 103 36 125 44 117)(3 104 28 126 45 109)(4 105 29 118 37 110)(5 106 30 119 38 111)(6 107 31 120 39 112)(7 108 32 121 40 113)(8 100 33 122 41 114)(9 101 34 123 42 115)(10 350 25 368 431 354)(11 351 26 369 432 355)(12 343 27 361 424 356)(13 344 19 362 425 357)(14 345 20 363 426 358)(15 346 21 364 427 359)(16 347 22 365 428 360)(17 348 23 366 429 352)(18 349 24 367 430 353)(46 87 68 73 60 95)(47 88 69 74 61 96)(48 89 70 75 62 97)(49 90 71 76 63 98)(50 82 72 77 55 99)(51 83 64 78 56 91)(52 84 65 79 57 92)(53 85 66 80 58 93)(54 86 67 81 59 94)(127 234 149 208 141 219)(128 226 150 209 142 220)(129 227 151 210 143 221)(130 228 152 211 144 222)(131 229 153 212 136 223)(132 230 145 213 137 224)(133 231 146 214 138 225)(134 232 147 215 139 217)(135 233 148 216 140 218)(154 195 176 181 168 203)(155 196 177 182 169 204)(156 197 178 183 170 205)(157 198 179 184 171 206)(158 190 180 185 163 207)(159 191 172 186 164 199)(160 192 173 187 165 200)(161 193 174 188 166 201)(162 194 175 189 167 202)(235 329 253 316 248 334)(236 330 254 317 249 335)(237 331 255 318 250 336)(238 332 256 319 251 337)(239 333 257 320 252 338)(240 325 258 321 244 339)(241 326 259 322 245 340)(242 327 260 323 246 341)(243 328 261 324 247 342)(262 302 280 289 275 307)(263 303 281 290 276 308)(264 304 282 291 277 309)(265 305 283 292 278 310)(266 306 284 293 279 311)(267 298 285 294 271 312)(268 299 286 295 272 313)(269 300 287 296 273 314)(270 301 288 297 274 315)(370 410 388 397 383 415)(371 411 389 398 384 416)(372 412 390 399 385 417)(373 413 391 400 386 418)(374 414 392 401 387 419)(375 406 393 402 379 420)(376 407 394 403 380 421)(377 408 395 404 381 422)(378 409 396 405 382 423)
(1 406 124 379)(2 414 125 387)(3 413 126 386)(4 412 118 385)(5 411 119 384)(6 410 120 383)(7 409 121 382)(8 408 122 381)(9 407 123 380)(10 87 368 60)(11 86 369 59)(12 85 361 58)(13 84 362 57)(14 83 363 56)(15 82 364 55)(16 90 365 63)(17 89 366 62)(18 88 367 61)(19 92 357 65)(20 91 358 64)(21 99 359 72)(22 98 360 71)(23 97 352 70)(24 96 353 69)(25 95 354 68)(26 94 355 67)(27 93 356 66)(28 418 109 391)(29 417 110 390)(30 416 111 389)(31 415 112 388)(32 423 113 396)(33 422 114 395)(34 421 115 394)(35 420 116 393)(36 419 117 392)(37 399 105 372)(38 398 106 371)(39 397 107 370)(40 405 108 378)(41 404 100 377)(42 403 101 376)(43 402 102 375)(44 401 103 374)(45 400 104 373)(46 431 73 350)(47 430 74 349)(48 429 75 348)(49 428 76 347)(50 427 77 346)(51 426 78 345)(52 425 79 344)(53 424 80 343)(54 432 81 351)(127 323 208 242)(128 322 209 241)(129 321 210 240)(130 320 211 239)(131 319 212 238)(132 318 213 237)(133 317 214 236)(134 316 215 235)(135 324 216 243)(136 337 229 256)(137 336 230 255)(138 335 231 254)(139 334 232 253)(140 342 233 261)(141 341 234 260)(142 340 226 259)(143 339 227 258)(144 338 228 257)(145 331 224 250)(146 330 225 249)(147 329 217 248)(148 328 218 247)(149 327 219 246)(150 326 220 245)(151 325 221 244)(152 333 222 252)(153 332 223 251)(154 296 181 269)(155 295 182 268)(156 294 183 267)(157 293 184 266)(158 292 185 265)(159 291 186 264)(160 290 187 263)(161 289 188 262)(162 297 189 270)(163 310 190 283)(164 309 191 282)(165 308 192 281)(166 307 193 280)(167 315 194 288)(168 314 195 287)(169 313 196 286)(170 312 197 285)(171 311 198 284)(172 304 199 277)(173 303 200 276)(174 302 201 275)(175 301 202 274)(176 300 203 273)(177 299 204 272)(178 298 205 271)(179 306 206 279)(180 305 207 278)
G:=sub<Sym(432)| (1,178,70,151)(2,179,71,152)(3,180,72,153)(4,172,64,145)(5,173,65,146)(6,174,66,147)(7,175,67,148)(8,176,68,149)(9,177,69,150)(10,341,422,314)(11,342,423,315)(12,334,415,307)(13,335,416,308)(14,336,417,309)(15,337,418,310)(16,338,419,311)(17,339,420,312)(18,340,421,313)(19,330,411,303)(20,331,412,304)(21,332,413,305)(22,333,414,306)(23,325,406,298)(24,326,407,299)(25,327,408,300)(26,328,409,301)(27,329,410,302)(28,163,55,136)(29,164,56,137)(30,165,57,138)(31,166,58,139)(32,167,59,140)(33,168,60,141)(34,169,61,142)(35,170,62,143)(36,171,63,144)(37,159,51,132)(38,160,52,133)(39,161,53,134)(40,162,54,135)(41,154,46,127)(42,155,47,128)(43,156,48,129)(44,157,49,130)(45,158,50,131)(73,208,100,181)(74,209,101,182)(75,210,102,183)(76,211,103,184)(77,212,104,185)(78,213,105,186)(79,214,106,187)(80,215,107,188)(81,216,108,189)(82,229,109,190)(83,230,110,191)(84,231,111,192)(85,232,112,193)(86,233,113,194)(87,234,114,195)(88,226,115,196)(89,227,116,197)(90,228,117,198)(91,224,118,199)(92,225,119,200)(93,217,120,201)(94,218,121,202)(95,219,122,203)(96,220,123,204)(97,221,124,205)(98,222,125,206)(99,223,126,207)(235,370,262,343)(236,371,263,344)(237,372,264,345)(238,373,265,346)(239,374,266,347)(240,375,267,348)(241,376,268,349)(242,377,269,350)(243,378,270,351)(244,379,271,352)(245,380,272,353)(246,381,273,354)(247,382,274,355)(248,383,275,356)(249,384,276,357)(250,385,277,358)(251,386,278,359)(252,387,279,360)(253,388,280,361)(254,389,281,362)(255,390,282,363)(256,391,283,364)(257,392,284,365)(258,393,285,366)(259,394,286,367)(260,395,287,368)(261,396,288,369)(289,424,316,397)(290,425,317,398)(291,426,318,399)(292,427,319,400)(293,428,320,401)(294,429,321,402)(295,430,322,403)(296,431,323,404)(297,432,324,405), (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), (1,102,35,124,43,116)(2,103,36,125,44,117)(3,104,28,126,45,109)(4,105,29,118,37,110)(5,106,30,119,38,111)(6,107,31,120,39,112)(7,108,32,121,40,113)(8,100,33,122,41,114)(9,101,34,123,42,115)(10,350,25,368,431,354)(11,351,26,369,432,355)(12,343,27,361,424,356)(13,344,19,362,425,357)(14,345,20,363,426,358)(15,346,21,364,427,359)(16,347,22,365,428,360)(17,348,23,366,429,352)(18,349,24,367,430,353)(46,87,68,73,60,95)(47,88,69,74,61,96)(48,89,70,75,62,97)(49,90,71,76,63,98)(50,82,72,77,55,99)(51,83,64,78,56,91)(52,84,65,79,57,92)(53,85,66,80,58,93)(54,86,67,81,59,94)(127,234,149,208,141,219)(128,226,150,209,142,220)(129,227,151,210,143,221)(130,228,152,211,144,222)(131,229,153,212,136,223)(132,230,145,213,137,224)(133,231,146,214,138,225)(134,232,147,215,139,217)(135,233,148,216,140,218)(154,195,176,181,168,203)(155,196,177,182,169,204)(156,197,178,183,170,205)(157,198,179,184,171,206)(158,190,180,185,163,207)(159,191,172,186,164,199)(160,192,173,187,165,200)(161,193,174,188,166,201)(162,194,175,189,167,202)(235,329,253,316,248,334)(236,330,254,317,249,335)(237,331,255,318,250,336)(238,332,256,319,251,337)(239,333,257,320,252,338)(240,325,258,321,244,339)(241,326,259,322,245,340)(242,327,260,323,246,341)(243,328,261,324,247,342)(262,302,280,289,275,307)(263,303,281,290,276,308)(264,304,282,291,277,309)(265,305,283,292,278,310)(266,306,284,293,279,311)(267,298,285,294,271,312)(268,299,286,295,272,313)(269,300,287,296,273,314)(270,301,288,297,274,315)(370,410,388,397,383,415)(371,411,389,398,384,416)(372,412,390,399,385,417)(373,413,391,400,386,418)(374,414,392,401,387,419)(375,406,393,402,379,420)(376,407,394,403,380,421)(377,408,395,404,381,422)(378,409,396,405,382,423), (1,406,124,379)(2,414,125,387)(3,413,126,386)(4,412,118,385)(5,411,119,384)(6,410,120,383)(7,409,121,382)(8,408,122,381)(9,407,123,380)(10,87,368,60)(11,86,369,59)(12,85,361,58)(13,84,362,57)(14,83,363,56)(15,82,364,55)(16,90,365,63)(17,89,366,62)(18,88,367,61)(19,92,357,65)(20,91,358,64)(21,99,359,72)(22,98,360,71)(23,97,352,70)(24,96,353,69)(25,95,354,68)(26,94,355,67)(27,93,356,66)(28,418,109,391)(29,417,110,390)(30,416,111,389)(31,415,112,388)(32,423,113,396)(33,422,114,395)(34,421,115,394)(35,420,116,393)(36,419,117,392)(37,399,105,372)(38,398,106,371)(39,397,107,370)(40,405,108,378)(41,404,100,377)(42,403,101,376)(43,402,102,375)(44,401,103,374)(45,400,104,373)(46,431,73,350)(47,430,74,349)(48,429,75,348)(49,428,76,347)(50,427,77,346)(51,426,78,345)(52,425,79,344)(53,424,80,343)(54,432,81,351)(127,323,208,242)(128,322,209,241)(129,321,210,240)(130,320,211,239)(131,319,212,238)(132,318,213,237)(133,317,214,236)(134,316,215,235)(135,324,216,243)(136,337,229,256)(137,336,230,255)(138,335,231,254)(139,334,232,253)(140,342,233,261)(141,341,234,260)(142,340,226,259)(143,339,227,258)(144,338,228,257)(145,331,224,250)(146,330,225,249)(147,329,217,248)(148,328,218,247)(149,327,219,246)(150,326,220,245)(151,325,221,244)(152,333,222,252)(153,332,223,251)(154,296,181,269)(155,295,182,268)(156,294,183,267)(157,293,184,266)(158,292,185,265)(159,291,186,264)(160,290,187,263)(161,289,188,262)(162,297,189,270)(163,310,190,283)(164,309,191,282)(165,308,192,281)(166,307,193,280)(167,315,194,288)(168,314,195,287)(169,313,196,286)(170,312,197,285)(171,311,198,284)(172,304,199,277)(173,303,200,276)(174,302,201,275)(175,301,202,274)(176,300,203,273)(177,299,204,272)(178,298,205,271)(179,306,206,279)(180,305,207,278)>;
G:=Group( (1,178,70,151)(2,179,71,152)(3,180,72,153)(4,172,64,145)(5,173,65,146)(6,174,66,147)(7,175,67,148)(8,176,68,149)(9,177,69,150)(10,341,422,314)(11,342,423,315)(12,334,415,307)(13,335,416,308)(14,336,417,309)(15,337,418,310)(16,338,419,311)(17,339,420,312)(18,340,421,313)(19,330,411,303)(20,331,412,304)(21,332,413,305)(22,333,414,306)(23,325,406,298)(24,326,407,299)(25,327,408,300)(26,328,409,301)(27,329,410,302)(28,163,55,136)(29,164,56,137)(30,165,57,138)(31,166,58,139)(32,167,59,140)(33,168,60,141)(34,169,61,142)(35,170,62,143)(36,171,63,144)(37,159,51,132)(38,160,52,133)(39,161,53,134)(40,162,54,135)(41,154,46,127)(42,155,47,128)(43,156,48,129)(44,157,49,130)(45,158,50,131)(73,208,100,181)(74,209,101,182)(75,210,102,183)(76,211,103,184)(77,212,104,185)(78,213,105,186)(79,214,106,187)(80,215,107,188)(81,216,108,189)(82,229,109,190)(83,230,110,191)(84,231,111,192)(85,232,112,193)(86,233,113,194)(87,234,114,195)(88,226,115,196)(89,227,116,197)(90,228,117,198)(91,224,118,199)(92,225,119,200)(93,217,120,201)(94,218,121,202)(95,219,122,203)(96,220,123,204)(97,221,124,205)(98,222,125,206)(99,223,126,207)(235,370,262,343)(236,371,263,344)(237,372,264,345)(238,373,265,346)(239,374,266,347)(240,375,267,348)(241,376,268,349)(242,377,269,350)(243,378,270,351)(244,379,271,352)(245,380,272,353)(246,381,273,354)(247,382,274,355)(248,383,275,356)(249,384,276,357)(250,385,277,358)(251,386,278,359)(252,387,279,360)(253,388,280,361)(254,389,281,362)(255,390,282,363)(256,391,283,364)(257,392,284,365)(258,393,285,366)(259,394,286,367)(260,395,287,368)(261,396,288,369)(289,424,316,397)(290,425,317,398)(291,426,318,399)(292,427,319,400)(293,428,320,401)(294,429,321,402)(295,430,322,403)(296,431,323,404)(297,432,324,405), (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), (1,102,35,124,43,116)(2,103,36,125,44,117)(3,104,28,126,45,109)(4,105,29,118,37,110)(5,106,30,119,38,111)(6,107,31,120,39,112)(7,108,32,121,40,113)(8,100,33,122,41,114)(9,101,34,123,42,115)(10,350,25,368,431,354)(11,351,26,369,432,355)(12,343,27,361,424,356)(13,344,19,362,425,357)(14,345,20,363,426,358)(15,346,21,364,427,359)(16,347,22,365,428,360)(17,348,23,366,429,352)(18,349,24,367,430,353)(46,87,68,73,60,95)(47,88,69,74,61,96)(48,89,70,75,62,97)(49,90,71,76,63,98)(50,82,72,77,55,99)(51,83,64,78,56,91)(52,84,65,79,57,92)(53,85,66,80,58,93)(54,86,67,81,59,94)(127,234,149,208,141,219)(128,226,150,209,142,220)(129,227,151,210,143,221)(130,228,152,211,144,222)(131,229,153,212,136,223)(132,230,145,213,137,224)(133,231,146,214,138,225)(134,232,147,215,139,217)(135,233,148,216,140,218)(154,195,176,181,168,203)(155,196,177,182,169,204)(156,197,178,183,170,205)(157,198,179,184,171,206)(158,190,180,185,163,207)(159,191,172,186,164,199)(160,192,173,187,165,200)(161,193,174,188,166,201)(162,194,175,189,167,202)(235,329,253,316,248,334)(236,330,254,317,249,335)(237,331,255,318,250,336)(238,332,256,319,251,337)(239,333,257,320,252,338)(240,325,258,321,244,339)(241,326,259,322,245,340)(242,327,260,323,246,341)(243,328,261,324,247,342)(262,302,280,289,275,307)(263,303,281,290,276,308)(264,304,282,291,277,309)(265,305,283,292,278,310)(266,306,284,293,279,311)(267,298,285,294,271,312)(268,299,286,295,272,313)(269,300,287,296,273,314)(270,301,288,297,274,315)(370,410,388,397,383,415)(371,411,389,398,384,416)(372,412,390,399,385,417)(373,413,391,400,386,418)(374,414,392,401,387,419)(375,406,393,402,379,420)(376,407,394,403,380,421)(377,408,395,404,381,422)(378,409,396,405,382,423), (1,406,124,379)(2,414,125,387)(3,413,126,386)(4,412,118,385)(5,411,119,384)(6,410,120,383)(7,409,121,382)(8,408,122,381)(9,407,123,380)(10,87,368,60)(11,86,369,59)(12,85,361,58)(13,84,362,57)(14,83,363,56)(15,82,364,55)(16,90,365,63)(17,89,366,62)(18,88,367,61)(19,92,357,65)(20,91,358,64)(21,99,359,72)(22,98,360,71)(23,97,352,70)(24,96,353,69)(25,95,354,68)(26,94,355,67)(27,93,356,66)(28,418,109,391)(29,417,110,390)(30,416,111,389)(31,415,112,388)(32,423,113,396)(33,422,114,395)(34,421,115,394)(35,420,116,393)(36,419,117,392)(37,399,105,372)(38,398,106,371)(39,397,107,370)(40,405,108,378)(41,404,100,377)(42,403,101,376)(43,402,102,375)(44,401,103,374)(45,400,104,373)(46,431,73,350)(47,430,74,349)(48,429,75,348)(49,428,76,347)(50,427,77,346)(51,426,78,345)(52,425,79,344)(53,424,80,343)(54,432,81,351)(127,323,208,242)(128,322,209,241)(129,321,210,240)(130,320,211,239)(131,319,212,238)(132,318,213,237)(133,317,214,236)(134,316,215,235)(135,324,216,243)(136,337,229,256)(137,336,230,255)(138,335,231,254)(139,334,232,253)(140,342,233,261)(141,341,234,260)(142,340,226,259)(143,339,227,258)(144,338,228,257)(145,331,224,250)(146,330,225,249)(147,329,217,248)(148,328,218,247)(149,327,219,246)(150,326,220,245)(151,325,221,244)(152,333,222,252)(153,332,223,251)(154,296,181,269)(155,295,182,268)(156,294,183,267)(157,293,184,266)(158,292,185,265)(159,291,186,264)(160,290,187,263)(161,289,188,262)(162,297,189,270)(163,310,190,283)(164,309,191,282)(165,308,192,281)(166,307,193,280)(167,315,194,288)(168,314,195,287)(169,313,196,286)(170,312,197,285)(171,311,198,284)(172,304,199,277)(173,303,200,276)(174,302,201,275)(175,301,202,274)(176,300,203,273)(177,299,204,272)(178,298,205,271)(179,306,206,279)(180,305,207,278) );
G=PermutationGroup([[(1,178,70,151),(2,179,71,152),(3,180,72,153),(4,172,64,145),(5,173,65,146),(6,174,66,147),(7,175,67,148),(8,176,68,149),(9,177,69,150),(10,341,422,314),(11,342,423,315),(12,334,415,307),(13,335,416,308),(14,336,417,309),(15,337,418,310),(16,338,419,311),(17,339,420,312),(18,340,421,313),(19,330,411,303),(20,331,412,304),(21,332,413,305),(22,333,414,306),(23,325,406,298),(24,326,407,299),(25,327,408,300),(26,328,409,301),(27,329,410,302),(28,163,55,136),(29,164,56,137),(30,165,57,138),(31,166,58,139),(32,167,59,140),(33,168,60,141),(34,169,61,142),(35,170,62,143),(36,171,63,144),(37,159,51,132),(38,160,52,133),(39,161,53,134),(40,162,54,135),(41,154,46,127),(42,155,47,128),(43,156,48,129),(44,157,49,130),(45,158,50,131),(73,208,100,181),(74,209,101,182),(75,210,102,183),(76,211,103,184),(77,212,104,185),(78,213,105,186),(79,214,106,187),(80,215,107,188),(81,216,108,189),(82,229,109,190),(83,230,110,191),(84,231,111,192),(85,232,112,193),(86,233,113,194),(87,234,114,195),(88,226,115,196),(89,227,116,197),(90,228,117,198),(91,224,118,199),(92,225,119,200),(93,217,120,201),(94,218,121,202),(95,219,122,203),(96,220,123,204),(97,221,124,205),(98,222,125,206),(99,223,126,207),(235,370,262,343),(236,371,263,344),(237,372,264,345),(238,373,265,346),(239,374,266,347),(240,375,267,348),(241,376,268,349),(242,377,269,350),(243,378,270,351),(244,379,271,352),(245,380,272,353),(246,381,273,354),(247,382,274,355),(248,383,275,356),(249,384,276,357),(250,385,277,358),(251,386,278,359),(252,387,279,360),(253,388,280,361),(254,389,281,362),(255,390,282,363),(256,391,283,364),(257,392,284,365),(258,393,285,366),(259,394,286,367),(260,395,287,368),(261,396,288,369),(289,424,316,397),(290,425,317,398),(291,426,318,399),(292,427,319,400),(293,428,320,401),(294,429,321,402),(295,430,322,403),(296,431,323,404),(297,432,324,405)], [(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)], [(1,102,35,124,43,116),(2,103,36,125,44,117),(3,104,28,126,45,109),(4,105,29,118,37,110),(5,106,30,119,38,111),(6,107,31,120,39,112),(7,108,32,121,40,113),(8,100,33,122,41,114),(9,101,34,123,42,115),(10,350,25,368,431,354),(11,351,26,369,432,355),(12,343,27,361,424,356),(13,344,19,362,425,357),(14,345,20,363,426,358),(15,346,21,364,427,359),(16,347,22,365,428,360),(17,348,23,366,429,352),(18,349,24,367,430,353),(46,87,68,73,60,95),(47,88,69,74,61,96),(48,89,70,75,62,97),(49,90,71,76,63,98),(50,82,72,77,55,99),(51,83,64,78,56,91),(52,84,65,79,57,92),(53,85,66,80,58,93),(54,86,67,81,59,94),(127,234,149,208,141,219),(128,226,150,209,142,220),(129,227,151,210,143,221),(130,228,152,211,144,222),(131,229,153,212,136,223),(132,230,145,213,137,224),(133,231,146,214,138,225),(134,232,147,215,139,217),(135,233,148,216,140,218),(154,195,176,181,168,203),(155,196,177,182,169,204),(156,197,178,183,170,205),(157,198,179,184,171,206),(158,190,180,185,163,207),(159,191,172,186,164,199),(160,192,173,187,165,200),(161,193,174,188,166,201),(162,194,175,189,167,202),(235,329,253,316,248,334),(236,330,254,317,249,335),(237,331,255,318,250,336),(238,332,256,319,251,337),(239,333,257,320,252,338),(240,325,258,321,244,339),(241,326,259,322,245,340),(242,327,260,323,246,341),(243,328,261,324,247,342),(262,302,280,289,275,307),(263,303,281,290,276,308),(264,304,282,291,277,309),(265,305,283,292,278,310),(266,306,284,293,279,311),(267,298,285,294,271,312),(268,299,286,295,272,313),(269,300,287,296,273,314),(270,301,288,297,274,315),(370,410,388,397,383,415),(371,411,389,398,384,416),(372,412,390,399,385,417),(373,413,391,400,386,418),(374,414,392,401,387,419),(375,406,393,402,379,420),(376,407,394,403,380,421),(377,408,395,404,381,422),(378,409,396,405,382,423)], [(1,406,124,379),(2,414,125,387),(3,413,126,386),(4,412,118,385),(5,411,119,384),(6,410,120,383),(7,409,121,382),(8,408,122,381),(9,407,123,380),(10,87,368,60),(11,86,369,59),(12,85,361,58),(13,84,362,57),(14,83,363,56),(15,82,364,55),(16,90,365,63),(17,89,366,62),(18,88,367,61),(19,92,357,65),(20,91,358,64),(21,99,359,72),(22,98,360,71),(23,97,352,70),(24,96,353,69),(25,95,354,68),(26,94,355,67),(27,93,356,66),(28,418,109,391),(29,417,110,390),(30,416,111,389),(31,415,112,388),(32,423,113,396),(33,422,114,395),(34,421,115,394),(35,420,116,393),(36,419,117,392),(37,399,105,372),(38,398,106,371),(39,397,107,370),(40,405,108,378),(41,404,100,377),(42,403,101,376),(43,402,102,375),(44,401,103,374),(45,400,104,373),(46,431,73,350),(47,430,74,349),(48,429,75,348),(49,428,76,347),(50,427,77,346),(51,426,78,345),(52,425,79,344),(53,424,80,343),(54,432,81,351),(127,323,208,242),(128,322,209,241),(129,321,210,240),(130,320,211,239),(131,319,212,238),(132,318,213,237),(133,317,214,236),(134,316,215,235),(135,324,216,243),(136,337,229,256),(137,336,230,255),(138,335,231,254),(139,334,232,253),(140,342,233,261),(141,341,234,260),(142,340,226,259),(143,339,227,258),(144,338,228,257),(145,331,224,250),(146,330,225,249),(147,329,217,248),(148,328,218,247),(149,327,219,246),(150,326,220,245),(151,325,221,244),(152,333,222,252),(153,332,223,251),(154,296,181,269),(155,295,182,268),(156,294,183,267),(157,293,184,266),(158,292,185,265),(159,291,186,264),(160,290,187,263),(161,289,188,262),(162,297,189,270),(163,310,190,283),(164,309,191,282),(165,308,192,281),(166,307,193,280),(167,315,194,288),(168,314,195,287),(169,313,196,286),(170,312,197,285),(171,311,198,284),(172,304,199,277),(173,303,200,276),(174,302,201,275),(175,301,202,274),(176,300,203,273),(177,299,204,272),(178,298,205,271),(179,306,206,279),(180,305,207,278)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 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 | 1 | 1 | 1 | 1 | 27 | ··· | 27 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | S3 | Dic3 | D6 | Dic3 | D6 | D9 | C4×S3 | C4×S3 | Dic9 | D18 | C4×D9 |
| kernel | C4×C9⋊Dic3 | C2×C9⋊Dic3 | C6×C36 | C9⋊Dic3 | C3×C36 | C2×C36 | C6×C12 | C36 | C2×C18 | C3×C12 | C62 | C2×C12 | C18 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 6 | 3 | 2 | 1 | 9 | 12 | 4 | 18 | 9 | 36 |
Matrix representation of C4×C9⋊Dic3 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 36 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 20 |
| 0 | 0 | 17 | 31 |
| 0 | 36 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 36 |
| 0 | 0 | 1 | 0 |
| 7 | 14 | 0 | 0 |
| 7 | 30 | 0 | 0 |
| 0 | 0 | 6 | 31 |
| 0 | 0 | 0 | 31 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[36,1,0,0,36,0,0,0,0,0,11,17,0,0,20,31],[0,1,0,0,36,1,0,0,0,0,1,1,0,0,36,0],[7,7,0,0,14,30,0,0,0,0,6,0,0,0,31,31] >;
C4×C9⋊Dic3 in GAP, Magma, Sage, TeX
C_4\times C_9\rtimes {\rm Dic}_3 % in TeX
G:=Group("C4xC9:Dic3"); // GroupNames label
G:=SmallGroup(432,180);
// by ID
G=gap.SmallGroup(432,180);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,64,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^9=c^6=1,d^2=c^3,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