direct product, metacyclic, nilpotent (class 3), monomial
Aliases: Q16×C3×C9, C72.9C6, C24.4C18, C12.23C62, C8.(C3×C18), (C3×C72).7C2, C4.3(C6×C18), C24.5(C3×C6), C6.22(D4×C9), (C3×C24).17C6, C36.48(C2×C6), (C3×C18).56D4, C18.38(C3×D4), Q8.3(C3×C18), (Q8×C9).12C6, (C3×Q8).8C18, C12.25(C2×C18), C6.20(D4×C32), C32.4(C3×Q16), C3.1(C32×Q16), (C3×C36).78C22, (Q8×C32).25C6, (C3×Q16).1C32, (C32×Q16).2C3, C2.5(D4×C3×C9), (Q8×C3×C9).4C2, (C3×C6).78(C3×D4), (C3×Q8).17(C3×C6), (C3×C12).103(C2×C6), SmallGroup(432,221)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C3×C9
G = < a,b,c,d | a3=b9=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 110 in 90 conjugacy classes, 70 normal (20 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C32, C12, C12, C12, Q16, C18, C3×C6, C24, C24, C3×Q8, C3×C9, C36, C36, C3×C12, C3×C12, C3×Q16, C3×Q16, C3×C18, C72, Q8×C9, C3×C24, Q8×C32, C3×C36, C3×C36, C9×Q16, C32×Q16, C3×C72, Q8×C3×C9, Q16×C3×C9
Quotients: C1, C2, C3, C22, C6, D4, C9, C32, C2×C6, Q16, C18, C3×C6, C3×D4, C3×C9, C2×C18, C62, C3×Q16, C3×C18, D4×C9, D4×C32, C6×C18, C9×Q16, C32×Q16, D4×C3×C9, Q16×C3×C9
►Regular action on 432 points
Generators in S432
(1 92 111)(2 93 112)(3 94 113)(4 95 114)(5 96 115)(6 97 116)(7 98 117)(8 99 109)(9 91 110)(10 60 101)(11 61 102)(12 62 103)(13 63 104)(14 55 105)(15 56 106)(16 57 107)(17 58 108)(18 59 100)(19 65 88)(20 66 89)(21 67 90)(22 68 82)(23 69 83)(24 70 84)(25 71 85)(26 72 86)(27 64 87)(28 148 52)(29 149 53)(30 150 54)(31 151 46)(32 152 47)(33 153 48)(34 145 49)(35 146 50)(36 147 51)(37 341 418)(38 342 419)(39 334 420)(40 335 421)(41 336 422)(42 337 423)(43 338 415)(44 339 416)(45 340 417)(73 377 353)(74 378 354)(75 370 355)(76 371 356)(77 372 357)(78 373 358)(79 374 359)(80 375 360)(81 376 352)(118 204 223)(119 205 224)(120 206 225)(121 207 217)(122 199 218)(123 200 219)(124 201 220)(125 202 221)(126 203 222)(127 173 196)(128 174 197)(129 175 198)(130 176 190)(131 177 191)(132 178 192)(133 179 193)(134 180 194)(135 172 195)(136 256 160)(137 257 161)(138 258 162)(139 259 154)(140 260 155)(141 261 156)(142 253 157)(143 254 158)(144 255 159)(163 213 182)(164 214 183)(165 215 184)(166 216 185)(167 208 186)(168 209 187)(169 210 188)(170 211 189)(171 212 181)(226 312 331)(227 313 332)(228 314 333)(229 315 325)(230 307 326)(231 308 327)(232 309 328)(233 310 329)(234 311 330)(235 281 304)(236 282 305)(237 283 306)(238 284 298)(239 285 299)(240 286 300)(241 287 301)(242 288 302)(243 280 303)(244 364 268)(245 365 269)(246 366 270)(247 367 262)(248 368 263)(249 369 264)(250 361 265)(251 362 266)(252 363 267)(271 321 290)(272 322 291)(273 323 292)(274 324 293)(275 316 294)(276 317 295)(277 318 296)(278 319 297)(279 320 289)(343 389 412)(344 390 413)(345 391 414)(346 392 406)(347 393 407)(348 394 408)(349 395 409)(350 396 410)(351 388 411)(379 429 398)(380 430 399)(381 431 400)(382 432 401)(383 424 402)(384 425 403)(385 426 404)(386 427 405)(387 428 397)
(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)
(1 393 69 387 63 378 54 339)(2 394 70 379 55 370 46 340)(3 395 71 380 56 371 47 341)(4 396 72 381 57 372 48 342)(5 388 64 382 58 373 49 334)(6 389 65 383 59 374 50 335)(7 390 66 384 60 375 51 336)(8 391 67 385 61 376 52 337)(9 392 68 386 62 377 53 338)(10 80 147 41 117 344 20 403)(11 81 148 42 109 345 21 404)(12 73 149 43 110 346 22 405)(13 74 150 44 111 347 23 397)(14 75 151 45 112 348 24 398)(15 76 152 37 113 349 25 399)(16 77 153 38 114 350 26 400)(17 78 145 39 115 351 27 401)(18 79 146 40 116 343 19 402)(28 423 99 414 90 426 102 352)(29 415 91 406 82 427 103 353)(30 416 92 407 83 428 104 354)(31 417 93 408 84 429 105 355)(32 418 94 409 85 430 106 356)(33 419 95 410 86 431 107 357)(34 420 96 411 87 432 108 358)(35 421 97 412 88 424 100 359)(36 422 98 413 89 425 101 360)(118 280 157 226 166 265 172 274)(119 281 158 227 167 266 173 275)(120 282 159 228 168 267 174 276)(121 283 160 229 169 268 175 277)(122 284 161 230 170 269 176 278)(123 285 162 231 171 270 177 279)(124 286 154 232 163 262 178 271)(125 287 155 233 164 263 179 272)(126 288 156 234 165 264 180 273)(127 294 224 235 254 332 186 362)(128 295 225 236 255 333 187 363)(129 296 217 237 256 325 188 364)(130 297 218 238 257 326 189 365)(131 289 219 239 258 327 181 366)(132 290 220 240 259 328 182 367)(133 291 221 241 260 329 183 368)(134 292 222 242 261 330 184 369)(135 293 223 243 253 331 185 361)(136 315 210 244 198 318 207 306)(137 307 211 245 190 319 199 298)(138 308 212 246 191 320 200 299)(139 309 213 247 192 321 201 300)(140 310 214 248 193 322 202 301)(141 311 215 249 194 323 203 302)(142 312 216 250 195 324 204 303)(143 313 208 251 196 316 205 304)(144 314 209 252 197 317 206 305)
(1 171 63 123)(2 163 55 124)(3 164 56 125)(4 165 57 126)(5 166 58 118)(6 167 59 119)(7 168 60 120)(8 169 61 121)(9 170 62 122)(10 225 117 187)(11 217 109 188)(12 218 110 189)(13 219 111 181)(14 220 112 182)(15 221 113 183)(16 222 114 184)(17 223 115 185)(18 224 116 186)(19 254 146 127)(20 255 147 128)(21 256 148 129)(22 257 149 130)(23 258 150 131)(24 259 151 132)(25 260 152 133)(26 261 153 134)(27 253 145 135)(28 198 90 136)(29 190 82 137)(30 191 83 138)(31 192 84 139)(32 193 85 140)(33 194 86 141)(34 195 87 142)(35 196 88 143)(36 197 89 144)(37 368 399 241)(38 369 400 242)(39 361 401 243)(40 362 402 235)(41 363 403 236)(42 364 404 237)(43 365 405 238)(44 366 397 239)(45 367 398 240)(46 178 70 154)(47 179 71 155)(48 180 72 156)(49 172 64 157)(50 173 65 158)(51 174 66 159)(52 175 67 160)(53 176 68 161)(54 177 69 162)(73 297 346 326)(74 289 347 327)(75 290 348 328)(76 291 349 329)(77 292 350 330)(78 293 351 331)(79 294 343 332)(80 295 344 333)(81 296 345 325)(91 211 103 199)(92 212 104 200)(93 213 105 201)(94 214 106 202)(95 215 107 203)(96 216 108 204)(97 208 100 205)(98 209 101 206)(99 210 102 207)(226 373 274 388)(227 374 275 389)(228 375 276 390)(229 376 277 391)(230 377 278 392)(231 378 279 393)(232 370 271 394)(233 371 272 395)(234 372 273 396)(244 426 306 423)(245 427 298 415)(246 428 299 416)(247 429 300 417)(248 430 301 418)(249 431 302 419)(250 432 303 420)(251 424 304 421)(252 425 305 422)(262 379 286 340)(263 380 287 341)(264 381 288 342)(265 382 280 334)(266 383 281 335)(267 384 282 336)(268 385 283 337)(269 386 284 338)(270 387 285 339)(307 353 319 406)(308 354 320 407)(309 355 321 408)(310 356 322 409)(311 357 323 410)(312 358 324 411)(313 359 316 412)(314 360 317 413)(315 352 318 414)
G:=sub<Sym(432)| (1,92,111)(2,93,112)(3,94,113)(4,95,114)(5,96,115)(6,97,116)(7,98,117)(8,99,109)(9,91,110)(10,60,101)(11,61,102)(12,62,103)(13,63,104)(14,55,105)(15,56,106)(16,57,107)(17,58,108)(18,59,100)(19,65,88)(20,66,89)(21,67,90)(22,68,82)(23,69,83)(24,70,84)(25,71,85)(26,72,86)(27,64,87)(28,148,52)(29,149,53)(30,150,54)(31,151,46)(32,152,47)(33,153,48)(34,145,49)(35,146,50)(36,147,51)(37,341,418)(38,342,419)(39,334,420)(40,335,421)(41,336,422)(42,337,423)(43,338,415)(44,339,416)(45,340,417)(73,377,353)(74,378,354)(75,370,355)(76,371,356)(77,372,357)(78,373,358)(79,374,359)(80,375,360)(81,376,352)(118,204,223)(119,205,224)(120,206,225)(121,207,217)(122,199,218)(123,200,219)(124,201,220)(125,202,221)(126,203,222)(127,173,196)(128,174,197)(129,175,198)(130,176,190)(131,177,191)(132,178,192)(133,179,193)(134,180,194)(135,172,195)(136,256,160)(137,257,161)(138,258,162)(139,259,154)(140,260,155)(141,261,156)(142,253,157)(143,254,158)(144,255,159)(163,213,182)(164,214,183)(165,215,184)(166,216,185)(167,208,186)(168,209,187)(169,210,188)(170,211,189)(171,212,181)(226,312,331)(227,313,332)(228,314,333)(229,315,325)(230,307,326)(231,308,327)(232,309,328)(233,310,329)(234,311,330)(235,281,304)(236,282,305)(237,283,306)(238,284,298)(239,285,299)(240,286,300)(241,287,301)(242,288,302)(243,280,303)(244,364,268)(245,365,269)(246,366,270)(247,367,262)(248,368,263)(249,369,264)(250,361,265)(251,362,266)(252,363,267)(271,321,290)(272,322,291)(273,323,292)(274,324,293)(275,316,294)(276,317,295)(277,318,296)(278,319,297)(279,320,289)(343,389,412)(344,390,413)(345,391,414)(346,392,406)(347,393,407)(348,394,408)(349,395,409)(350,396,410)(351,388,411)(379,429,398)(380,430,399)(381,431,400)(382,432,401)(383,424,402)(384,425,403)(385,426,404)(386,427,405)(387,428,397), (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), (1,393,69,387,63,378,54,339)(2,394,70,379,55,370,46,340)(3,395,71,380,56,371,47,341)(4,396,72,381,57,372,48,342)(5,388,64,382,58,373,49,334)(6,389,65,383,59,374,50,335)(7,390,66,384,60,375,51,336)(8,391,67,385,61,376,52,337)(9,392,68,386,62,377,53,338)(10,80,147,41,117,344,20,403)(11,81,148,42,109,345,21,404)(12,73,149,43,110,346,22,405)(13,74,150,44,111,347,23,397)(14,75,151,45,112,348,24,398)(15,76,152,37,113,349,25,399)(16,77,153,38,114,350,26,400)(17,78,145,39,115,351,27,401)(18,79,146,40,116,343,19,402)(28,423,99,414,90,426,102,352)(29,415,91,406,82,427,103,353)(30,416,92,407,83,428,104,354)(31,417,93,408,84,429,105,355)(32,418,94,409,85,430,106,356)(33,419,95,410,86,431,107,357)(34,420,96,411,87,432,108,358)(35,421,97,412,88,424,100,359)(36,422,98,413,89,425,101,360)(118,280,157,226,166,265,172,274)(119,281,158,227,167,266,173,275)(120,282,159,228,168,267,174,276)(121,283,160,229,169,268,175,277)(122,284,161,230,170,269,176,278)(123,285,162,231,171,270,177,279)(124,286,154,232,163,262,178,271)(125,287,155,233,164,263,179,272)(126,288,156,234,165,264,180,273)(127,294,224,235,254,332,186,362)(128,295,225,236,255,333,187,363)(129,296,217,237,256,325,188,364)(130,297,218,238,257,326,189,365)(131,289,219,239,258,327,181,366)(132,290,220,240,259,328,182,367)(133,291,221,241,260,329,183,368)(134,292,222,242,261,330,184,369)(135,293,223,243,253,331,185,361)(136,315,210,244,198,318,207,306)(137,307,211,245,190,319,199,298)(138,308,212,246,191,320,200,299)(139,309,213,247,192,321,201,300)(140,310,214,248,193,322,202,301)(141,311,215,249,194,323,203,302)(142,312,216,250,195,324,204,303)(143,313,208,251,196,316,205,304)(144,314,209,252,197,317,206,305), (1,171,63,123)(2,163,55,124)(3,164,56,125)(4,165,57,126)(5,166,58,118)(6,167,59,119)(7,168,60,120)(8,169,61,121)(9,170,62,122)(10,225,117,187)(11,217,109,188)(12,218,110,189)(13,219,111,181)(14,220,112,182)(15,221,113,183)(16,222,114,184)(17,223,115,185)(18,224,116,186)(19,254,146,127)(20,255,147,128)(21,256,148,129)(22,257,149,130)(23,258,150,131)(24,259,151,132)(25,260,152,133)(26,261,153,134)(27,253,145,135)(28,198,90,136)(29,190,82,137)(30,191,83,138)(31,192,84,139)(32,193,85,140)(33,194,86,141)(34,195,87,142)(35,196,88,143)(36,197,89,144)(37,368,399,241)(38,369,400,242)(39,361,401,243)(40,362,402,235)(41,363,403,236)(42,364,404,237)(43,365,405,238)(44,366,397,239)(45,367,398,240)(46,178,70,154)(47,179,71,155)(48,180,72,156)(49,172,64,157)(50,173,65,158)(51,174,66,159)(52,175,67,160)(53,176,68,161)(54,177,69,162)(73,297,346,326)(74,289,347,327)(75,290,348,328)(76,291,349,329)(77,292,350,330)(78,293,351,331)(79,294,343,332)(80,295,344,333)(81,296,345,325)(91,211,103,199)(92,212,104,200)(93,213,105,201)(94,214,106,202)(95,215,107,203)(96,216,108,204)(97,208,100,205)(98,209,101,206)(99,210,102,207)(226,373,274,388)(227,374,275,389)(228,375,276,390)(229,376,277,391)(230,377,278,392)(231,378,279,393)(232,370,271,394)(233,371,272,395)(234,372,273,396)(244,426,306,423)(245,427,298,415)(246,428,299,416)(247,429,300,417)(248,430,301,418)(249,431,302,419)(250,432,303,420)(251,424,304,421)(252,425,305,422)(262,379,286,340)(263,380,287,341)(264,381,288,342)(265,382,280,334)(266,383,281,335)(267,384,282,336)(268,385,283,337)(269,386,284,338)(270,387,285,339)(307,353,319,406)(308,354,320,407)(309,355,321,408)(310,356,322,409)(311,357,323,410)(312,358,324,411)(313,359,316,412)(314,360,317,413)(315,352,318,414)>;
G:=Group( (1,92,111)(2,93,112)(3,94,113)(4,95,114)(5,96,115)(6,97,116)(7,98,117)(8,99,109)(9,91,110)(10,60,101)(11,61,102)(12,62,103)(13,63,104)(14,55,105)(15,56,106)(16,57,107)(17,58,108)(18,59,100)(19,65,88)(20,66,89)(21,67,90)(22,68,82)(23,69,83)(24,70,84)(25,71,85)(26,72,86)(27,64,87)(28,148,52)(29,149,53)(30,150,54)(31,151,46)(32,152,47)(33,153,48)(34,145,49)(35,146,50)(36,147,51)(37,341,418)(38,342,419)(39,334,420)(40,335,421)(41,336,422)(42,337,423)(43,338,415)(44,339,416)(45,340,417)(73,377,353)(74,378,354)(75,370,355)(76,371,356)(77,372,357)(78,373,358)(79,374,359)(80,375,360)(81,376,352)(118,204,223)(119,205,224)(120,206,225)(121,207,217)(122,199,218)(123,200,219)(124,201,220)(125,202,221)(126,203,222)(127,173,196)(128,174,197)(129,175,198)(130,176,190)(131,177,191)(132,178,192)(133,179,193)(134,180,194)(135,172,195)(136,256,160)(137,257,161)(138,258,162)(139,259,154)(140,260,155)(141,261,156)(142,253,157)(143,254,158)(144,255,159)(163,213,182)(164,214,183)(165,215,184)(166,216,185)(167,208,186)(168,209,187)(169,210,188)(170,211,189)(171,212,181)(226,312,331)(227,313,332)(228,314,333)(229,315,325)(230,307,326)(231,308,327)(232,309,328)(233,310,329)(234,311,330)(235,281,304)(236,282,305)(237,283,306)(238,284,298)(239,285,299)(240,286,300)(241,287,301)(242,288,302)(243,280,303)(244,364,268)(245,365,269)(246,366,270)(247,367,262)(248,368,263)(249,369,264)(250,361,265)(251,362,266)(252,363,267)(271,321,290)(272,322,291)(273,323,292)(274,324,293)(275,316,294)(276,317,295)(277,318,296)(278,319,297)(279,320,289)(343,389,412)(344,390,413)(345,391,414)(346,392,406)(347,393,407)(348,394,408)(349,395,409)(350,396,410)(351,388,411)(379,429,398)(380,430,399)(381,431,400)(382,432,401)(383,424,402)(384,425,403)(385,426,404)(386,427,405)(387,428,397), (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), (1,393,69,387,63,378,54,339)(2,394,70,379,55,370,46,340)(3,395,71,380,56,371,47,341)(4,396,72,381,57,372,48,342)(5,388,64,382,58,373,49,334)(6,389,65,383,59,374,50,335)(7,390,66,384,60,375,51,336)(8,391,67,385,61,376,52,337)(9,392,68,386,62,377,53,338)(10,80,147,41,117,344,20,403)(11,81,148,42,109,345,21,404)(12,73,149,43,110,346,22,405)(13,74,150,44,111,347,23,397)(14,75,151,45,112,348,24,398)(15,76,152,37,113,349,25,399)(16,77,153,38,114,350,26,400)(17,78,145,39,115,351,27,401)(18,79,146,40,116,343,19,402)(28,423,99,414,90,426,102,352)(29,415,91,406,82,427,103,353)(30,416,92,407,83,428,104,354)(31,417,93,408,84,429,105,355)(32,418,94,409,85,430,106,356)(33,419,95,410,86,431,107,357)(34,420,96,411,87,432,108,358)(35,421,97,412,88,424,100,359)(36,422,98,413,89,425,101,360)(118,280,157,226,166,265,172,274)(119,281,158,227,167,266,173,275)(120,282,159,228,168,267,174,276)(121,283,160,229,169,268,175,277)(122,284,161,230,170,269,176,278)(123,285,162,231,171,270,177,279)(124,286,154,232,163,262,178,271)(125,287,155,233,164,263,179,272)(126,288,156,234,165,264,180,273)(127,294,224,235,254,332,186,362)(128,295,225,236,255,333,187,363)(129,296,217,237,256,325,188,364)(130,297,218,238,257,326,189,365)(131,289,219,239,258,327,181,366)(132,290,220,240,259,328,182,367)(133,291,221,241,260,329,183,368)(134,292,222,242,261,330,184,369)(135,293,223,243,253,331,185,361)(136,315,210,244,198,318,207,306)(137,307,211,245,190,319,199,298)(138,308,212,246,191,320,200,299)(139,309,213,247,192,321,201,300)(140,310,214,248,193,322,202,301)(141,311,215,249,194,323,203,302)(142,312,216,250,195,324,204,303)(143,313,208,251,196,316,205,304)(144,314,209,252,197,317,206,305), (1,171,63,123)(2,163,55,124)(3,164,56,125)(4,165,57,126)(5,166,58,118)(6,167,59,119)(7,168,60,120)(8,169,61,121)(9,170,62,122)(10,225,117,187)(11,217,109,188)(12,218,110,189)(13,219,111,181)(14,220,112,182)(15,221,113,183)(16,222,114,184)(17,223,115,185)(18,224,116,186)(19,254,146,127)(20,255,147,128)(21,256,148,129)(22,257,149,130)(23,258,150,131)(24,259,151,132)(25,260,152,133)(26,261,153,134)(27,253,145,135)(28,198,90,136)(29,190,82,137)(30,191,83,138)(31,192,84,139)(32,193,85,140)(33,194,86,141)(34,195,87,142)(35,196,88,143)(36,197,89,144)(37,368,399,241)(38,369,400,242)(39,361,401,243)(40,362,402,235)(41,363,403,236)(42,364,404,237)(43,365,405,238)(44,366,397,239)(45,367,398,240)(46,178,70,154)(47,179,71,155)(48,180,72,156)(49,172,64,157)(50,173,65,158)(51,174,66,159)(52,175,67,160)(53,176,68,161)(54,177,69,162)(73,297,346,326)(74,289,347,327)(75,290,348,328)(76,291,349,329)(77,292,350,330)(78,293,351,331)(79,294,343,332)(80,295,344,333)(81,296,345,325)(91,211,103,199)(92,212,104,200)(93,213,105,201)(94,214,106,202)(95,215,107,203)(96,216,108,204)(97,208,100,205)(98,209,101,206)(99,210,102,207)(226,373,274,388)(227,374,275,389)(228,375,276,390)(229,376,277,391)(230,377,278,392)(231,378,279,393)(232,370,271,394)(233,371,272,395)(234,372,273,396)(244,426,306,423)(245,427,298,415)(246,428,299,416)(247,429,300,417)(248,430,301,418)(249,431,302,419)(250,432,303,420)(251,424,304,421)(252,425,305,422)(262,379,286,340)(263,380,287,341)(264,381,288,342)(265,382,280,334)(266,383,281,335)(267,384,282,336)(268,385,283,337)(269,386,284,338)(270,387,285,339)(307,353,319,406)(308,354,320,407)(309,355,321,408)(310,356,322,409)(311,357,323,410)(312,358,324,411)(313,359,316,412)(314,360,317,413)(315,352,318,414) );
G=PermutationGroup([[(1,92,111),(2,93,112),(3,94,113),(4,95,114),(5,96,115),(6,97,116),(7,98,117),(8,99,109),(9,91,110),(10,60,101),(11,61,102),(12,62,103),(13,63,104),(14,55,105),(15,56,106),(16,57,107),(17,58,108),(18,59,100),(19,65,88),(20,66,89),(21,67,90),(22,68,82),(23,69,83),(24,70,84),(25,71,85),(26,72,86),(27,64,87),(28,148,52),(29,149,53),(30,150,54),(31,151,46),(32,152,47),(33,153,48),(34,145,49),(35,146,50),(36,147,51),(37,341,418),(38,342,419),(39,334,420),(40,335,421),(41,336,422),(42,337,423),(43,338,415),(44,339,416),(45,340,417),(73,377,353),(74,378,354),(75,370,355),(76,371,356),(77,372,357),(78,373,358),(79,374,359),(80,375,360),(81,376,352),(118,204,223),(119,205,224),(120,206,225),(121,207,217),(122,199,218),(123,200,219),(124,201,220),(125,202,221),(126,203,222),(127,173,196),(128,174,197),(129,175,198),(130,176,190),(131,177,191),(132,178,192),(133,179,193),(134,180,194),(135,172,195),(136,256,160),(137,257,161),(138,258,162),(139,259,154),(140,260,155),(141,261,156),(142,253,157),(143,254,158),(144,255,159),(163,213,182),(164,214,183),(165,215,184),(166,216,185),(167,208,186),(168,209,187),(169,210,188),(170,211,189),(171,212,181),(226,312,331),(227,313,332),(228,314,333),(229,315,325),(230,307,326),(231,308,327),(232,309,328),(233,310,329),(234,311,330),(235,281,304),(236,282,305),(237,283,306),(238,284,298),(239,285,299),(240,286,300),(241,287,301),(242,288,302),(243,280,303),(244,364,268),(245,365,269),(246,366,270),(247,367,262),(248,368,263),(249,369,264),(250,361,265),(251,362,266),(252,363,267),(271,321,290),(272,322,291),(273,323,292),(274,324,293),(275,316,294),(276,317,295),(277,318,296),(278,319,297),(279,320,289),(343,389,412),(344,390,413),(345,391,414),(346,392,406),(347,393,407),(348,394,408),(349,395,409),(350,396,410),(351,388,411),(379,429,398),(380,430,399),(381,431,400),(382,432,401),(383,424,402),(384,425,403),(385,426,404),(386,427,405),(387,428,397)], [(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)], [(1,393,69,387,63,378,54,339),(2,394,70,379,55,370,46,340),(3,395,71,380,56,371,47,341),(4,396,72,381,57,372,48,342),(5,388,64,382,58,373,49,334),(6,389,65,383,59,374,50,335),(7,390,66,384,60,375,51,336),(8,391,67,385,61,376,52,337),(9,392,68,386,62,377,53,338),(10,80,147,41,117,344,20,403),(11,81,148,42,109,345,21,404),(12,73,149,43,110,346,22,405),(13,74,150,44,111,347,23,397),(14,75,151,45,112,348,24,398),(15,76,152,37,113,349,25,399),(16,77,153,38,114,350,26,400),(17,78,145,39,115,351,27,401),(18,79,146,40,116,343,19,402),(28,423,99,414,90,426,102,352),(29,415,91,406,82,427,103,353),(30,416,92,407,83,428,104,354),(31,417,93,408,84,429,105,355),(32,418,94,409,85,430,106,356),(33,419,95,410,86,431,107,357),(34,420,96,411,87,432,108,358),(35,421,97,412,88,424,100,359),(36,422,98,413,89,425,101,360),(118,280,157,226,166,265,172,274),(119,281,158,227,167,266,173,275),(120,282,159,228,168,267,174,276),(121,283,160,229,169,268,175,277),(122,284,161,230,170,269,176,278),(123,285,162,231,171,270,177,279),(124,286,154,232,163,262,178,271),(125,287,155,233,164,263,179,272),(126,288,156,234,165,264,180,273),(127,294,224,235,254,332,186,362),(128,295,225,236,255,333,187,363),(129,296,217,237,256,325,188,364),(130,297,218,238,257,326,189,365),(131,289,219,239,258,327,181,366),(132,290,220,240,259,328,182,367),(133,291,221,241,260,329,183,368),(134,292,222,242,261,330,184,369),(135,293,223,243,253,331,185,361),(136,315,210,244,198,318,207,306),(137,307,211,245,190,319,199,298),(138,308,212,246,191,320,200,299),(139,309,213,247,192,321,201,300),(140,310,214,248,193,322,202,301),(141,311,215,249,194,323,203,302),(142,312,216,250,195,324,204,303),(143,313,208,251,196,316,205,304),(144,314,209,252,197,317,206,305)], [(1,171,63,123),(2,163,55,124),(3,164,56,125),(4,165,57,126),(5,166,58,118),(6,167,59,119),(7,168,60,120),(8,169,61,121),(9,170,62,122),(10,225,117,187),(11,217,109,188),(12,218,110,189),(13,219,111,181),(14,220,112,182),(15,221,113,183),(16,222,114,184),(17,223,115,185),(18,224,116,186),(19,254,146,127),(20,255,147,128),(21,256,148,129),(22,257,149,130),(23,258,150,131),(24,259,151,132),(25,260,152,133),(26,261,153,134),(27,253,145,135),(28,198,90,136),(29,190,82,137),(30,191,83,138),(31,192,84,139),(32,193,85,140),(33,194,86,141),(34,195,87,142),(35,196,88,143),(36,197,89,144),(37,368,399,241),(38,369,400,242),(39,361,401,243),(40,362,402,235),(41,363,403,236),(42,364,404,237),(43,365,405,238),(44,366,397,239),(45,367,398,240),(46,178,70,154),(47,179,71,155),(48,180,72,156),(49,172,64,157),(50,173,65,158),(51,174,66,159),(52,175,67,160),(53,176,68,161),(54,177,69,162),(73,297,346,326),(74,289,347,327),(75,290,348,328),(76,291,349,329),(77,292,350,330),(78,293,351,331),(79,294,343,332),(80,295,344,333),(81,296,345,325),(91,211,103,199),(92,212,104,200),(93,213,105,201),(94,214,106,202),(95,215,107,203),(96,216,108,204),(97,208,100,205),(98,209,101,206),(99,210,102,207),(226,373,274,388),(227,374,275,389),(228,375,276,390),(229,376,277,391),(230,377,278,392),(231,378,279,393),(232,370,271,394),(233,371,272,395),(234,372,273,396),(244,426,306,423),(245,427,298,415),(246,428,299,416),(247,429,300,417),(248,430,301,418),(249,431,302,419),(250,432,303,420),(251,424,304,421),(252,425,305,422),(262,379,286,340),(263,380,287,341),(264,381,288,342),(265,382,280,334),(266,383,281,335),(267,384,282,336),(268,385,283,337),(269,386,284,338),(270,387,285,339),(307,353,319,406),(308,354,320,407),(309,355,321,408),(310,356,322,409),(311,357,323,410),(312,358,324,411),(313,359,316,412),(314,360,317,413),(315,352,318,414)]])
189 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 4C | 6A | ··· | 6H | 8A | 8B | 9A | ··· | 9R | 12A | ··· | 12H | 12I | ··· | 12X | 18A | ··· | 18R | 24A | ··· | 24P | 36A | ··· | 36R | 36S | ··· | 36BB | 72A | ··· | 72AJ |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
189 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 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | D4 | Q16 | C3×D4 | C3×D4 | C3×Q16 | C3×Q16 | D4×C9 | C9×Q16 |
| kernel | Q16×C3×C9 | C3×C72 | Q8×C3×C9 | C9×Q16 | C32×Q16 | C72 | Q8×C9 | C3×C24 | Q8×C32 | C3×Q16 | C24 | C3×Q8 | C3×C18 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 6 | 2 | 6 | 12 | 2 | 4 | 18 | 18 | 36 | 1 | 2 | 6 | 2 | 12 | 4 | 18 | 36 |
Matrix representation of Q16×C3×C9 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 41 | 41 |
| 0 | 16 | 0 |
| 72 | 0 | 0 |
| 0 | 11 | 71 |
| 0 | 61 | 62 |
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[1,0,0,0,4,0,0,0,4],[1,0,0,0,41,16,0,41,0],[72,0,0,0,11,61,0,71,62] >;
Q16×C3×C9 in GAP, Magma, Sage, TeX
Q_{16}\times C_3\times C_9 % in TeX
G:=Group("Q16xC3xC9"); // GroupNames label
G:=SmallGroup(432,221);
// by ID
G=gap.SmallGroup(432,221);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,1512,533,1520,394,8824,4421,242]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^8=1,d^2=c^4,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