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## G = C6×Q8⋊C9order 432 = 24·33

### Direct product of C6 and Q8⋊C9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C6×Q8⋊C9
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — C3×Q8⋊C9 — C6×Q8⋊C9
 Lower central Q8 — C6×Q8⋊C9
 Upper central C1 — C62

Generators and relations for C6×Q8⋊C9
G = < a,b,c,d | a6=b4=d9=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Subgroups: 202 in 94 conjugacy classes, 50 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, Q8, Q8, C9, C32, C12, C2×C6, C2×C6, C2×Q8, C18, C3×C6, C3×C6, C2×C12, C3×Q8, C3×Q8, C3×Q8, C3×C9, C2×C18, C3×C12, C62, C6×Q8, C6×Q8, C3×C18, Q8⋊C9, C6×C12, Q8×C32, Q8×C32, C6×C18, C2×Q8⋊C9, Q8×C3×C6, C3×Q8⋊C9, C6×Q8⋊C9
Quotients: C1, C2, C3, C6, C9, C32, A4, C18, C3×C6, SL2(𝔽3), C2×A4, C3×C9, C3.A4, C3×A4, C2×SL2(𝔽3), C3×C18, Q8⋊C9, C2×C3.A4, C3×SL2(𝔽3), C6×A4, C3×C3.A4, C2×Q8⋊C9, C6×SL2(𝔽3), C3×Q8⋊C9, C6×C3.A4, C6×Q8⋊C9

Smallest permutation representation of C6×Q8⋊C9
Regular action on 432 points
Generators in S432
(1 107 62 298 119 227)(2 108 63 299 120 228)(3 100 55 300 121 229)(4 101 56 301 122 230)(5 102 57 302 123 231)(6 103 58 303 124 232)(7 104 59 304 125 233)(8 105 60 305 126 234)(9 106 61 306 118 226)(10 353 282 141 324 174)(11 354 283 142 316 175)(12 355 284 143 317 176)(13 356 285 144 318 177)(14 357 286 136 319 178)(15 358 287 137 320 179)(16 359 288 138 321 180)(17 360 280 139 322 172)(18 352 281 140 323 173)(19 312 380 417 168 112)(20 313 381 418 169 113)(21 314 382 419 170 114)(22 315 383 420 171 115)(23 307 384 421 163 116)(24 308 385 422 164 117)(25 309 386 423 165 109)(26 310 387 415 166 110)(27 311 379 416 167 111)(28 428 86 277 249 412)(29 429 87 278 250 413)(30 430 88 279 251 414)(31 431 89 271 252 406)(32 432 90 272 244 407)(33 424 82 273 245 408)(34 425 83 274 246 409)(35 426 84 275 247 410)(36 427 85 276 248 411)(37 207 79 270 258 327)(38 199 80 262 259 328)(39 200 81 263 260 329)(40 201 73 264 261 330)(41 202 74 265 253 331)(42 203 75 266 254 332)(43 204 76 267 255 333)(44 205 77 268 256 325)(45 206 78 269 257 326)(46 291 346 220 134 91)(47 292 347 221 135 92)(48 293 348 222 127 93)(49 294 349 223 128 94)(50 295 350 224 129 95)(51 296 351 225 130 96)(52 297 343 217 131 97)(53 289 344 218 132 98)(54 290 345 219 133 99)(64 241 364 192 152 392)(65 242 365 193 153 393)(66 243 366 194 145 394)(67 235 367 195 146 395)(68 236 368 196 147 396)(69 237 369 197 148 388)(70 238 361 198 149 389)(71 239 362 190 150 390)(72 240 363 191 151 391)(154 209 188 400 375 338)(155 210 189 401 376 339)(156 211 181 402 377 340)(157 212 182 403 378 341)(158 213 183 404 370 342)(159 214 184 405 371 334)(160 215 185 397 372 335)(161 216 186 398 373 336)(162 208 187 399 374 337)
(1 54 31 81)(2 68 32 357)(3 376 33 379)(4 48 34 75)(5 71 35 360)(6 370 36 382)(7 51 28 78)(8 65 29 354)(9 373 30 385)(10 405 392 315)(11 234 393 413)(12 199 394 98)(13 399 395 309)(14 228 396 407)(15 202 388 92)(16 402 389 312)(17 231 390 410)(18 205 391 95)(19 180 181 149)(20 128 182 43)(21 124 183 248)(22 174 184 152)(23 131 185 37)(24 118 186 251)(25 177 187 146)(26 134 188 40)(27 121 189 245)(38 145 132 176)(39 119 133 252)(41 148 135 179)(42 122 127 246)(44 151 129 173)(45 125 130 249)(46 375 73 387)(47 358 74 69)(49 378 76 381)(50 352 77 72)(52 372 79 384)(53 355 80 66)(55 155 82 167)(56 348 83 254)(57 362 84 139)(58 158 85 170)(59 351 86 257)(60 365 87 142)(61 161 88 164)(62 345 89 260)(63 368 90 136)(64 383 353 371)(67 386 356 374)(70 380 359 377)(91 400 201 310)(93 409 203 230)(94 403 204 313)(96 412 206 233)(97 397 207 307)(99 406 200 227)(100 339 424 416)(101 293 425 266)(102 239 426 280)(103 342 427 419)(104 296 428 269)(105 242 429 283)(106 336 430 422)(107 290 431 263)(108 236 432 286)(109 318 208 195)(110 220 209 330)(111 300 210 273)(112 321 211 198)(113 223 212 333)(114 303 213 276)(115 324 214 192)(116 217 215 327)(117 306 216 279)(120 147 244 178)(123 150 247 172)(126 153 250 175)(137 253 369 347)(138 156 361 168)(140 256 363 350)(141 159 364 171)(143 259 366 344)(144 162 367 165)(154 261 166 346)(157 255 169 349)(160 258 163 343)(190 275 322 302)(191 224 323 325)(193 278 316 305)(194 218 317 328)(196 272 319 299)(197 221 320 331)(219 271 329 298)(222 274 332 301)(225 277 326 304)(226 398 414 308)(229 401 408 311)(232 404 411 314)(235 423 285 337)(237 292 287 265)(238 417 288 340)(240 295 281 268)(241 420 282 334)(243 289 284 262)(264 415 291 338)(267 418 294 341)(270 421 297 335)
(1 67 31 356)(2 375 32 387)(3 47 33 74)(4 70 34 359)(5 378 35 381)(6 50 36 77)(7 64 28 353)(8 372 29 384)(9 53 30 80)(10 233 392 412)(11 207 393 97)(12 398 394 308)(13 227 395 406)(14 201 396 91)(15 401 388 311)(16 230 389 409)(17 204 390 94)(18 404 391 314)(19 127 181 42)(20 123 182 247)(21 173 183 151)(22 130 184 45)(23 126 185 250)(24 176 186 145)(25 133 187 39)(26 120 188 244)(27 179 189 148)(37 153 131 175)(38 118 132 251)(40 147 134 178)(41 121 135 245)(43 150 128 172)(44 124 129 248)(46 357 73 68)(48 377 75 380)(49 360 76 71)(51 371 78 383)(52 354 79 65)(54 374 81 386)(55 347 82 253)(56 361 83 138)(57 157 84 169)(58 350 85 256)(59 364 86 141)(60 160 87 163)(61 344 88 259)(62 367 89 144)(63 154 90 166)(66 385 355 373)(69 379 358 376)(72 382 352 370)(92 408 202 229)(93 402 203 312)(95 411 205 232)(96 405 206 315)(98 414 199 226)(99 399 200 309)(100 292 424 265)(101 238 425 288)(102 341 426 418)(103 295 427 268)(104 241 428 282)(105 335 429 421)(106 289 430 262)(107 235 431 285)(108 338 432 415)(109 219 208 329)(110 299 209 272)(111 320 210 197)(112 222 211 332)(113 302 212 275)(114 323 213 191)(115 225 214 326)(116 305 215 278)(117 317 216 194)(119 146 252 177)(122 149 246 180)(125 152 249 174)(136 261 368 346)(137 155 369 167)(139 255 362 349)(140 158 363 170)(142 258 365 343)(143 161 366 164)(156 254 168 348)(159 257 171 351)(162 260 165 345)(190 223 322 333)(192 277 324 304)(193 217 316 327)(195 271 318 298)(196 220 319 330)(198 274 321 301)(218 279 328 306)(221 273 331 300)(224 276 325 303)(228 400 407 310)(231 403 410 313)(234 397 413 307)(236 291 286 264)(237 416 287 339)(239 294 280 267)(240 419 281 342)(242 297 283 270)(243 422 284 336)(263 423 290 337)(266 417 293 340)(269 420 296 334)
(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)

G:=sub<Sym(432)| (1,107,62,298,119,227)(2,108,63,299,120,228)(3,100,55,300,121,229)(4,101,56,301,122,230)(5,102,57,302,123,231)(6,103,58,303,124,232)(7,104,59,304,125,233)(8,105,60,305,126,234)(9,106,61,306,118,226)(10,353,282,141,324,174)(11,354,283,142,316,175)(12,355,284,143,317,176)(13,356,285,144,318,177)(14,357,286,136,319,178)(15,358,287,137,320,179)(16,359,288,138,321,180)(17,360,280,139,322,172)(18,352,281,140,323,173)(19,312,380,417,168,112)(20,313,381,418,169,113)(21,314,382,419,170,114)(22,315,383,420,171,115)(23,307,384,421,163,116)(24,308,385,422,164,117)(25,309,386,423,165,109)(26,310,387,415,166,110)(27,311,379,416,167,111)(28,428,86,277,249,412)(29,429,87,278,250,413)(30,430,88,279,251,414)(31,431,89,271,252,406)(32,432,90,272,244,407)(33,424,82,273,245,408)(34,425,83,274,246,409)(35,426,84,275,247,410)(36,427,85,276,248,411)(37,207,79,270,258,327)(38,199,80,262,259,328)(39,200,81,263,260,329)(40,201,73,264,261,330)(41,202,74,265,253,331)(42,203,75,266,254,332)(43,204,76,267,255,333)(44,205,77,268,256,325)(45,206,78,269,257,326)(46,291,346,220,134,91)(47,292,347,221,135,92)(48,293,348,222,127,93)(49,294,349,223,128,94)(50,295,350,224,129,95)(51,296,351,225,130,96)(52,297,343,217,131,97)(53,289,344,218,132,98)(54,290,345,219,133,99)(64,241,364,192,152,392)(65,242,365,193,153,393)(66,243,366,194,145,394)(67,235,367,195,146,395)(68,236,368,196,147,396)(69,237,369,197,148,388)(70,238,361,198,149,389)(71,239,362,190,150,390)(72,240,363,191,151,391)(154,209,188,400,375,338)(155,210,189,401,376,339)(156,211,181,402,377,340)(157,212,182,403,378,341)(158,213,183,404,370,342)(159,214,184,405,371,334)(160,215,185,397,372,335)(161,216,186,398,373,336)(162,208,187,399,374,337), (1,54,31,81)(2,68,32,357)(3,376,33,379)(4,48,34,75)(5,71,35,360)(6,370,36,382)(7,51,28,78)(8,65,29,354)(9,373,30,385)(10,405,392,315)(11,234,393,413)(12,199,394,98)(13,399,395,309)(14,228,396,407)(15,202,388,92)(16,402,389,312)(17,231,390,410)(18,205,391,95)(19,180,181,149)(20,128,182,43)(21,124,183,248)(22,174,184,152)(23,131,185,37)(24,118,186,251)(25,177,187,146)(26,134,188,40)(27,121,189,245)(38,145,132,176)(39,119,133,252)(41,148,135,179)(42,122,127,246)(44,151,129,173)(45,125,130,249)(46,375,73,387)(47,358,74,69)(49,378,76,381)(50,352,77,72)(52,372,79,384)(53,355,80,66)(55,155,82,167)(56,348,83,254)(57,362,84,139)(58,158,85,170)(59,351,86,257)(60,365,87,142)(61,161,88,164)(62,345,89,260)(63,368,90,136)(64,383,353,371)(67,386,356,374)(70,380,359,377)(91,400,201,310)(93,409,203,230)(94,403,204,313)(96,412,206,233)(97,397,207,307)(99,406,200,227)(100,339,424,416)(101,293,425,266)(102,239,426,280)(103,342,427,419)(104,296,428,269)(105,242,429,283)(106,336,430,422)(107,290,431,263)(108,236,432,286)(109,318,208,195)(110,220,209,330)(111,300,210,273)(112,321,211,198)(113,223,212,333)(114,303,213,276)(115,324,214,192)(116,217,215,327)(117,306,216,279)(120,147,244,178)(123,150,247,172)(126,153,250,175)(137,253,369,347)(138,156,361,168)(140,256,363,350)(141,159,364,171)(143,259,366,344)(144,162,367,165)(154,261,166,346)(157,255,169,349)(160,258,163,343)(190,275,322,302)(191,224,323,325)(193,278,316,305)(194,218,317,328)(196,272,319,299)(197,221,320,331)(219,271,329,298)(222,274,332,301)(225,277,326,304)(226,398,414,308)(229,401,408,311)(232,404,411,314)(235,423,285,337)(237,292,287,265)(238,417,288,340)(240,295,281,268)(241,420,282,334)(243,289,284,262)(264,415,291,338)(267,418,294,341)(270,421,297,335), (1,67,31,356)(2,375,32,387)(3,47,33,74)(4,70,34,359)(5,378,35,381)(6,50,36,77)(7,64,28,353)(8,372,29,384)(9,53,30,80)(10,233,392,412)(11,207,393,97)(12,398,394,308)(13,227,395,406)(14,201,396,91)(15,401,388,311)(16,230,389,409)(17,204,390,94)(18,404,391,314)(19,127,181,42)(20,123,182,247)(21,173,183,151)(22,130,184,45)(23,126,185,250)(24,176,186,145)(25,133,187,39)(26,120,188,244)(27,179,189,148)(37,153,131,175)(38,118,132,251)(40,147,134,178)(41,121,135,245)(43,150,128,172)(44,124,129,248)(46,357,73,68)(48,377,75,380)(49,360,76,71)(51,371,78,383)(52,354,79,65)(54,374,81,386)(55,347,82,253)(56,361,83,138)(57,157,84,169)(58,350,85,256)(59,364,86,141)(60,160,87,163)(61,344,88,259)(62,367,89,144)(63,154,90,166)(66,385,355,373)(69,379,358,376)(72,382,352,370)(92,408,202,229)(93,402,203,312)(95,411,205,232)(96,405,206,315)(98,414,199,226)(99,399,200,309)(100,292,424,265)(101,238,425,288)(102,341,426,418)(103,295,427,268)(104,241,428,282)(105,335,429,421)(106,289,430,262)(107,235,431,285)(108,338,432,415)(109,219,208,329)(110,299,209,272)(111,320,210,197)(112,222,211,332)(113,302,212,275)(114,323,213,191)(115,225,214,326)(116,305,215,278)(117,317,216,194)(119,146,252,177)(122,149,246,180)(125,152,249,174)(136,261,368,346)(137,155,369,167)(139,255,362,349)(140,158,363,170)(142,258,365,343)(143,161,366,164)(156,254,168,348)(159,257,171,351)(162,260,165,345)(190,223,322,333)(192,277,324,304)(193,217,316,327)(195,271,318,298)(196,220,319,330)(198,274,321,301)(218,279,328,306)(221,273,331,300)(224,276,325,303)(228,400,407,310)(231,403,410,313)(234,397,413,307)(236,291,286,264)(237,416,287,339)(239,294,280,267)(240,419,281,342)(242,297,283,270)(243,422,284,336)(263,423,290,337)(266,417,293,340)(269,420,296,334), (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)>;

G:=Group( (1,107,62,298,119,227)(2,108,63,299,120,228)(3,100,55,300,121,229)(4,101,56,301,122,230)(5,102,57,302,123,231)(6,103,58,303,124,232)(7,104,59,304,125,233)(8,105,60,305,126,234)(9,106,61,306,118,226)(10,353,282,141,324,174)(11,354,283,142,316,175)(12,355,284,143,317,176)(13,356,285,144,318,177)(14,357,286,136,319,178)(15,358,287,137,320,179)(16,359,288,138,321,180)(17,360,280,139,322,172)(18,352,281,140,323,173)(19,312,380,417,168,112)(20,313,381,418,169,113)(21,314,382,419,170,114)(22,315,383,420,171,115)(23,307,384,421,163,116)(24,308,385,422,164,117)(25,309,386,423,165,109)(26,310,387,415,166,110)(27,311,379,416,167,111)(28,428,86,277,249,412)(29,429,87,278,250,413)(30,430,88,279,251,414)(31,431,89,271,252,406)(32,432,90,272,244,407)(33,424,82,273,245,408)(34,425,83,274,246,409)(35,426,84,275,247,410)(36,427,85,276,248,411)(37,207,79,270,258,327)(38,199,80,262,259,328)(39,200,81,263,260,329)(40,201,73,264,261,330)(41,202,74,265,253,331)(42,203,75,266,254,332)(43,204,76,267,255,333)(44,205,77,268,256,325)(45,206,78,269,257,326)(46,291,346,220,134,91)(47,292,347,221,135,92)(48,293,348,222,127,93)(49,294,349,223,128,94)(50,295,350,224,129,95)(51,296,351,225,130,96)(52,297,343,217,131,97)(53,289,344,218,132,98)(54,290,345,219,133,99)(64,241,364,192,152,392)(65,242,365,193,153,393)(66,243,366,194,145,394)(67,235,367,195,146,395)(68,236,368,196,147,396)(69,237,369,197,148,388)(70,238,361,198,149,389)(71,239,362,190,150,390)(72,240,363,191,151,391)(154,209,188,400,375,338)(155,210,189,401,376,339)(156,211,181,402,377,340)(157,212,182,403,378,341)(158,213,183,404,370,342)(159,214,184,405,371,334)(160,215,185,397,372,335)(161,216,186,398,373,336)(162,208,187,399,374,337), (1,54,31,81)(2,68,32,357)(3,376,33,379)(4,48,34,75)(5,71,35,360)(6,370,36,382)(7,51,28,78)(8,65,29,354)(9,373,30,385)(10,405,392,315)(11,234,393,413)(12,199,394,98)(13,399,395,309)(14,228,396,407)(15,202,388,92)(16,402,389,312)(17,231,390,410)(18,205,391,95)(19,180,181,149)(20,128,182,43)(21,124,183,248)(22,174,184,152)(23,131,185,37)(24,118,186,251)(25,177,187,146)(26,134,188,40)(27,121,189,245)(38,145,132,176)(39,119,133,252)(41,148,135,179)(42,122,127,246)(44,151,129,173)(45,125,130,249)(46,375,73,387)(47,358,74,69)(49,378,76,381)(50,352,77,72)(52,372,79,384)(53,355,80,66)(55,155,82,167)(56,348,83,254)(57,362,84,139)(58,158,85,170)(59,351,86,257)(60,365,87,142)(61,161,88,164)(62,345,89,260)(63,368,90,136)(64,383,353,371)(67,386,356,374)(70,380,359,377)(91,400,201,310)(93,409,203,230)(94,403,204,313)(96,412,206,233)(97,397,207,307)(99,406,200,227)(100,339,424,416)(101,293,425,266)(102,239,426,280)(103,342,427,419)(104,296,428,269)(105,242,429,283)(106,336,430,422)(107,290,431,263)(108,236,432,286)(109,318,208,195)(110,220,209,330)(111,300,210,273)(112,321,211,198)(113,223,212,333)(114,303,213,276)(115,324,214,192)(116,217,215,327)(117,306,216,279)(120,147,244,178)(123,150,247,172)(126,153,250,175)(137,253,369,347)(138,156,361,168)(140,256,363,350)(141,159,364,171)(143,259,366,344)(144,162,367,165)(154,261,166,346)(157,255,169,349)(160,258,163,343)(190,275,322,302)(191,224,323,325)(193,278,316,305)(194,218,317,328)(196,272,319,299)(197,221,320,331)(219,271,329,298)(222,274,332,301)(225,277,326,304)(226,398,414,308)(229,401,408,311)(232,404,411,314)(235,423,285,337)(237,292,287,265)(238,417,288,340)(240,295,281,268)(241,420,282,334)(243,289,284,262)(264,415,291,338)(267,418,294,341)(270,421,297,335), (1,67,31,356)(2,375,32,387)(3,47,33,74)(4,70,34,359)(5,378,35,381)(6,50,36,77)(7,64,28,353)(8,372,29,384)(9,53,30,80)(10,233,392,412)(11,207,393,97)(12,398,394,308)(13,227,395,406)(14,201,396,91)(15,401,388,311)(16,230,389,409)(17,204,390,94)(18,404,391,314)(19,127,181,42)(20,123,182,247)(21,173,183,151)(22,130,184,45)(23,126,185,250)(24,176,186,145)(25,133,187,39)(26,120,188,244)(27,179,189,148)(37,153,131,175)(38,118,132,251)(40,147,134,178)(41,121,135,245)(43,150,128,172)(44,124,129,248)(46,357,73,68)(48,377,75,380)(49,360,76,71)(51,371,78,383)(52,354,79,65)(54,374,81,386)(55,347,82,253)(56,361,83,138)(57,157,84,169)(58,350,85,256)(59,364,86,141)(60,160,87,163)(61,344,88,259)(62,367,89,144)(63,154,90,166)(66,385,355,373)(69,379,358,376)(72,382,352,370)(92,408,202,229)(93,402,203,312)(95,411,205,232)(96,405,206,315)(98,414,199,226)(99,399,200,309)(100,292,424,265)(101,238,425,288)(102,341,426,418)(103,295,427,268)(104,241,428,282)(105,335,429,421)(106,289,430,262)(107,235,431,285)(108,338,432,415)(109,219,208,329)(110,299,209,272)(111,320,210,197)(112,222,211,332)(113,302,212,275)(114,323,213,191)(115,225,214,326)(116,305,215,278)(117,317,216,194)(119,146,252,177)(122,149,246,180)(125,152,249,174)(136,261,368,346)(137,155,369,167)(139,255,362,349)(140,158,363,170)(142,258,365,343)(143,161,366,164)(156,254,168,348)(159,257,171,351)(162,260,165,345)(190,223,322,333)(192,277,324,304)(193,217,316,327)(195,271,318,298)(196,220,319,330)(198,274,321,301)(218,279,328,306)(221,273,331,300)(224,276,325,303)(228,400,407,310)(231,403,410,313)(234,397,413,307)(236,291,286,264)(237,416,287,339)(239,294,280,267)(240,419,281,342)(242,297,283,270)(243,422,284,336)(263,423,290,337)(266,417,293,340)(269,420,296,334), (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) );

G=PermutationGroup([[(1,107,62,298,119,227),(2,108,63,299,120,228),(3,100,55,300,121,229),(4,101,56,301,122,230),(5,102,57,302,123,231),(6,103,58,303,124,232),(7,104,59,304,125,233),(8,105,60,305,126,234),(9,106,61,306,118,226),(10,353,282,141,324,174),(11,354,283,142,316,175),(12,355,284,143,317,176),(13,356,285,144,318,177),(14,357,286,136,319,178),(15,358,287,137,320,179),(16,359,288,138,321,180),(17,360,280,139,322,172),(18,352,281,140,323,173),(19,312,380,417,168,112),(20,313,381,418,169,113),(21,314,382,419,170,114),(22,315,383,420,171,115),(23,307,384,421,163,116),(24,308,385,422,164,117),(25,309,386,423,165,109),(26,310,387,415,166,110),(27,311,379,416,167,111),(28,428,86,277,249,412),(29,429,87,278,250,413),(30,430,88,279,251,414),(31,431,89,271,252,406),(32,432,90,272,244,407),(33,424,82,273,245,408),(34,425,83,274,246,409),(35,426,84,275,247,410),(36,427,85,276,248,411),(37,207,79,270,258,327),(38,199,80,262,259,328),(39,200,81,263,260,329),(40,201,73,264,261,330),(41,202,74,265,253,331),(42,203,75,266,254,332),(43,204,76,267,255,333),(44,205,77,268,256,325),(45,206,78,269,257,326),(46,291,346,220,134,91),(47,292,347,221,135,92),(48,293,348,222,127,93),(49,294,349,223,128,94),(50,295,350,224,129,95),(51,296,351,225,130,96),(52,297,343,217,131,97),(53,289,344,218,132,98),(54,290,345,219,133,99),(64,241,364,192,152,392),(65,242,365,193,153,393),(66,243,366,194,145,394),(67,235,367,195,146,395),(68,236,368,196,147,396),(69,237,369,197,148,388),(70,238,361,198,149,389),(71,239,362,190,150,390),(72,240,363,191,151,391),(154,209,188,400,375,338),(155,210,189,401,376,339),(156,211,181,402,377,340),(157,212,182,403,378,341),(158,213,183,404,370,342),(159,214,184,405,371,334),(160,215,185,397,372,335),(161,216,186,398,373,336),(162,208,187,399,374,337)], [(1,54,31,81),(2,68,32,357),(3,376,33,379),(4,48,34,75),(5,71,35,360),(6,370,36,382),(7,51,28,78),(8,65,29,354),(9,373,30,385),(10,405,392,315),(11,234,393,413),(12,199,394,98),(13,399,395,309),(14,228,396,407),(15,202,388,92),(16,402,389,312),(17,231,390,410),(18,205,391,95),(19,180,181,149),(20,128,182,43),(21,124,183,248),(22,174,184,152),(23,131,185,37),(24,118,186,251),(25,177,187,146),(26,134,188,40),(27,121,189,245),(38,145,132,176),(39,119,133,252),(41,148,135,179),(42,122,127,246),(44,151,129,173),(45,125,130,249),(46,375,73,387),(47,358,74,69),(49,378,76,381),(50,352,77,72),(52,372,79,384),(53,355,80,66),(55,155,82,167),(56,348,83,254),(57,362,84,139),(58,158,85,170),(59,351,86,257),(60,365,87,142),(61,161,88,164),(62,345,89,260),(63,368,90,136),(64,383,353,371),(67,386,356,374),(70,380,359,377),(91,400,201,310),(93,409,203,230),(94,403,204,313),(96,412,206,233),(97,397,207,307),(99,406,200,227),(100,339,424,416),(101,293,425,266),(102,239,426,280),(103,342,427,419),(104,296,428,269),(105,242,429,283),(106,336,430,422),(107,290,431,263),(108,236,432,286),(109,318,208,195),(110,220,209,330),(111,300,210,273),(112,321,211,198),(113,223,212,333),(114,303,213,276),(115,324,214,192),(116,217,215,327),(117,306,216,279),(120,147,244,178),(123,150,247,172),(126,153,250,175),(137,253,369,347),(138,156,361,168),(140,256,363,350),(141,159,364,171),(143,259,366,344),(144,162,367,165),(154,261,166,346),(157,255,169,349),(160,258,163,343),(190,275,322,302),(191,224,323,325),(193,278,316,305),(194,218,317,328),(196,272,319,299),(197,221,320,331),(219,271,329,298),(222,274,332,301),(225,277,326,304),(226,398,414,308),(229,401,408,311),(232,404,411,314),(235,423,285,337),(237,292,287,265),(238,417,288,340),(240,295,281,268),(241,420,282,334),(243,289,284,262),(264,415,291,338),(267,418,294,341),(270,421,297,335)], [(1,67,31,356),(2,375,32,387),(3,47,33,74),(4,70,34,359),(5,378,35,381),(6,50,36,77),(7,64,28,353),(8,372,29,384),(9,53,30,80),(10,233,392,412),(11,207,393,97),(12,398,394,308),(13,227,395,406),(14,201,396,91),(15,401,388,311),(16,230,389,409),(17,204,390,94),(18,404,391,314),(19,127,181,42),(20,123,182,247),(21,173,183,151),(22,130,184,45),(23,126,185,250),(24,176,186,145),(25,133,187,39),(26,120,188,244),(27,179,189,148),(37,153,131,175),(38,118,132,251),(40,147,134,178),(41,121,135,245),(43,150,128,172),(44,124,129,248),(46,357,73,68),(48,377,75,380),(49,360,76,71),(51,371,78,383),(52,354,79,65),(54,374,81,386),(55,347,82,253),(56,361,83,138),(57,157,84,169),(58,350,85,256),(59,364,86,141),(60,160,87,163),(61,344,88,259),(62,367,89,144),(63,154,90,166),(66,385,355,373),(69,379,358,376),(72,382,352,370),(92,408,202,229),(93,402,203,312),(95,411,205,232),(96,405,206,315),(98,414,199,226),(99,399,200,309),(100,292,424,265),(101,238,425,288),(102,341,426,418),(103,295,427,268),(104,241,428,282),(105,335,429,421),(106,289,430,262),(107,235,431,285),(108,338,432,415),(109,219,208,329),(110,299,209,272),(111,320,210,197),(112,222,211,332),(113,302,212,275),(114,323,213,191),(115,225,214,326),(116,305,215,278),(117,317,216,194),(119,146,252,177),(122,149,246,180),(125,152,249,174),(136,261,368,346),(137,155,369,167),(139,255,362,349),(140,158,363,170),(142,258,365,343),(143,161,366,164),(156,254,168,348),(159,257,171,351),(162,260,165,345),(190,223,322,333),(192,277,324,304),(193,217,316,327),(195,271,318,298),(196,220,319,330),(198,274,321,301),(218,279,328,306),(221,273,331,300),(224,276,325,303),(228,400,407,310),(231,403,410,313),(234,397,413,307),(236,291,286,264),(237,416,287,339),(239,294,280,267),(240,419,281,342),(242,297,283,270),(243,422,284,336),(263,423,290,337),(266,417,293,340),(269,420,296,334)], [(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)]])

126 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 6A ··· 6X 9A ··· 9R 12A ··· 12P 18A ··· 18BB order 1 2 2 2 3 ··· 3 4 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 1 1 ··· 1 6 6 1 ··· 1 4 ··· 4 6 ··· 6 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 type + + - + + image C1 C2 C3 C3 C6 C6 C9 C18 SL2(𝔽3) SL2(𝔽3) Q8⋊C9 C3×SL2(𝔽3) A4 C2×A4 C3.A4 C3×A4 C2×C3.A4 C6×A4 kernel C6×Q8⋊C9 C3×Q8⋊C9 C2×Q8⋊C9 Q8×C3×C6 Q8⋊C9 Q8×C32 C6×Q8 C3×Q8 C3×C6 C3×C6 C6 C6 C62 C3×C6 C2×C6 C2×C6 C6 C6 # reps 1 1 6 2 6 2 18 18 2 4 36 12 1 1 6 2 6 2

Matrix representation of C6×Q8⋊C9 in GL3(𝔽37) generated by

 36 0 0 0 10 0 0 0 10
,
 1 0 0 0 0 36 0 1 0
,
 1 0 0 0 10 26 0 26 27
,
 12 0 0 0 0 36 0 26 27
G:=sub<GL(3,GF(37))| [36,0,0,0,10,0,0,0,10],[1,0,0,0,0,1,0,36,0],[1,0,0,0,10,26,0,26,27],[12,0,0,0,0,26,0,36,27] >;

C6×Q8⋊C9 in GAP, Magma, Sage, TeX

C_6\times Q_8\rtimes C_9
% in TeX

G:=Group("C6xQ8:C9");
// GroupNames label

G:=SmallGroup(432,334);
// by ID

G=gap.SmallGroup(432,334);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,134,1901,172,3414,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=d^9=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

׿
×
𝔽