direct product, non-abelian, soluble
Aliases: C2×Q8⋊C27, Q8⋊C54, C18.SL2(𝔽3), (C2×Q8)⋊C27, C6.(Q8⋊C9), (C6×Q8).C9, (Q8×C18).C3, (C2×C18).6A4, (Q8×C9).5C6, C18.11(C2×A4), (C3×Q8).2C18, C9.(C2×SL2(𝔽3)), C22.2(C9.A4), C3.(C2×Q8⋊C9), C2.2(C2×C9.A4), C6.5(C2×C3.A4), (C2×C6).4(C3.A4), SmallGroup(432,41)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C2×Q8⋊C27 |
Generators and relations for C2×Q8⋊C27
G = < a,b,c,d | a2=b4=d27=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Smallest permutation representation of C2×Q8⋊C27
►Regular action on 432 points
Generators in S432
(1 131)(2 132)(3 133)(4 134)(5 135)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 130)(28 428)(29 429)(30 430)(31 431)(32 432)(33 406)(34 407)(35 408)(36 409)(37 410)(38 411)(39 412)(40 413)(41 414)(42 415)(43 416)(44 417)(45 418)(46 419)(47 420)(48 421)(49 422)(50 423)(51 424)(52 425)(53 426)(54 427)(55 367)(56 368)(57 369)(58 370)(59 371)(60 372)(61 373)(62 374)(63 375)(64 376)(65 377)(66 378)(67 352)(68 353)(69 354)(70 355)(71 356)(72 357)(73 358)(74 359)(75 360)(76 361)(77 362)(78 363)(79 364)(80 365)(81 366)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 136)(105 137)(106 138)(107 139)(108 140)(163 325)(164 326)(165 327)(166 328)(167 329)(168 330)(169 331)(170 332)(171 333)(172 334)(173 335)(174 336)(175 337)(176 338)(177 339)(178 340)(179 341)(180 342)(181 343)(182 344)(183 345)(184 346)(185 347)(186 348)(187 349)(188 350)(189 351)(190 231)(191 232)(192 233)(193 234)(194 235)(195 236)(196 237)(197 238)(198 239)(199 240)(200 241)(201 242)(202 243)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)(211 225)(212 226)(213 227)(214 228)(215 229)(216 230)(244 316)(245 317)(246 318)(247 319)(248 320)(249 321)(250 322)(251 323)(252 324)(253 298)(254 299)(255 300)(256 301)(257 302)(258 303)(259 304)(260 305)(261 306)(262 307)(263 308)(264 309)(265 310)(266 311)(267 312)(268 313)(269 314)(270 315)(271 400)(272 401)(273 402)(274 403)(275 404)(276 405)(277 379)(278 380)(279 381)(280 382)(281 383)(282 384)(283 385)(284 386)(285 387)(286 388)(287 389)(288 390)(289 391)(290 392)(291 393)(292 394)(293 395)(294 396)(295 397)(296 398)(297 399)
(1 86 223 294)(2 343 224 54)(3 249 225 61)(4 89 226 297)(5 346 227 30)(6 252 228 64)(7 92 229 273)(8 349 230 33)(9 255 231 67)(10 95 232 276)(11 325 233 36)(12 258 234 70)(13 98 235 279)(14 328 236 39)(15 261 237 73)(16 101 238 282)(17 331 239 42)(18 264 240 76)(19 104 241 285)(20 334 242 45)(21 267 243 79)(22 107 217 288)(23 337 218 48)(24 270 219 55)(25 83 220 291)(26 340 221 51)(27 246 222 58)(28 296 344 88)(29 250 345 62)(31 272 347 91)(32 253 348 65)(34 275 350 94)(35 256 351 68)(37 278 326 97)(38 259 327 71)(40 281 329 100)(41 262 330 74)(43 284 332 103)(44 265 333 77)(46 287 335 106)(47 268 336 80)(49 290 338 82)(50 244 339 56)(52 293 341 85)(53 247 342 59)(57 84 245 292)(60 87 248 295)(63 90 251 271)(66 93 254 274)(69 96 257 277)(72 99 260 280)(75 102 263 283)(78 105 266 286)(81 108 269 289)(109 324 214 376)(110 151 215 402)(111 187 216 406)(112 300 190 352)(113 154 191 405)(114 163 192 409)(115 303 193 355)(116 157 194 381)(117 166 195 412)(118 306 196 358)(119 160 197 384)(120 169 198 415)(121 309 199 361)(122 136 200 387)(123 172 201 418)(124 312 202 364)(125 139 203 390)(126 175 204 421)(127 315 205 367)(128 142 206 393)(129 178 207 424)(130 318 208 370)(131 145 209 396)(132 181 210 427)(133 321 211 373)(134 148 212 399)(135 184 213 430)(137 311 388 363)(138 419 389 173)(140 314 391 366)(141 422 392 176)(143 317 394 369)(144 425 395 179)(146 320 397 372)(147 428 398 182)(149 323 400 375)(150 431 401 185)(152 299 403 378)(153 407 404 188)(155 302 379 354)(156 410 380 164)(158 305 382 357)(159 413 383 167)(161 308 385 360)(162 416 386 170)(165 356 411 304)(168 359 414 307)(171 362 417 310)(174 365 420 313)(177 368 423 316)(180 371 426 319)(183 374 429 322)(186 377 432 298)(189 353 408 301)
(1 342 223 53)(2 248 224 60)(3 88 225 296)(4 345 226 29)(5 251 227 63)(6 91 228 272)(7 348 229 32)(8 254 230 66)(9 94 231 275)(10 351 232 35)(11 257 233 69)(12 97 234 278)(13 327 235 38)(14 260 236 72)(15 100 237 281)(16 330 238 41)(17 263 239 75)(18 103 240 284)(19 333 241 44)(20 266 242 78)(21 106 243 287)(22 336 217 47)(23 269 218 81)(24 82 219 290)(25 339 220 50)(26 245 221 57)(27 85 222 293)(28 249 344 61)(30 271 346 90)(31 252 347 64)(33 274 349 93)(34 255 350 67)(36 277 325 96)(37 258 326 70)(39 280 328 99)(40 261 329 73)(42 283 331 102)(43 264 332 76)(45 286 334 105)(46 267 335 79)(48 289 337 108)(49 270 338 55)(51 292 340 84)(52 246 341 58)(54 295 343 87)(56 83 244 291)(59 86 247 294)(62 89 250 297)(65 92 253 273)(68 95 256 276)(71 98 259 279)(74 101 262 282)(77 104 265 285)(80 107 268 288)(109 150 214 401)(110 186 215 432)(111 299 216 378)(112 153 190 404)(113 189 191 408)(114 302 192 354)(115 156 193 380)(116 165 194 411)(117 305 195 357)(118 159 196 383)(119 168 197 414)(120 308 198 360)(121 162 199 386)(122 171 200 417)(123 311 201 363)(124 138 202 389)(125 174 203 420)(126 314 204 366)(127 141 205 392)(128 177 206 423)(129 317 207 369)(130 144 208 395)(131 180 209 426)(132 320 210 372)(133 147 211 398)(134 183 212 429)(135 323 213 375)(136 310 387 362)(137 418 388 172)(139 313 390 365)(140 421 391 175)(142 316 393 368)(143 424 394 178)(145 319 396 371)(146 427 397 181)(148 322 399 374)(149 430 400 184)(151 298 402 377)(152 406 403 187)(154 301 405 353)(155 409 379 163)(157 304 381 356)(158 412 382 166)(160 307 384 359)(161 415 385 169)(164 355 410 303)(167 358 413 306)(170 361 416 309)(173 364 419 312)(176 367 422 315)(179 370 425 318)(182 373 428 321)(185 376 431 324)(188 352 407 300)
(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)| (1,131)(2,132)(3,133)(4,134)(5,135)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,130)(28,428)(29,429)(30,430)(31,431)(32,432)(33,406)(34,407)(35,408)(36,409)(37,410)(38,411)(39,412)(40,413)(41,414)(42,415)(43,416)(44,417)(45,418)(46,419)(47,420)(48,421)(49,422)(50,423)(51,424)(52,425)(53,426)(54,427)(55,367)(56,368)(57,369)(58,370)(59,371)(60,372)(61,373)(62,374)(63,375)(64,376)(65,377)(66,378)(67,352)(68,353)(69,354)(70,355)(71,356)(72,357)(73,358)(74,359)(75,360)(76,361)(77,362)(78,363)(79,364)(80,365)(81,366)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,136)(105,137)(106,138)(107,139)(108,140)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224)(211,225)(212,226)(213,227)(214,228)(215,229)(216,230)(244,316)(245,317)(246,318)(247,319)(248,320)(249,321)(250,322)(251,323)(252,324)(253,298)(254,299)(255,300)(256,301)(257,302)(258,303)(259,304)(260,305)(261,306)(262,307)(263,308)(264,309)(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(271,400)(272,401)(273,402)(274,403)(275,404)(276,405)(277,379)(278,380)(279,381)(280,382)(281,383)(282,384)(283,385)(284,386)(285,387)(286,388)(287,389)(288,390)(289,391)(290,392)(291,393)(292,394)(293,395)(294,396)(295,397)(296,398)(297,399), (1,86,223,294)(2,343,224,54)(3,249,225,61)(4,89,226,297)(5,346,227,30)(6,252,228,64)(7,92,229,273)(8,349,230,33)(9,255,231,67)(10,95,232,276)(11,325,233,36)(12,258,234,70)(13,98,235,279)(14,328,236,39)(15,261,237,73)(16,101,238,282)(17,331,239,42)(18,264,240,76)(19,104,241,285)(20,334,242,45)(21,267,243,79)(22,107,217,288)(23,337,218,48)(24,270,219,55)(25,83,220,291)(26,340,221,51)(27,246,222,58)(28,296,344,88)(29,250,345,62)(31,272,347,91)(32,253,348,65)(34,275,350,94)(35,256,351,68)(37,278,326,97)(38,259,327,71)(40,281,329,100)(41,262,330,74)(43,284,332,103)(44,265,333,77)(46,287,335,106)(47,268,336,80)(49,290,338,82)(50,244,339,56)(52,293,341,85)(53,247,342,59)(57,84,245,292)(60,87,248,295)(63,90,251,271)(66,93,254,274)(69,96,257,277)(72,99,260,280)(75,102,263,283)(78,105,266,286)(81,108,269,289)(109,324,214,376)(110,151,215,402)(111,187,216,406)(112,300,190,352)(113,154,191,405)(114,163,192,409)(115,303,193,355)(116,157,194,381)(117,166,195,412)(118,306,196,358)(119,160,197,384)(120,169,198,415)(121,309,199,361)(122,136,200,387)(123,172,201,418)(124,312,202,364)(125,139,203,390)(126,175,204,421)(127,315,205,367)(128,142,206,393)(129,178,207,424)(130,318,208,370)(131,145,209,396)(132,181,210,427)(133,321,211,373)(134,148,212,399)(135,184,213,430)(137,311,388,363)(138,419,389,173)(140,314,391,366)(141,422,392,176)(143,317,394,369)(144,425,395,179)(146,320,397,372)(147,428,398,182)(149,323,400,375)(150,431,401,185)(152,299,403,378)(153,407,404,188)(155,302,379,354)(156,410,380,164)(158,305,382,357)(159,413,383,167)(161,308,385,360)(162,416,386,170)(165,356,411,304)(168,359,414,307)(171,362,417,310)(174,365,420,313)(177,368,423,316)(180,371,426,319)(183,374,429,322)(186,377,432,298)(189,353,408,301), (1,342,223,53)(2,248,224,60)(3,88,225,296)(4,345,226,29)(5,251,227,63)(6,91,228,272)(7,348,229,32)(8,254,230,66)(9,94,231,275)(10,351,232,35)(11,257,233,69)(12,97,234,278)(13,327,235,38)(14,260,236,72)(15,100,237,281)(16,330,238,41)(17,263,239,75)(18,103,240,284)(19,333,241,44)(20,266,242,78)(21,106,243,287)(22,336,217,47)(23,269,218,81)(24,82,219,290)(25,339,220,50)(26,245,221,57)(27,85,222,293)(28,249,344,61)(30,271,346,90)(31,252,347,64)(33,274,349,93)(34,255,350,67)(36,277,325,96)(37,258,326,70)(39,280,328,99)(40,261,329,73)(42,283,331,102)(43,264,332,76)(45,286,334,105)(46,267,335,79)(48,289,337,108)(49,270,338,55)(51,292,340,84)(52,246,341,58)(54,295,343,87)(56,83,244,291)(59,86,247,294)(62,89,250,297)(65,92,253,273)(68,95,256,276)(71,98,259,279)(74,101,262,282)(77,104,265,285)(80,107,268,288)(109,150,214,401)(110,186,215,432)(111,299,216,378)(112,153,190,404)(113,189,191,408)(114,302,192,354)(115,156,193,380)(116,165,194,411)(117,305,195,357)(118,159,196,383)(119,168,197,414)(120,308,198,360)(121,162,199,386)(122,171,200,417)(123,311,201,363)(124,138,202,389)(125,174,203,420)(126,314,204,366)(127,141,205,392)(128,177,206,423)(129,317,207,369)(130,144,208,395)(131,180,209,426)(132,320,210,372)(133,147,211,398)(134,183,212,429)(135,323,213,375)(136,310,387,362)(137,418,388,172)(139,313,390,365)(140,421,391,175)(142,316,393,368)(143,424,394,178)(145,319,396,371)(146,427,397,181)(148,322,399,374)(149,430,400,184)(151,298,402,377)(152,406,403,187)(154,301,405,353)(155,409,379,163)(157,304,381,356)(158,412,382,166)(160,307,384,359)(161,415,385,169)(164,355,410,303)(167,358,413,306)(170,361,416,309)(173,364,419,312)(176,367,422,315)(179,370,425,318)(182,373,428,321)(185,376,431,324)(188,352,407,300), (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( (1,131)(2,132)(3,133)(4,134)(5,135)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,130)(28,428)(29,429)(30,430)(31,431)(32,432)(33,406)(34,407)(35,408)(36,409)(37,410)(38,411)(39,412)(40,413)(41,414)(42,415)(43,416)(44,417)(45,418)(46,419)(47,420)(48,421)(49,422)(50,423)(51,424)(52,425)(53,426)(54,427)(55,367)(56,368)(57,369)(58,370)(59,371)(60,372)(61,373)(62,374)(63,375)(64,376)(65,377)(66,378)(67,352)(68,353)(69,354)(70,355)(71,356)(72,357)(73,358)(74,359)(75,360)(76,361)(77,362)(78,363)(79,364)(80,365)(81,366)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,136)(105,137)(106,138)(107,139)(108,140)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224)(211,225)(212,226)(213,227)(214,228)(215,229)(216,230)(244,316)(245,317)(246,318)(247,319)(248,320)(249,321)(250,322)(251,323)(252,324)(253,298)(254,299)(255,300)(256,301)(257,302)(258,303)(259,304)(260,305)(261,306)(262,307)(263,308)(264,309)(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(271,400)(272,401)(273,402)(274,403)(275,404)(276,405)(277,379)(278,380)(279,381)(280,382)(281,383)(282,384)(283,385)(284,386)(285,387)(286,388)(287,389)(288,390)(289,391)(290,392)(291,393)(292,394)(293,395)(294,396)(295,397)(296,398)(297,399), (1,86,223,294)(2,343,224,54)(3,249,225,61)(4,89,226,297)(5,346,227,30)(6,252,228,64)(7,92,229,273)(8,349,230,33)(9,255,231,67)(10,95,232,276)(11,325,233,36)(12,258,234,70)(13,98,235,279)(14,328,236,39)(15,261,237,73)(16,101,238,282)(17,331,239,42)(18,264,240,76)(19,104,241,285)(20,334,242,45)(21,267,243,79)(22,107,217,288)(23,337,218,48)(24,270,219,55)(25,83,220,291)(26,340,221,51)(27,246,222,58)(28,296,344,88)(29,250,345,62)(31,272,347,91)(32,253,348,65)(34,275,350,94)(35,256,351,68)(37,278,326,97)(38,259,327,71)(40,281,329,100)(41,262,330,74)(43,284,332,103)(44,265,333,77)(46,287,335,106)(47,268,336,80)(49,290,338,82)(50,244,339,56)(52,293,341,85)(53,247,342,59)(57,84,245,292)(60,87,248,295)(63,90,251,271)(66,93,254,274)(69,96,257,277)(72,99,260,280)(75,102,263,283)(78,105,266,286)(81,108,269,289)(109,324,214,376)(110,151,215,402)(111,187,216,406)(112,300,190,352)(113,154,191,405)(114,163,192,409)(115,303,193,355)(116,157,194,381)(117,166,195,412)(118,306,196,358)(119,160,197,384)(120,169,198,415)(121,309,199,361)(122,136,200,387)(123,172,201,418)(124,312,202,364)(125,139,203,390)(126,175,204,421)(127,315,205,367)(128,142,206,393)(129,178,207,424)(130,318,208,370)(131,145,209,396)(132,181,210,427)(133,321,211,373)(134,148,212,399)(135,184,213,430)(137,311,388,363)(138,419,389,173)(140,314,391,366)(141,422,392,176)(143,317,394,369)(144,425,395,179)(146,320,397,372)(147,428,398,182)(149,323,400,375)(150,431,401,185)(152,299,403,378)(153,407,404,188)(155,302,379,354)(156,410,380,164)(158,305,382,357)(159,413,383,167)(161,308,385,360)(162,416,386,170)(165,356,411,304)(168,359,414,307)(171,362,417,310)(174,365,420,313)(177,368,423,316)(180,371,426,319)(183,374,429,322)(186,377,432,298)(189,353,408,301), (1,342,223,53)(2,248,224,60)(3,88,225,296)(4,345,226,29)(5,251,227,63)(6,91,228,272)(7,348,229,32)(8,254,230,66)(9,94,231,275)(10,351,232,35)(11,257,233,69)(12,97,234,278)(13,327,235,38)(14,260,236,72)(15,100,237,281)(16,330,238,41)(17,263,239,75)(18,103,240,284)(19,333,241,44)(20,266,242,78)(21,106,243,287)(22,336,217,47)(23,269,218,81)(24,82,219,290)(25,339,220,50)(26,245,221,57)(27,85,222,293)(28,249,344,61)(30,271,346,90)(31,252,347,64)(33,274,349,93)(34,255,350,67)(36,277,325,96)(37,258,326,70)(39,280,328,99)(40,261,329,73)(42,283,331,102)(43,264,332,76)(45,286,334,105)(46,267,335,79)(48,289,337,108)(49,270,338,55)(51,292,340,84)(52,246,341,58)(54,295,343,87)(56,83,244,291)(59,86,247,294)(62,89,250,297)(65,92,253,273)(68,95,256,276)(71,98,259,279)(74,101,262,282)(77,104,265,285)(80,107,268,288)(109,150,214,401)(110,186,215,432)(111,299,216,378)(112,153,190,404)(113,189,191,408)(114,302,192,354)(115,156,193,380)(116,165,194,411)(117,305,195,357)(118,159,196,383)(119,168,197,414)(120,308,198,360)(121,162,199,386)(122,171,200,417)(123,311,201,363)(124,138,202,389)(125,174,203,420)(126,314,204,366)(127,141,205,392)(128,177,206,423)(129,317,207,369)(130,144,208,395)(131,180,209,426)(132,320,210,372)(133,147,211,398)(134,183,212,429)(135,323,213,375)(136,310,387,362)(137,418,388,172)(139,313,390,365)(140,421,391,175)(142,316,393,368)(143,424,394,178)(145,319,396,371)(146,427,397,181)(148,322,399,374)(149,430,400,184)(151,298,402,377)(152,406,403,187)(154,301,405,353)(155,409,379,163)(157,304,381,356)(158,412,382,166)(160,307,384,359)(161,415,385,169)(164,355,410,303)(167,358,413,306)(170,361,416,309)(173,364,419,312)(176,367,422,315)(179,370,425,318)(182,373,428,321)(185,376,431,324)(188,352,407,300), (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([[(1,131),(2,132),(3,133),(4,134),(5,135),(6,109),(7,110),(8,111),(9,112),(10,113),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,121),(19,122),(20,123),(21,124),(22,125),(23,126),(24,127),(25,128),(26,129),(27,130),(28,428),(29,429),(30,430),(31,431),(32,432),(33,406),(34,407),(35,408),(36,409),(37,410),(38,411),(39,412),(40,413),(41,414),(42,415),(43,416),(44,417),(45,418),(46,419),(47,420),(48,421),(49,422),(50,423),(51,424),(52,425),(53,426),(54,427),(55,367),(56,368),(57,369),(58,370),(59,371),(60,372),(61,373),(62,374),(63,375),(64,376),(65,377),(66,378),(67,352),(68,353),(69,354),(70,355),(71,356),(72,357),(73,358),(74,359),(75,360),(76,361),(77,362),(78,363),(79,364),(80,365),(81,366),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,136),(105,137),(106,138),(107,139),(108,140),(163,325),(164,326),(165,327),(166,328),(167,329),(168,330),(169,331),(170,332),(171,333),(172,334),(173,335),(174,336),(175,337),(176,338),(177,339),(178,340),(179,341),(180,342),(181,343),(182,344),(183,345),(184,346),(185,347),(186,348),(187,349),(188,350),(189,351),(190,231),(191,232),(192,233),(193,234),(194,235),(195,236),(196,237),(197,238),(198,239),(199,240),(200,241),(201,242),(202,243),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224),(211,225),(212,226),(213,227),(214,228),(215,229),(216,230),(244,316),(245,317),(246,318),(247,319),(248,320),(249,321),(250,322),(251,323),(252,324),(253,298),(254,299),(255,300),(256,301),(257,302),(258,303),(259,304),(260,305),(261,306),(262,307),(263,308),(264,309),(265,310),(266,311),(267,312),(268,313),(269,314),(270,315),(271,400),(272,401),(273,402),(274,403),(275,404),(276,405),(277,379),(278,380),(279,381),(280,382),(281,383),(282,384),(283,385),(284,386),(285,387),(286,388),(287,389),(288,390),(289,391),(290,392),(291,393),(292,394),(293,395),(294,396),(295,397),(296,398),(297,399)], [(1,86,223,294),(2,343,224,54),(3,249,225,61),(4,89,226,297),(5,346,227,30),(6,252,228,64),(7,92,229,273),(8,349,230,33),(9,255,231,67),(10,95,232,276),(11,325,233,36),(12,258,234,70),(13,98,235,279),(14,328,236,39),(15,261,237,73),(16,101,238,282),(17,331,239,42),(18,264,240,76),(19,104,241,285),(20,334,242,45),(21,267,243,79),(22,107,217,288),(23,337,218,48),(24,270,219,55),(25,83,220,291),(26,340,221,51),(27,246,222,58),(28,296,344,88),(29,250,345,62),(31,272,347,91),(32,253,348,65),(34,275,350,94),(35,256,351,68),(37,278,326,97),(38,259,327,71),(40,281,329,100),(41,262,330,74),(43,284,332,103),(44,265,333,77),(46,287,335,106),(47,268,336,80),(49,290,338,82),(50,244,339,56),(52,293,341,85),(53,247,342,59),(57,84,245,292),(60,87,248,295),(63,90,251,271),(66,93,254,274),(69,96,257,277),(72,99,260,280),(75,102,263,283),(78,105,266,286),(81,108,269,289),(109,324,214,376),(110,151,215,402),(111,187,216,406),(112,300,190,352),(113,154,191,405),(114,163,192,409),(115,303,193,355),(116,157,194,381),(117,166,195,412),(118,306,196,358),(119,160,197,384),(120,169,198,415),(121,309,199,361),(122,136,200,387),(123,172,201,418),(124,312,202,364),(125,139,203,390),(126,175,204,421),(127,315,205,367),(128,142,206,393),(129,178,207,424),(130,318,208,370),(131,145,209,396),(132,181,210,427),(133,321,211,373),(134,148,212,399),(135,184,213,430),(137,311,388,363),(138,419,389,173),(140,314,391,366),(141,422,392,176),(143,317,394,369),(144,425,395,179),(146,320,397,372),(147,428,398,182),(149,323,400,375),(150,431,401,185),(152,299,403,378),(153,407,404,188),(155,302,379,354),(156,410,380,164),(158,305,382,357),(159,413,383,167),(161,308,385,360),(162,416,386,170),(165,356,411,304),(168,359,414,307),(171,362,417,310),(174,365,420,313),(177,368,423,316),(180,371,426,319),(183,374,429,322),(186,377,432,298),(189,353,408,301)], [(1,342,223,53),(2,248,224,60),(3,88,225,296),(4,345,226,29),(5,251,227,63),(6,91,228,272),(7,348,229,32),(8,254,230,66),(9,94,231,275),(10,351,232,35),(11,257,233,69),(12,97,234,278),(13,327,235,38),(14,260,236,72),(15,100,237,281),(16,330,238,41),(17,263,239,75),(18,103,240,284),(19,333,241,44),(20,266,242,78),(21,106,243,287),(22,336,217,47),(23,269,218,81),(24,82,219,290),(25,339,220,50),(26,245,221,57),(27,85,222,293),(28,249,344,61),(30,271,346,90),(31,252,347,64),(33,274,349,93),(34,255,350,67),(36,277,325,96),(37,258,326,70),(39,280,328,99),(40,261,329,73),(42,283,331,102),(43,264,332,76),(45,286,334,105),(46,267,335,79),(48,289,337,108),(49,270,338,55),(51,292,340,84),(52,246,341,58),(54,295,343,87),(56,83,244,291),(59,86,247,294),(62,89,250,297),(65,92,253,273),(68,95,256,276),(71,98,259,279),(74,101,262,282),(77,104,265,285),(80,107,268,288),(109,150,214,401),(110,186,215,432),(111,299,216,378),(112,153,190,404),(113,189,191,408),(114,302,192,354),(115,156,193,380),(116,165,194,411),(117,305,195,357),(118,159,196,383),(119,168,197,414),(120,308,198,360),(121,162,199,386),(122,171,200,417),(123,311,201,363),(124,138,202,389),(125,174,203,420),(126,314,204,366),(127,141,205,392),(128,177,206,423),(129,317,207,369),(130,144,208,395),(131,180,209,426),(132,320,210,372),(133,147,211,398),(134,183,212,429),(135,323,213,375),(136,310,387,362),(137,418,388,172),(139,313,390,365),(140,421,391,175),(142,316,393,368),(143,424,394,178),(145,319,396,371),(146,427,397,181),(148,322,399,374),(149,430,400,184),(151,298,402,377),(152,406,403,187),(154,301,405,353),(155,409,379,163),(157,304,381,356),(158,412,382,166),(160,307,384,359),(161,415,385,169),(164,355,410,303),(167,358,413,306),(170,361,416,309),(173,364,419,312),(176,367,422,315),(179,370,425,318),(182,373,428,321),(185,376,431,324),(188,352,407,300)], [(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 | 3B | 4A | 4B | 6A | ··· | 6F | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18R | 27A | ··· | 27R | 36A | ··· | 36L | 54A | ··· | 54BB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 1 | ··· | 1 | 6 | 6 | 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 | C6 | C9 | C18 | C27 | C54 | SL2(𝔽3) | SL2(𝔽3) | Q8⋊C9 | Q8⋊C27 | A4 | C2×A4 | C3.A4 | C2×C3.A4 | C9.A4 | C2×C9.A4 |
| kernel | C2×Q8⋊C27 | Q8⋊C27 | Q8×C18 | Q8×C9 | C6×Q8 | C3×Q8 | C2×Q8 | Q8 | C18 | C18 | C6 | C2 | C2×C18 | C18 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 2 | 4 | 12 | 36 | 1 | 1 | 2 | 2 | 6 | 6 |
Matrix representation of C2×Q8⋊C27 ►in GL4(𝔽109) generated by
| 1 | 0 | 0 | 0 |
| 0 | 108 | 0 | 0 |
| 0 | 0 | 108 | 0 |
| 0 | 0 | 0 | 108 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 96 | 87 |
| 0 | 0 | 87 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 108 |
| 0 | 0 | 1 | 0 |
| 49 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 44 |
| 0 | 0 | 90 | 34 |
G:=sub<GL(4,GF(109))| [1,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[1,0,0,0,0,1,0,0,0,0,96,87,0,0,87,13],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,108,0],[49,0,0,0,0,1,0,0,0,0,12,90,0,0,44,34] >;
C2×Q8⋊C27 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes C_{27} % in TeX
G:=Group("C2xQ8:C27"); // GroupNames label
G:=SmallGroup(432,41);
// by ID
G=gap.SmallGroup(432,41);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,50,79,1901,172,3414,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^27=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
Export