direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C4⋊C4×C3×C9, C12⋊3C36, C36⋊7C12, C4⋊(C3×C36), (C3×C36)⋊7C4, C6.5(Q8×C9), C2.2(C6×C36), (C6×C36).6C2, C6.19(D4×C9), C18.8(C3×Q8), (C6×C12).24C6, C12.4(C3×C12), C6.13(C6×C12), (C2×C36).15C6, C6.14(C2×C36), (C3×C18).53D4, C18.35(C3×D4), (C3×C18).10Q8, (C2×C12).13C18, C18.28(C2×C12), (C3×C12).26C12, C6.3(Q8×C32), C62.81(C2×C6), (C2×C6).29C62, C22.3(C6×C18), C6.15(D4×C32), (C6×C18).53C22, C2.(Q8×C3×C9), C2.2(D4×C3×C9), (C2×C12).4(C3×C6), (C2×C4).1(C3×C18), (C3×C6).75(C3×D4), C32.4(C3×C4⋊C4), C3.1(C32×C4⋊C4), (C3×C6).16(C3×Q8), (C2×C18).37(C2×C6), (C3×C6).67(C2×C12), (C3×C18).42(C2×C4), (C2×C6).20(C2×C18), (C32×C4⋊C4).2C3, (C3×C4⋊C4).1C32, SmallGroup(432,206)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C3×C9
G = < a,b,c,d | a3=b9=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 150 in 130 conjugacy classes, 110 normal (28 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C4⋊C4, C18, C3×C6, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C3×C12, C3×C12, C62, C3×C4⋊C4, C3×C4⋊C4, C3×C18, C2×C36, C6×C12, C6×C12, C3×C36, C3×C36, C6×C18, C9×C4⋊C4, C32×C4⋊C4, C6×C36, C6×C36, C4⋊C4×C3×C9
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C9, C32, C12, C2×C6, C4⋊C4, C18, C3×C6, C2×C12, C3×D4, C3×Q8, C3×C9, C36, C2×C18, C3×C12, C62, C3×C4⋊C4, C3×C18, C2×C36, D4×C9, Q8×C9, C6×C12, D4×C32, Q8×C32, C3×C36, C6×C18, C9×C4⋊C4, C32×C4⋊C4, C6×C36, D4×C3×C9, Q8×C3×C9, C4⋊C4×C3×C9
►Regular action on 432 points
Generators in S432
(1 95 114)(2 96 115)(3 97 116)(4 98 117)(5 99 109)(6 91 110)(7 92 111)(8 93 112)(9 94 113)(10 55 104)(11 56 105)(12 57 106)(13 58 107)(14 59 108)(15 60 100)(16 61 101)(17 62 102)(18 63 103)(19 66 85)(20 67 86)(21 68 87)(22 69 88)(23 70 89)(24 71 90)(25 72 82)(26 64 83)(27 65 84)(28 149 50)(29 150 51)(30 151 52)(31 152 53)(32 153 54)(33 145 46)(34 146 47)(35 147 48)(36 148 49)(37 340 420)(38 341 421)(39 342 422)(40 334 423)(41 335 415)(42 336 416)(43 337 417)(44 338 418)(45 339 419)(73 376 354)(74 377 355)(75 378 356)(76 370 357)(77 371 358)(78 372 359)(79 373 360)(80 374 352)(81 375 353)(118 207 217)(119 199 218)(120 200 219)(121 201 220)(122 202 221)(123 203 222)(124 204 223)(125 205 224)(126 206 225)(127 174 193)(128 175 194)(129 176 195)(130 177 196)(131 178 197)(132 179 198)(133 180 190)(134 172 191)(135 173 192)(136 257 158)(137 258 159)(138 259 160)(139 260 161)(140 261 162)(141 253 154)(142 254 155)(143 255 156)(144 256 157)(163 212 187)(164 213 188)(165 214 189)(166 215 181)(167 216 182)(168 208 183)(169 209 184)(170 210 185)(171 211 186)(226 315 325)(227 307 326)(228 308 327)(229 309 328)(230 310 329)(231 311 330)(232 312 331)(233 313 332)(234 314 333)(235 282 301)(236 283 302)(237 284 303)(238 285 304)(239 286 305)(240 287 306)(241 288 298)(242 280 299)(243 281 300)(244 365 266)(245 366 267)(246 367 268)(247 368 269)(248 369 270)(249 361 262)(250 362 263)(251 363 264)(252 364 265)(271 320 295)(272 321 296)(273 322 297)(274 323 289)(275 324 290)(276 316 291)(277 317 292)(278 318 293)(279 319 294)(343 390 409)(344 391 410)(345 392 411)(346 393 412)(347 394 413)(348 395 414)(349 396 406)(350 388 407)(351 389 408)(379 428 403)(380 429 404)(381 430 405)(382 431 397)(383 432 398)(384 424 399)(385 425 400)(386 426 401)(387 427 402)
(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)
(1 160 64 163)(2 161 65 164)(3 162 66 165)(4 154 67 166)(5 155 68 167)(6 156 69 168)(7 157 70 169)(8 158 71 170)(9 159 72 171)(10 134 151 222)(11 135 152 223)(12 127 153 224)(13 128 145 225)(14 129 146 217)(15 130 147 218)(16 131 148 219)(17 132 149 220)(18 133 150 221)(19 189 116 261)(20 181 117 253)(21 182 109 254)(22 183 110 255)(23 184 111 256)(24 185 112 257)(25 186 113 258)(26 187 114 259)(27 188 115 260)(28 201 102 198)(29 202 103 190)(30 203 104 191)(31 204 105 192)(32 205 106 193)(33 206 107 194)(34 207 108 195)(35 199 100 196)(36 200 101 197)(37 368 351 296)(38 369 343 297)(39 361 344 289)(40 362 345 290)(41 363 346 291)(42 364 347 292)(43 365 348 293)(44 366 349 294)(45 367 350 295)(46 126 58 175)(47 118 59 176)(48 119 60 177)(49 120 61 178)(50 121 62 179)(51 122 63 180)(52 123 55 172)(53 124 56 173)(54 125 57 174)(73 330 403 242)(74 331 404 243)(75 332 405 235)(76 333 397 236)(77 325 398 237)(78 326 399 238)(79 327 400 239)(80 328 401 240)(81 329 402 241)(82 211 94 137)(83 212 95 138)(84 213 96 139)(85 214 97 140)(86 215 98 141)(87 216 99 142)(88 208 91 143)(89 209 92 144)(90 210 93 136)(226 383 284 371)(227 384 285 372)(228 385 286 373)(229 386 287 374)(230 387 288 375)(231 379 280 376)(232 380 281 377)(233 381 282 378)(234 382 283 370)(244 414 318 417)(245 406 319 418)(246 407 320 419)(247 408 321 420)(248 409 322 421)(249 410 323 422)(250 411 324 423)(251 412 316 415)(252 413 317 416)(262 391 274 342)(263 392 275 334)(264 393 276 335)(265 394 277 336)(266 395 278 337)(267 396 279 338)(268 388 271 339)(269 389 272 340)(270 390 273 341)(298 353 310 427)(299 354 311 428)(300 355 312 429)(301 356 313 430)(302 357 314 431)(303 358 315 432)(304 359 307 424)(305 360 308 425)(306 352 309 426)
(1 271 55 231)(2 272 56 232)(3 273 57 233)(4 274 58 234)(5 275 59 226)(6 276 60 227)(7 277 61 228)(8 278 62 229)(9 279 63 230)(10 330 114 295)(11 331 115 296)(12 332 116 297)(13 333 117 289)(14 325 109 290)(15 326 110 291)(16 327 111 292)(17 328 112 293)(18 329 113 294)(19 369 153 235)(20 361 145 236)(21 362 146 237)(22 363 147 238)(23 364 148 239)(24 365 149 240)(25 366 150 241)(26 367 151 242)(27 368 152 243)(28 306 90 244)(29 298 82 245)(30 299 83 246)(31 300 84 247)(32 301 85 248)(33 302 86 249)(34 303 87 250)(35 304 88 251)(36 305 89 252)(37 223 404 188)(38 224 405 189)(39 225 397 181)(40 217 398 182)(41 218 399 183)(42 219 400 184)(43 220 401 185)(44 221 402 186)(45 222 403 187)(46 283 67 262)(47 284 68 263)(48 285 69 264)(49 286 70 265)(50 287 71 266)(51 288 72 267)(52 280 64 268)(53 281 65 269)(54 282 66 270)(73 259 350 134)(74 260 351 135)(75 261 343 127)(76 253 344 128)(77 254 345 129)(78 255 346 130)(79 256 347 131)(80 257 348 132)(81 258 349 133)(91 316 100 307)(92 317 101 308)(93 318 102 309)(94 319 103 310)(95 320 104 311)(96 321 105 312)(97 322 106 313)(98 323 107 314)(99 324 108 315)(118 383 167 334)(119 384 168 335)(120 385 169 336)(121 386 170 337)(122 387 171 338)(123 379 163 339)(124 380 164 340)(125 381 165 341)(126 382 166 342)(136 414 198 352)(137 406 190 353)(138 407 191 354)(139 408 192 355)(140 409 193 356)(141 410 194 357)(142 411 195 358)(143 412 196 359)(144 413 197 360)(154 391 175 370)(155 392 176 371)(156 393 177 372)(157 394 178 373)(158 395 179 374)(159 396 180 375)(160 388 172 376)(161 389 173 377)(162 390 174 378)(199 424 208 415)(200 425 209 416)(201 426 210 417)(202 427 211 418)(203 428 212 419)(204 429 213 420)(205 430 214 421)(206 431 215 422)(207 432 216 423)
G:=sub<Sym(432)| (1,95,114)(2,96,115)(3,97,116)(4,98,117)(5,99,109)(6,91,110)(7,92,111)(8,93,112)(9,94,113)(10,55,104)(11,56,105)(12,57,106)(13,58,107)(14,59,108)(15,60,100)(16,61,101)(17,62,102)(18,63,103)(19,66,85)(20,67,86)(21,68,87)(22,69,88)(23,70,89)(24,71,90)(25,72,82)(26,64,83)(27,65,84)(28,149,50)(29,150,51)(30,151,52)(31,152,53)(32,153,54)(33,145,46)(34,146,47)(35,147,48)(36,148,49)(37,340,420)(38,341,421)(39,342,422)(40,334,423)(41,335,415)(42,336,416)(43,337,417)(44,338,418)(45,339,419)(73,376,354)(74,377,355)(75,378,356)(76,370,357)(77,371,358)(78,372,359)(79,373,360)(80,374,352)(81,375,353)(118,207,217)(119,199,218)(120,200,219)(121,201,220)(122,202,221)(123,203,222)(124,204,223)(125,205,224)(126,206,225)(127,174,193)(128,175,194)(129,176,195)(130,177,196)(131,178,197)(132,179,198)(133,180,190)(134,172,191)(135,173,192)(136,257,158)(137,258,159)(138,259,160)(139,260,161)(140,261,162)(141,253,154)(142,254,155)(143,255,156)(144,256,157)(163,212,187)(164,213,188)(165,214,189)(166,215,181)(167,216,182)(168,208,183)(169,209,184)(170,210,185)(171,211,186)(226,315,325)(227,307,326)(228,308,327)(229,309,328)(230,310,329)(231,311,330)(232,312,331)(233,313,332)(234,314,333)(235,282,301)(236,283,302)(237,284,303)(238,285,304)(239,286,305)(240,287,306)(241,288,298)(242,280,299)(243,281,300)(244,365,266)(245,366,267)(246,367,268)(247,368,269)(248,369,270)(249,361,262)(250,362,263)(251,363,264)(252,364,265)(271,320,295)(272,321,296)(273,322,297)(274,323,289)(275,324,290)(276,316,291)(277,317,292)(278,318,293)(279,319,294)(343,390,409)(344,391,410)(345,392,411)(346,393,412)(347,394,413)(348,395,414)(349,396,406)(350,388,407)(351,389,408)(379,428,403)(380,429,404)(381,430,405)(382,431,397)(383,432,398)(384,424,399)(385,425,400)(386,426,401)(387,427,402), (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), (1,160,64,163)(2,161,65,164)(3,162,66,165)(4,154,67,166)(5,155,68,167)(6,156,69,168)(7,157,70,169)(8,158,71,170)(9,159,72,171)(10,134,151,222)(11,135,152,223)(12,127,153,224)(13,128,145,225)(14,129,146,217)(15,130,147,218)(16,131,148,219)(17,132,149,220)(18,133,150,221)(19,189,116,261)(20,181,117,253)(21,182,109,254)(22,183,110,255)(23,184,111,256)(24,185,112,257)(25,186,113,258)(26,187,114,259)(27,188,115,260)(28,201,102,198)(29,202,103,190)(30,203,104,191)(31,204,105,192)(32,205,106,193)(33,206,107,194)(34,207,108,195)(35,199,100,196)(36,200,101,197)(37,368,351,296)(38,369,343,297)(39,361,344,289)(40,362,345,290)(41,363,346,291)(42,364,347,292)(43,365,348,293)(44,366,349,294)(45,367,350,295)(46,126,58,175)(47,118,59,176)(48,119,60,177)(49,120,61,178)(50,121,62,179)(51,122,63,180)(52,123,55,172)(53,124,56,173)(54,125,57,174)(73,330,403,242)(74,331,404,243)(75,332,405,235)(76,333,397,236)(77,325,398,237)(78,326,399,238)(79,327,400,239)(80,328,401,240)(81,329,402,241)(82,211,94,137)(83,212,95,138)(84,213,96,139)(85,214,97,140)(86,215,98,141)(87,216,99,142)(88,208,91,143)(89,209,92,144)(90,210,93,136)(226,383,284,371)(227,384,285,372)(228,385,286,373)(229,386,287,374)(230,387,288,375)(231,379,280,376)(232,380,281,377)(233,381,282,378)(234,382,283,370)(244,414,318,417)(245,406,319,418)(246,407,320,419)(247,408,321,420)(248,409,322,421)(249,410,323,422)(250,411,324,423)(251,412,316,415)(252,413,317,416)(262,391,274,342)(263,392,275,334)(264,393,276,335)(265,394,277,336)(266,395,278,337)(267,396,279,338)(268,388,271,339)(269,389,272,340)(270,390,273,341)(298,353,310,427)(299,354,311,428)(300,355,312,429)(301,356,313,430)(302,357,314,431)(303,358,315,432)(304,359,307,424)(305,360,308,425)(306,352,309,426), (1,271,55,231)(2,272,56,232)(3,273,57,233)(4,274,58,234)(5,275,59,226)(6,276,60,227)(7,277,61,228)(8,278,62,229)(9,279,63,230)(10,330,114,295)(11,331,115,296)(12,332,116,297)(13,333,117,289)(14,325,109,290)(15,326,110,291)(16,327,111,292)(17,328,112,293)(18,329,113,294)(19,369,153,235)(20,361,145,236)(21,362,146,237)(22,363,147,238)(23,364,148,239)(24,365,149,240)(25,366,150,241)(26,367,151,242)(27,368,152,243)(28,306,90,244)(29,298,82,245)(30,299,83,246)(31,300,84,247)(32,301,85,248)(33,302,86,249)(34,303,87,250)(35,304,88,251)(36,305,89,252)(37,223,404,188)(38,224,405,189)(39,225,397,181)(40,217,398,182)(41,218,399,183)(42,219,400,184)(43,220,401,185)(44,221,402,186)(45,222,403,187)(46,283,67,262)(47,284,68,263)(48,285,69,264)(49,286,70,265)(50,287,71,266)(51,288,72,267)(52,280,64,268)(53,281,65,269)(54,282,66,270)(73,259,350,134)(74,260,351,135)(75,261,343,127)(76,253,344,128)(77,254,345,129)(78,255,346,130)(79,256,347,131)(80,257,348,132)(81,258,349,133)(91,316,100,307)(92,317,101,308)(93,318,102,309)(94,319,103,310)(95,320,104,311)(96,321,105,312)(97,322,106,313)(98,323,107,314)(99,324,108,315)(118,383,167,334)(119,384,168,335)(120,385,169,336)(121,386,170,337)(122,387,171,338)(123,379,163,339)(124,380,164,340)(125,381,165,341)(126,382,166,342)(136,414,198,352)(137,406,190,353)(138,407,191,354)(139,408,192,355)(140,409,193,356)(141,410,194,357)(142,411,195,358)(143,412,196,359)(144,413,197,360)(154,391,175,370)(155,392,176,371)(156,393,177,372)(157,394,178,373)(158,395,179,374)(159,396,180,375)(160,388,172,376)(161,389,173,377)(162,390,174,378)(199,424,208,415)(200,425,209,416)(201,426,210,417)(202,427,211,418)(203,428,212,419)(204,429,213,420)(205,430,214,421)(206,431,215,422)(207,432,216,423)>;
G:=Group( (1,95,114)(2,96,115)(3,97,116)(4,98,117)(5,99,109)(6,91,110)(7,92,111)(8,93,112)(9,94,113)(10,55,104)(11,56,105)(12,57,106)(13,58,107)(14,59,108)(15,60,100)(16,61,101)(17,62,102)(18,63,103)(19,66,85)(20,67,86)(21,68,87)(22,69,88)(23,70,89)(24,71,90)(25,72,82)(26,64,83)(27,65,84)(28,149,50)(29,150,51)(30,151,52)(31,152,53)(32,153,54)(33,145,46)(34,146,47)(35,147,48)(36,148,49)(37,340,420)(38,341,421)(39,342,422)(40,334,423)(41,335,415)(42,336,416)(43,337,417)(44,338,418)(45,339,419)(73,376,354)(74,377,355)(75,378,356)(76,370,357)(77,371,358)(78,372,359)(79,373,360)(80,374,352)(81,375,353)(118,207,217)(119,199,218)(120,200,219)(121,201,220)(122,202,221)(123,203,222)(124,204,223)(125,205,224)(126,206,225)(127,174,193)(128,175,194)(129,176,195)(130,177,196)(131,178,197)(132,179,198)(133,180,190)(134,172,191)(135,173,192)(136,257,158)(137,258,159)(138,259,160)(139,260,161)(140,261,162)(141,253,154)(142,254,155)(143,255,156)(144,256,157)(163,212,187)(164,213,188)(165,214,189)(166,215,181)(167,216,182)(168,208,183)(169,209,184)(170,210,185)(171,211,186)(226,315,325)(227,307,326)(228,308,327)(229,309,328)(230,310,329)(231,311,330)(232,312,331)(233,313,332)(234,314,333)(235,282,301)(236,283,302)(237,284,303)(238,285,304)(239,286,305)(240,287,306)(241,288,298)(242,280,299)(243,281,300)(244,365,266)(245,366,267)(246,367,268)(247,368,269)(248,369,270)(249,361,262)(250,362,263)(251,363,264)(252,364,265)(271,320,295)(272,321,296)(273,322,297)(274,323,289)(275,324,290)(276,316,291)(277,317,292)(278,318,293)(279,319,294)(343,390,409)(344,391,410)(345,392,411)(346,393,412)(347,394,413)(348,395,414)(349,396,406)(350,388,407)(351,389,408)(379,428,403)(380,429,404)(381,430,405)(382,431,397)(383,432,398)(384,424,399)(385,425,400)(386,426,401)(387,427,402), (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), (1,160,64,163)(2,161,65,164)(3,162,66,165)(4,154,67,166)(5,155,68,167)(6,156,69,168)(7,157,70,169)(8,158,71,170)(9,159,72,171)(10,134,151,222)(11,135,152,223)(12,127,153,224)(13,128,145,225)(14,129,146,217)(15,130,147,218)(16,131,148,219)(17,132,149,220)(18,133,150,221)(19,189,116,261)(20,181,117,253)(21,182,109,254)(22,183,110,255)(23,184,111,256)(24,185,112,257)(25,186,113,258)(26,187,114,259)(27,188,115,260)(28,201,102,198)(29,202,103,190)(30,203,104,191)(31,204,105,192)(32,205,106,193)(33,206,107,194)(34,207,108,195)(35,199,100,196)(36,200,101,197)(37,368,351,296)(38,369,343,297)(39,361,344,289)(40,362,345,290)(41,363,346,291)(42,364,347,292)(43,365,348,293)(44,366,349,294)(45,367,350,295)(46,126,58,175)(47,118,59,176)(48,119,60,177)(49,120,61,178)(50,121,62,179)(51,122,63,180)(52,123,55,172)(53,124,56,173)(54,125,57,174)(73,330,403,242)(74,331,404,243)(75,332,405,235)(76,333,397,236)(77,325,398,237)(78,326,399,238)(79,327,400,239)(80,328,401,240)(81,329,402,241)(82,211,94,137)(83,212,95,138)(84,213,96,139)(85,214,97,140)(86,215,98,141)(87,216,99,142)(88,208,91,143)(89,209,92,144)(90,210,93,136)(226,383,284,371)(227,384,285,372)(228,385,286,373)(229,386,287,374)(230,387,288,375)(231,379,280,376)(232,380,281,377)(233,381,282,378)(234,382,283,370)(244,414,318,417)(245,406,319,418)(246,407,320,419)(247,408,321,420)(248,409,322,421)(249,410,323,422)(250,411,324,423)(251,412,316,415)(252,413,317,416)(262,391,274,342)(263,392,275,334)(264,393,276,335)(265,394,277,336)(266,395,278,337)(267,396,279,338)(268,388,271,339)(269,389,272,340)(270,390,273,341)(298,353,310,427)(299,354,311,428)(300,355,312,429)(301,356,313,430)(302,357,314,431)(303,358,315,432)(304,359,307,424)(305,360,308,425)(306,352,309,426), (1,271,55,231)(2,272,56,232)(3,273,57,233)(4,274,58,234)(5,275,59,226)(6,276,60,227)(7,277,61,228)(8,278,62,229)(9,279,63,230)(10,330,114,295)(11,331,115,296)(12,332,116,297)(13,333,117,289)(14,325,109,290)(15,326,110,291)(16,327,111,292)(17,328,112,293)(18,329,113,294)(19,369,153,235)(20,361,145,236)(21,362,146,237)(22,363,147,238)(23,364,148,239)(24,365,149,240)(25,366,150,241)(26,367,151,242)(27,368,152,243)(28,306,90,244)(29,298,82,245)(30,299,83,246)(31,300,84,247)(32,301,85,248)(33,302,86,249)(34,303,87,250)(35,304,88,251)(36,305,89,252)(37,223,404,188)(38,224,405,189)(39,225,397,181)(40,217,398,182)(41,218,399,183)(42,219,400,184)(43,220,401,185)(44,221,402,186)(45,222,403,187)(46,283,67,262)(47,284,68,263)(48,285,69,264)(49,286,70,265)(50,287,71,266)(51,288,72,267)(52,280,64,268)(53,281,65,269)(54,282,66,270)(73,259,350,134)(74,260,351,135)(75,261,343,127)(76,253,344,128)(77,254,345,129)(78,255,346,130)(79,256,347,131)(80,257,348,132)(81,258,349,133)(91,316,100,307)(92,317,101,308)(93,318,102,309)(94,319,103,310)(95,320,104,311)(96,321,105,312)(97,322,106,313)(98,323,107,314)(99,324,108,315)(118,383,167,334)(119,384,168,335)(120,385,169,336)(121,386,170,337)(122,387,171,338)(123,379,163,339)(124,380,164,340)(125,381,165,341)(126,382,166,342)(136,414,198,352)(137,406,190,353)(138,407,191,354)(139,408,192,355)(140,409,193,356)(141,410,194,357)(142,411,195,358)(143,412,196,359)(144,413,197,360)(154,391,175,370)(155,392,176,371)(156,393,177,372)(157,394,178,373)(158,395,179,374)(159,396,180,375)(160,388,172,376)(161,389,173,377)(162,390,174,378)(199,424,208,415)(200,425,209,416)(201,426,210,417)(202,427,211,418)(203,428,212,419)(204,429,213,420)(205,430,214,421)(206,431,215,422)(207,432,216,423) );
G=PermutationGroup([[(1,95,114),(2,96,115),(3,97,116),(4,98,117),(5,99,109),(6,91,110),(7,92,111),(8,93,112),(9,94,113),(10,55,104),(11,56,105),(12,57,106),(13,58,107),(14,59,108),(15,60,100),(16,61,101),(17,62,102),(18,63,103),(19,66,85),(20,67,86),(21,68,87),(22,69,88),(23,70,89),(24,71,90),(25,72,82),(26,64,83),(27,65,84),(28,149,50),(29,150,51),(30,151,52),(31,152,53),(32,153,54),(33,145,46),(34,146,47),(35,147,48),(36,148,49),(37,340,420),(38,341,421),(39,342,422),(40,334,423),(41,335,415),(42,336,416),(43,337,417),(44,338,418),(45,339,419),(73,376,354),(74,377,355),(75,378,356),(76,370,357),(77,371,358),(78,372,359),(79,373,360),(80,374,352),(81,375,353),(118,207,217),(119,199,218),(120,200,219),(121,201,220),(122,202,221),(123,203,222),(124,204,223),(125,205,224),(126,206,225),(127,174,193),(128,175,194),(129,176,195),(130,177,196),(131,178,197),(132,179,198),(133,180,190),(134,172,191),(135,173,192),(136,257,158),(137,258,159),(138,259,160),(139,260,161),(140,261,162),(141,253,154),(142,254,155),(143,255,156),(144,256,157),(163,212,187),(164,213,188),(165,214,189),(166,215,181),(167,216,182),(168,208,183),(169,209,184),(170,210,185),(171,211,186),(226,315,325),(227,307,326),(228,308,327),(229,309,328),(230,310,329),(231,311,330),(232,312,331),(233,313,332),(234,314,333),(235,282,301),(236,283,302),(237,284,303),(238,285,304),(239,286,305),(240,287,306),(241,288,298),(242,280,299),(243,281,300),(244,365,266),(245,366,267),(246,367,268),(247,368,269),(248,369,270),(249,361,262),(250,362,263),(251,363,264),(252,364,265),(271,320,295),(272,321,296),(273,322,297),(274,323,289),(275,324,290),(276,316,291),(277,317,292),(278,318,293),(279,319,294),(343,390,409),(344,391,410),(345,392,411),(346,393,412),(347,394,413),(348,395,414),(349,396,406),(350,388,407),(351,389,408),(379,428,403),(380,429,404),(381,430,405),(382,431,397),(383,432,398),(384,424,399),(385,425,400),(386,426,401),(387,427,402)], [(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)], [(1,160,64,163),(2,161,65,164),(3,162,66,165),(4,154,67,166),(5,155,68,167),(6,156,69,168),(7,157,70,169),(8,158,71,170),(9,159,72,171),(10,134,151,222),(11,135,152,223),(12,127,153,224),(13,128,145,225),(14,129,146,217),(15,130,147,218),(16,131,148,219),(17,132,149,220),(18,133,150,221),(19,189,116,261),(20,181,117,253),(21,182,109,254),(22,183,110,255),(23,184,111,256),(24,185,112,257),(25,186,113,258),(26,187,114,259),(27,188,115,260),(28,201,102,198),(29,202,103,190),(30,203,104,191),(31,204,105,192),(32,205,106,193),(33,206,107,194),(34,207,108,195),(35,199,100,196),(36,200,101,197),(37,368,351,296),(38,369,343,297),(39,361,344,289),(40,362,345,290),(41,363,346,291),(42,364,347,292),(43,365,348,293),(44,366,349,294),(45,367,350,295),(46,126,58,175),(47,118,59,176),(48,119,60,177),(49,120,61,178),(50,121,62,179),(51,122,63,180),(52,123,55,172),(53,124,56,173),(54,125,57,174),(73,330,403,242),(74,331,404,243),(75,332,405,235),(76,333,397,236),(77,325,398,237),(78,326,399,238),(79,327,400,239),(80,328,401,240),(81,329,402,241),(82,211,94,137),(83,212,95,138),(84,213,96,139),(85,214,97,140),(86,215,98,141),(87,216,99,142),(88,208,91,143),(89,209,92,144),(90,210,93,136),(226,383,284,371),(227,384,285,372),(228,385,286,373),(229,386,287,374),(230,387,288,375),(231,379,280,376),(232,380,281,377),(233,381,282,378),(234,382,283,370),(244,414,318,417),(245,406,319,418),(246,407,320,419),(247,408,321,420),(248,409,322,421),(249,410,323,422),(250,411,324,423),(251,412,316,415),(252,413,317,416),(262,391,274,342),(263,392,275,334),(264,393,276,335),(265,394,277,336),(266,395,278,337),(267,396,279,338),(268,388,271,339),(269,389,272,340),(270,390,273,341),(298,353,310,427),(299,354,311,428),(300,355,312,429),(301,356,313,430),(302,357,314,431),(303,358,315,432),(304,359,307,424),(305,360,308,425),(306,352,309,426)], [(1,271,55,231),(2,272,56,232),(3,273,57,233),(4,274,58,234),(5,275,59,226),(6,276,60,227),(7,277,61,228),(8,278,62,229),(9,279,63,230),(10,330,114,295),(11,331,115,296),(12,332,116,297),(13,333,117,289),(14,325,109,290),(15,326,110,291),(16,327,111,292),(17,328,112,293),(18,329,113,294),(19,369,153,235),(20,361,145,236),(21,362,146,237),(22,363,147,238),(23,364,148,239),(24,365,149,240),(25,366,150,241),(26,367,151,242),(27,368,152,243),(28,306,90,244),(29,298,82,245),(30,299,83,246),(31,300,84,247),(32,301,85,248),(33,302,86,249),(34,303,87,250),(35,304,88,251),(36,305,89,252),(37,223,404,188),(38,224,405,189),(39,225,397,181),(40,217,398,182),(41,218,399,183),(42,219,400,184),(43,220,401,185),(44,221,402,186),(45,222,403,187),(46,283,67,262),(47,284,68,263),(48,285,69,264),(49,286,70,265),(50,287,71,266),(51,288,72,267),(52,280,64,268),(53,281,65,269),(54,282,66,270),(73,259,350,134),(74,260,351,135),(75,261,343,127),(76,253,344,128),(77,254,345,129),(78,255,346,130),(79,256,347,131),(80,257,348,132),(81,258,349,133),(91,316,100,307),(92,317,101,308),(93,318,102,309),(94,319,103,310),(95,320,104,311),(96,321,105,312),(97,322,106,313),(98,323,107,314),(99,324,108,315),(118,383,167,334),(119,384,168,335),(120,385,169,336),(121,386,170,337),(122,387,171,338),(123,379,163,339),(124,380,164,340),(125,381,165,341),(126,382,166,342),(136,414,198,352),(137,406,190,353),(138,407,191,354),(139,408,192,355),(140,409,193,356),(141,410,194,357),(142,411,195,358),(143,412,196,359),(144,413,197,360),(154,391,175,370),(155,392,176,371),(156,393,177,372),(157,394,178,373),(158,395,179,374),(159,396,180,375),(160,388,172,376),(161,389,173,377),(162,390,174,378),(199,424,208,415),(200,425,209,416),(201,426,210,417),(202,427,211,418),(203,428,212,419),(204,429,213,420),(205,430,214,421),(206,431,215,422),(207,432,216,423)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | ··· | 4F | 6A | ··· | 6X | 9A | ··· | 9R | 12A | ··· | 12AV | 18A | ··· | 18BB | 36A | ··· | 36DD |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C36 | D4 | Q8 | C3×D4 | C3×Q8 | C3×D4 | C3×Q8 | D4×C9 | Q8×C9 |
| kernel | C4⋊C4×C3×C9 | C6×C36 | C9×C4⋊C4 | C32×C4⋊C4 | C3×C36 | C2×C36 | C6×C12 | C3×C4⋊C4 | C36 | C3×C12 | C2×C12 | C12 | C3×C18 | C3×C18 | C18 | C18 | C3×C6 | C3×C6 | C6 | C6 |
| # reps | 1 | 3 | 6 | 2 | 4 | 18 | 6 | 18 | 24 | 8 | 54 | 72 | 1 | 1 | 6 | 6 | 2 | 2 | 18 | 18 |
Matrix representation of C4⋊C4×C3×C9 ►in GL4(𝔽37) generated by
| 26 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 33 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 1 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 27 | 26 |
| 0 | 0 | 26 | 10 |
G:=sub<GL(4,GF(37))| [26,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[33,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,36,0],[36,0,0,0,0,6,0,0,0,0,27,26,0,0,26,10] >;
C4⋊C4×C3×C9 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_3\times C_9
% in TeX
G:=Group("C4:C4xC3xC9"); // GroupNames label
G:=SmallGroup(432,206);
// by ID
G=gap.SmallGroup(432,206);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,260,772]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations