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G = Q8×C3×C18order 432 = 24·33

Direct product of C3×C18 and Q8

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C3×C18
 Chief series C1 — C3 — C6 — C3×C6 — C3×C18 — C3×C36 — Q8×C3×C9 — Q8×C3×C18
 Lower central C1 — C2 — Q8×C3×C18
 Upper central C1 — C6×C18 — Q8×C3×C18

Generators and relations for Q8×C3×C18
G = < a,b,c,d | a3=b18=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 190, all normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, Q8, C9, C32, C12, C2×C6, C2×C6, C2×Q8, C18, C3×C6, C3×C6, C2×C12, C3×Q8, C3×C9, C36, C2×C18, C3×C12, C62, C6×Q8, C6×Q8, C3×C18, C3×C18, C2×C36, Q8×C9, C6×C12, Q8×C32, C3×C36, C6×C18, Q8×C18, Q8×C3×C6, C6×C36, Q8×C3×C9, Q8×C3×C18
Quotients: C1, C2, C3, C22, C6, Q8, C23, C9, C32, C2×C6, C2×Q8, C18, C3×C6, C3×Q8, C22×C6, C3×C9, C2×C18, C62, C6×Q8, C3×C18, Q8×C9, C22×C18, Q8×C32, C2×C62, C6×C18, Q8×C18, Q8×C3×C6, Q8×C3×C9, C2×C6×C18, Q8×C3×C18

Smallest permutation representation of Q8×C3×C18
Regular action on 432 points
Generators in S432
(1 413 24)(2 414 25)(3 397 26)(4 398 27)(5 399 28)(6 400 29)(7 401 30)(8 402 31)(9 403 32)(10 404 33)(11 405 34)(12 406 35)(13 407 36)(14 408 19)(15 409 20)(16 410 21)(17 411 22)(18 412 23)(37 257 276)(38 258 277)(39 259 278)(40 260 279)(41 261 280)(42 262 281)(43 263 282)(44 264 283)(45 265 284)(46 266 285)(47 267 286)(48 268 287)(49 269 288)(50 270 271)(51 253 272)(52 254 273)(53 255 274)(54 256 275)(55 340 428)(56 341 429)(57 342 430)(58 325 431)(59 326 432)(60 327 415)(61 328 416)(62 329 417)(63 330 418)(64 331 419)(65 332 420)(66 333 421)(67 334 422)(68 335 423)(69 336 424)(70 337 425)(71 338 426)(72 339 427)(73 234 172)(74 217 173)(75 218 174)(76 219 175)(77 220 176)(78 221 177)(79 222 178)(80 223 179)(81 224 180)(82 225 163)(83 226 164)(84 227 165)(85 228 166)(86 229 167)(87 230 168)(88 231 169)(89 232 170)(90 233 171)(91 123 345)(92 124 346)(93 125 347)(94 126 348)(95 109 349)(96 110 350)(97 111 351)(98 112 352)(99 113 353)(100 114 354)(101 115 355)(102 116 356)(103 117 357)(104 118 358)(105 119 359)(106 120 360)(107 121 343)(108 122 344)(127 196 371)(128 197 372)(129 198 373)(130 181 374)(131 182 375)(132 183 376)(133 184 377)(134 185 378)(135 186 361)(136 187 362)(137 188 363)(138 189 364)(139 190 365)(140 191 366)(141 192 367)(142 193 368)(143 194 369)(144 195 370)(145 214 320)(146 215 321)(147 216 322)(148 199 323)(149 200 324)(150 201 307)(151 202 308)(152 203 309)(153 204 310)(154 205 311)(155 206 312)(156 207 313)(157 208 314)(158 209 315)(159 210 316)(160 211 317)(161 212 318)(162 213 319)(235 380 300)(236 381 301)(237 382 302)(238 383 303)(239 384 304)(240 385 305)(241 386 306)(242 387 289)(243 388 290)(244 389 291)(245 390 292)(246 391 293)(247 392 294)(248 393 295)(249 394 296)(250 395 297)(251 396 298)(252 379 299)
(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)
(1 248 123 175)(2 249 124 176)(3 250 125 177)(4 251 126 178)(5 252 109 179)(6 235 110 180)(7 236 111 163)(8 237 112 164)(9 238 113 165)(10 239 114 166)(11 240 115 167)(12 241 116 168)(13 242 117 169)(14 243 118 170)(15 244 119 171)(16 245 120 172)(17 246 121 173)(18 247 122 174)(19 290 104 232)(20 291 105 233)(21 292 106 234)(22 293 107 217)(23 294 108 218)(24 295 91 219)(25 296 92 220)(26 297 93 221)(27 298 94 222)(28 299 95 223)(29 300 96 224)(30 301 97 225)(31 302 98 226)(32 303 99 227)(33 304 100 228)(34 305 101 229)(35 306 102 230)(36 289 103 231)(37 137 147 423)(38 138 148 424)(39 139 149 425)(40 140 150 426)(41 141 151 427)(42 142 152 428)(43 143 153 429)(44 144 154 430)(45 127 155 431)(46 128 156 432)(47 129 157 415)(48 130 158 416)(49 131 159 417)(50 132 160 418)(51 133 161 419)(52 134 162 420)(53 135 145 421)(54 136 146 422)(55 262 193 203)(56 263 194 204)(57 264 195 205)(58 265 196 206)(59 266 197 207)(60 267 198 208)(61 268 181 209)(62 269 182 210)(63 270 183 211)(64 253 184 212)(65 254 185 213)(66 255 186 214)(67 256 187 215)(68 257 188 216)(69 258 189 199)(70 259 190 200)(71 260 191 201)(72 261 192 202)(73 410 390 360)(74 411 391 343)(75 412 392 344)(76 413 393 345)(77 414 394 346)(78 397 395 347)(79 398 396 348)(80 399 379 349)(81 400 380 350)(82 401 381 351)(83 402 382 352)(84 403 383 353)(85 404 384 354)(86 405 385 355)(87 406 386 356)(88 407 387 357)(89 408 388 358)(90 409 389 359)(271 376 317 330)(272 377 318 331)(273 378 319 332)(274 361 320 333)(275 362 321 334)(276 363 322 335)(277 364 323 336)(278 365 324 337)(279 366 307 338)(280 367 308 339)(281 368 309 340)(282 369 310 341)(283 370 311 342)(284 371 312 325)(285 372 313 326)(286 373 314 327)(287 374 315 328)(288 375 316 329)
(1 315 123 287)(2 316 124 288)(3 317 125 271)(4 318 126 272)(5 319 109 273)(6 320 110 274)(7 321 111 275)(8 322 112 276)(9 323 113 277)(10 324 114 278)(11 307 115 279)(12 308 116 280)(13 309 117 281)(14 310 118 282)(15 311 119 283)(16 312 120 284)(17 313 121 285)(18 314 122 286)(19 204 104 263)(20 205 105 264)(21 206 106 265)(22 207 107 266)(23 208 108 267)(24 209 91 268)(25 210 92 269)(26 211 93 270)(27 212 94 253)(28 213 95 254)(29 214 96 255)(30 215 97 256)(31 216 98 257)(32 199 99 258)(33 200 100 259)(34 201 101 260)(35 202 102 261)(36 203 103 262)(37 402 147 352)(38 403 148 353)(39 404 149 354)(40 405 150 355)(41 406 151 356)(42 407 152 357)(43 408 153 358)(44 409 154 359)(45 410 155 360)(46 411 156 343)(47 412 157 344)(48 413 158 345)(49 414 159 346)(50 397 160 347)(51 398 161 348)(52 399 162 349)(53 400 145 350)(54 401 146 351)(55 289 193 231)(56 290 194 232)(57 291 195 233)(58 292 196 234)(59 293 197 217)(60 294 198 218)(61 295 181 219)(62 296 182 220)(63 297 183 221)(64 298 184 222)(65 299 185 223)(66 300 186 224)(67 301 187 225)(68 302 188 226)(69 303 189 227)(70 304 190 228)(71 305 191 229)(72 306 192 230)(73 431 390 127)(74 432 391 128)(75 415 392 129)(76 416 393 130)(77 417 394 131)(78 418 395 132)(79 419 396 133)(80 420 379 134)(81 421 380 135)(82 422 381 136)(83 423 382 137)(84 424 383 138)(85 425 384 139)(86 426 385 140)(87 427 386 141)(88 428 387 142)(89 429 388 143)(90 430 389 144)(163 334 236 362)(164 335 237 363)(165 336 238 364)(166 337 239 365)(167 338 240 366)(168 339 241 367)(169 340 242 368)(170 341 243 369)(171 342 244 370)(172 325 245 371)(173 326 246 372)(174 327 247 373)(175 328 248 374)(176 329 249 375)(177 330 250 376)(178 331 251 377)(179 332 252 378)(180 333 235 361)

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

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

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

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 type + + + - image C1 C2 C2 C3 C3 C6 C6 C6 C6 C9 C18 C18 Q8 C3×Q8 C3×Q8 Q8×C9 kernel Q8×C3×C18 C6×C36 Q8×C3×C9 Q8×C18 Q8×C3×C6 C2×C36 Q8×C9 C6×C12 Q8×C32 C6×Q8 C2×C12 C3×Q8 C3×C18 C18 C3×C6 C6 # reps 1 3 4 6 2 18 24 6 8 18 54 72 2 12 4 36

Matrix representation of Q8×C3×C18 in GL4(𝔽37) generated by

 1 0 0 0 0 26 0 0 0 0 10 0 0 0 0 10
,
 36 0 0 0 0 10 0 0 0 0 21 0 0 0 0 21
,
 36 0 0 0 0 36 0 0 0 0 0 1 0 0 36 0
,
 1 0 0 0 0 1 0 0 0 0 3 29 0 0 29 34
G:=sub<GL(4,GF(37))| [1,0,0,0,0,26,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,10,0,0,0,0,21,0,0,0,0,21],[36,0,0,0,0,36,0,0,0,0,0,36,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,3,29,0,0,29,34] >;

Q8×C3×C18 in GAP, Magma, Sage, TeX

Q_8\times C_3\times C_{18}
% in TeX

G:=Group("Q8xC3xC18");
// GroupNames label

G:=SmallGroup(432,406);
// by ID

G=gap.SmallGroup(432,406);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,1037,512,528]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^18=c^4=1,d^2=c^2,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

׿
×
𝔽