direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C54, C54.12C23, C108.20C22, C9.(C6×Q8), C3.(Q8×C18), C6.4(Q8×C9), (C2×C4).3C54, C4.4(C2×C54), (C6×Q8).2C9, C18.7(C3×Q8), (C2×C36).23C6, C36.44(C2×C6), (C2×C108).9C2, (Q8×C9).13C6, (C3×Q8).9C18, (Q8×C18).2C3, (C2×C12).11C18, C12.21(C2×C18), C22.4(C2×C54), C2.2(C22×C54), C6.12(C22×C18), C18.26(C22×C6), (C2×C54).15C22, (C2×C6).18(C2×C18), (C2×C18).35(C2×C6), SmallGroup(432,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C54
G = < a,b,c | a54=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 76, all normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, Q8, C9, C12, C2×C6, C2×Q8, C18, C18, C2×C12, C3×Q8, C27, C36, C2×C18, C6×Q8, C54, C54, C2×C36, Q8×C9, C108, C2×C54, Q8×C18, C2×C108, Q8×C27, Q8×C54
Quotients: C1, C2, C3, C22, C6, Q8, C23, C9, C2×C6, C2×Q8, C18, C3×Q8, C22×C6, C27, C2×C18, C6×Q8, C54, Q8×C9, C22×C18, C2×C54, Q8×C18, Q8×C27, C22×C54, Q8×C54
►Regular action on 432 points
Generators in S432
(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 97 117 264)(2 98 118 265)(3 99 119 266)(4 100 120 267)(5 101 121 268)(6 102 122 269)(7 103 123 270)(8 104 124 217)(9 105 125 218)(10 106 126 219)(11 107 127 220)(12 108 128 221)(13 55 129 222)(14 56 130 223)(15 57 131 224)(16 58 132 225)(17 59 133 226)(18 60 134 227)(19 61 135 228)(20 62 136 229)(21 63 137 230)(22 64 138 231)(23 65 139 232)(24 66 140 233)(25 67 141 234)(26 68 142 235)(27 69 143 236)(28 70 144 237)(29 71 145 238)(30 72 146 239)(31 73 147 240)(32 74 148 241)(33 75 149 242)(34 76 150 243)(35 77 151 244)(36 78 152 245)(37 79 153 246)(38 80 154 247)(39 81 155 248)(40 82 156 249)(41 83 157 250)(42 84 158 251)(43 85 159 252)(44 86 160 253)(45 87 161 254)(46 88 162 255)(47 89 109 256)(48 90 110 257)(49 91 111 258)(50 92 112 259)(51 93 113 260)(52 94 114 261)(53 95 115 262)(54 96 116 263)(163 301 384 375)(164 302 385 376)(165 303 386 377)(166 304 387 378)(167 305 388 325)(168 306 389 326)(169 307 390 327)(170 308 391 328)(171 309 392 329)(172 310 393 330)(173 311 394 331)(174 312 395 332)(175 313 396 333)(176 314 397 334)(177 315 398 335)(178 316 399 336)(179 317 400 337)(180 318 401 338)(181 319 402 339)(182 320 403 340)(183 321 404 341)(184 322 405 342)(185 323 406 343)(186 324 407 344)(187 271 408 345)(188 272 409 346)(189 273 410 347)(190 274 411 348)(191 275 412 349)(192 276 413 350)(193 277 414 351)(194 278 415 352)(195 279 416 353)(196 280 417 354)(197 281 418 355)(198 282 419 356)(199 283 420 357)(200 284 421 358)(201 285 422 359)(202 286 423 360)(203 287 424 361)(204 288 425 362)(205 289 426 363)(206 290 427 364)(207 291 428 365)(208 292 429 366)(209 293 430 367)(210 294 431 368)(211 295 432 369)(212 296 379 370)(213 297 380 371)(214 298 381 372)(215 299 382 373)(216 300 383 374)
(1 287 117 361)(2 288 118 362)(3 289 119 363)(4 290 120 364)(5 291 121 365)(6 292 122 366)(7 293 123 367)(8 294 124 368)(9 295 125 369)(10 296 126 370)(11 297 127 371)(12 298 128 372)(13 299 129 373)(14 300 130 374)(15 301 131 375)(16 302 132 376)(17 303 133 377)(18 304 134 378)(19 305 135 325)(20 306 136 326)(21 307 137 327)(22 308 138 328)(23 309 139 329)(24 310 140 330)(25 311 141 331)(26 312 142 332)(27 313 143 333)(28 314 144 334)(29 315 145 335)(30 316 146 336)(31 317 147 337)(32 318 148 338)(33 319 149 339)(34 320 150 340)(35 321 151 341)(36 322 152 342)(37 323 153 343)(38 324 154 344)(39 271 155 345)(40 272 156 346)(41 273 157 347)(42 274 158 348)(43 275 159 349)(44 276 160 350)(45 277 161 351)(46 278 162 352)(47 279 109 353)(48 280 110 354)(49 281 111 355)(50 282 112 356)(51 283 113 357)(52 284 114 358)(53 285 115 359)(54 286 116 360)(55 215 222 382)(56 216 223 383)(57 163 224 384)(58 164 225 385)(59 165 226 386)(60 166 227 387)(61 167 228 388)(62 168 229 389)(63 169 230 390)(64 170 231 391)(65 171 232 392)(66 172 233 393)(67 173 234 394)(68 174 235 395)(69 175 236 396)(70 176 237 397)(71 177 238 398)(72 178 239 399)(73 179 240 400)(74 180 241 401)(75 181 242 402)(76 182 243 403)(77 183 244 404)(78 184 245 405)(79 185 246 406)(80 186 247 407)(81 187 248 408)(82 188 249 409)(83 189 250 410)(84 190 251 411)(85 191 252 412)(86 192 253 413)(87 193 254 414)(88 194 255 415)(89 195 256 416)(90 196 257 417)(91 197 258 418)(92 198 259 419)(93 199 260 420)(94 200 261 421)(95 201 262 422)(96 202 263 423)(97 203 264 424)(98 204 265 425)(99 205 266 426)(100 206 267 427)(101 207 268 428)(102 208 269 429)(103 209 270 430)(104 210 217 431)(105 211 218 432)(106 212 219 379)(107 213 220 380)(108 214 221 381)
G:=sub<Sym(432)| (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), (1,97,117,264)(2,98,118,265)(3,99,119,266)(4,100,120,267)(5,101,121,268)(6,102,122,269)(7,103,123,270)(8,104,124,217)(9,105,125,218)(10,106,126,219)(11,107,127,220)(12,108,128,221)(13,55,129,222)(14,56,130,223)(15,57,131,224)(16,58,132,225)(17,59,133,226)(18,60,134,227)(19,61,135,228)(20,62,136,229)(21,63,137,230)(22,64,138,231)(23,65,139,232)(24,66,140,233)(25,67,141,234)(26,68,142,235)(27,69,143,236)(28,70,144,237)(29,71,145,238)(30,72,146,239)(31,73,147,240)(32,74,148,241)(33,75,149,242)(34,76,150,243)(35,77,151,244)(36,78,152,245)(37,79,153,246)(38,80,154,247)(39,81,155,248)(40,82,156,249)(41,83,157,250)(42,84,158,251)(43,85,159,252)(44,86,160,253)(45,87,161,254)(46,88,162,255)(47,89,109,256)(48,90,110,257)(49,91,111,258)(50,92,112,259)(51,93,113,260)(52,94,114,261)(53,95,115,262)(54,96,116,263)(163,301,384,375)(164,302,385,376)(165,303,386,377)(166,304,387,378)(167,305,388,325)(168,306,389,326)(169,307,390,327)(170,308,391,328)(171,309,392,329)(172,310,393,330)(173,311,394,331)(174,312,395,332)(175,313,396,333)(176,314,397,334)(177,315,398,335)(178,316,399,336)(179,317,400,337)(180,318,401,338)(181,319,402,339)(182,320,403,340)(183,321,404,341)(184,322,405,342)(185,323,406,343)(186,324,407,344)(187,271,408,345)(188,272,409,346)(189,273,410,347)(190,274,411,348)(191,275,412,349)(192,276,413,350)(193,277,414,351)(194,278,415,352)(195,279,416,353)(196,280,417,354)(197,281,418,355)(198,282,419,356)(199,283,420,357)(200,284,421,358)(201,285,422,359)(202,286,423,360)(203,287,424,361)(204,288,425,362)(205,289,426,363)(206,290,427,364)(207,291,428,365)(208,292,429,366)(209,293,430,367)(210,294,431,368)(211,295,432,369)(212,296,379,370)(213,297,380,371)(214,298,381,372)(215,299,382,373)(216,300,383,374), (1,287,117,361)(2,288,118,362)(3,289,119,363)(4,290,120,364)(5,291,121,365)(6,292,122,366)(7,293,123,367)(8,294,124,368)(9,295,125,369)(10,296,126,370)(11,297,127,371)(12,298,128,372)(13,299,129,373)(14,300,130,374)(15,301,131,375)(16,302,132,376)(17,303,133,377)(18,304,134,378)(19,305,135,325)(20,306,136,326)(21,307,137,327)(22,308,138,328)(23,309,139,329)(24,310,140,330)(25,311,141,331)(26,312,142,332)(27,313,143,333)(28,314,144,334)(29,315,145,335)(30,316,146,336)(31,317,147,337)(32,318,148,338)(33,319,149,339)(34,320,150,340)(35,321,151,341)(36,322,152,342)(37,323,153,343)(38,324,154,344)(39,271,155,345)(40,272,156,346)(41,273,157,347)(42,274,158,348)(43,275,159,349)(44,276,160,350)(45,277,161,351)(46,278,162,352)(47,279,109,353)(48,280,110,354)(49,281,111,355)(50,282,112,356)(51,283,113,357)(52,284,114,358)(53,285,115,359)(54,286,116,360)(55,215,222,382)(56,216,223,383)(57,163,224,384)(58,164,225,385)(59,165,226,386)(60,166,227,387)(61,167,228,388)(62,168,229,389)(63,169,230,390)(64,170,231,391)(65,171,232,392)(66,172,233,393)(67,173,234,394)(68,174,235,395)(69,175,236,396)(70,176,237,397)(71,177,238,398)(72,178,239,399)(73,179,240,400)(74,180,241,401)(75,181,242,402)(76,182,243,403)(77,183,244,404)(78,184,245,405)(79,185,246,406)(80,186,247,407)(81,187,248,408)(82,188,249,409)(83,189,250,410)(84,190,251,411)(85,191,252,412)(86,192,253,413)(87,193,254,414)(88,194,255,415)(89,195,256,416)(90,196,257,417)(91,197,258,418)(92,198,259,419)(93,199,260,420)(94,200,261,421)(95,201,262,422)(96,202,263,423)(97,203,264,424)(98,204,265,425)(99,205,266,426)(100,206,267,427)(101,207,268,428)(102,208,269,429)(103,209,270,430)(104,210,217,431)(105,211,218,432)(106,212,219,379)(107,213,220,380)(108,214,221,381)>;
G:=Group( (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), (1,97,117,264)(2,98,118,265)(3,99,119,266)(4,100,120,267)(5,101,121,268)(6,102,122,269)(7,103,123,270)(8,104,124,217)(9,105,125,218)(10,106,126,219)(11,107,127,220)(12,108,128,221)(13,55,129,222)(14,56,130,223)(15,57,131,224)(16,58,132,225)(17,59,133,226)(18,60,134,227)(19,61,135,228)(20,62,136,229)(21,63,137,230)(22,64,138,231)(23,65,139,232)(24,66,140,233)(25,67,141,234)(26,68,142,235)(27,69,143,236)(28,70,144,237)(29,71,145,238)(30,72,146,239)(31,73,147,240)(32,74,148,241)(33,75,149,242)(34,76,150,243)(35,77,151,244)(36,78,152,245)(37,79,153,246)(38,80,154,247)(39,81,155,248)(40,82,156,249)(41,83,157,250)(42,84,158,251)(43,85,159,252)(44,86,160,253)(45,87,161,254)(46,88,162,255)(47,89,109,256)(48,90,110,257)(49,91,111,258)(50,92,112,259)(51,93,113,260)(52,94,114,261)(53,95,115,262)(54,96,116,263)(163,301,384,375)(164,302,385,376)(165,303,386,377)(166,304,387,378)(167,305,388,325)(168,306,389,326)(169,307,390,327)(170,308,391,328)(171,309,392,329)(172,310,393,330)(173,311,394,331)(174,312,395,332)(175,313,396,333)(176,314,397,334)(177,315,398,335)(178,316,399,336)(179,317,400,337)(180,318,401,338)(181,319,402,339)(182,320,403,340)(183,321,404,341)(184,322,405,342)(185,323,406,343)(186,324,407,344)(187,271,408,345)(188,272,409,346)(189,273,410,347)(190,274,411,348)(191,275,412,349)(192,276,413,350)(193,277,414,351)(194,278,415,352)(195,279,416,353)(196,280,417,354)(197,281,418,355)(198,282,419,356)(199,283,420,357)(200,284,421,358)(201,285,422,359)(202,286,423,360)(203,287,424,361)(204,288,425,362)(205,289,426,363)(206,290,427,364)(207,291,428,365)(208,292,429,366)(209,293,430,367)(210,294,431,368)(211,295,432,369)(212,296,379,370)(213,297,380,371)(214,298,381,372)(215,299,382,373)(216,300,383,374), (1,287,117,361)(2,288,118,362)(3,289,119,363)(4,290,120,364)(5,291,121,365)(6,292,122,366)(7,293,123,367)(8,294,124,368)(9,295,125,369)(10,296,126,370)(11,297,127,371)(12,298,128,372)(13,299,129,373)(14,300,130,374)(15,301,131,375)(16,302,132,376)(17,303,133,377)(18,304,134,378)(19,305,135,325)(20,306,136,326)(21,307,137,327)(22,308,138,328)(23,309,139,329)(24,310,140,330)(25,311,141,331)(26,312,142,332)(27,313,143,333)(28,314,144,334)(29,315,145,335)(30,316,146,336)(31,317,147,337)(32,318,148,338)(33,319,149,339)(34,320,150,340)(35,321,151,341)(36,322,152,342)(37,323,153,343)(38,324,154,344)(39,271,155,345)(40,272,156,346)(41,273,157,347)(42,274,158,348)(43,275,159,349)(44,276,160,350)(45,277,161,351)(46,278,162,352)(47,279,109,353)(48,280,110,354)(49,281,111,355)(50,282,112,356)(51,283,113,357)(52,284,114,358)(53,285,115,359)(54,286,116,360)(55,215,222,382)(56,216,223,383)(57,163,224,384)(58,164,225,385)(59,165,226,386)(60,166,227,387)(61,167,228,388)(62,168,229,389)(63,169,230,390)(64,170,231,391)(65,171,232,392)(66,172,233,393)(67,173,234,394)(68,174,235,395)(69,175,236,396)(70,176,237,397)(71,177,238,398)(72,178,239,399)(73,179,240,400)(74,180,241,401)(75,181,242,402)(76,182,243,403)(77,183,244,404)(78,184,245,405)(79,185,246,406)(80,186,247,407)(81,187,248,408)(82,188,249,409)(83,189,250,410)(84,190,251,411)(85,191,252,412)(86,192,253,413)(87,193,254,414)(88,194,255,415)(89,195,256,416)(90,196,257,417)(91,197,258,418)(92,198,259,419)(93,199,260,420)(94,200,261,421)(95,201,262,422)(96,202,263,423)(97,203,264,424)(98,204,265,425)(99,205,266,426)(100,206,267,427)(101,207,268,428)(102,208,269,429)(103,209,270,430)(104,210,217,431)(105,211,218,432)(106,212,219,379)(107,213,220,380)(108,214,221,381) );
G=PermutationGroup([[(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)], [(1,97,117,264),(2,98,118,265),(3,99,119,266),(4,100,120,267),(5,101,121,268),(6,102,122,269),(7,103,123,270),(8,104,124,217),(9,105,125,218),(10,106,126,219),(11,107,127,220),(12,108,128,221),(13,55,129,222),(14,56,130,223),(15,57,131,224),(16,58,132,225),(17,59,133,226),(18,60,134,227),(19,61,135,228),(20,62,136,229),(21,63,137,230),(22,64,138,231),(23,65,139,232),(24,66,140,233),(25,67,141,234),(26,68,142,235),(27,69,143,236),(28,70,144,237),(29,71,145,238),(30,72,146,239),(31,73,147,240),(32,74,148,241),(33,75,149,242),(34,76,150,243),(35,77,151,244),(36,78,152,245),(37,79,153,246),(38,80,154,247),(39,81,155,248),(40,82,156,249),(41,83,157,250),(42,84,158,251),(43,85,159,252),(44,86,160,253),(45,87,161,254),(46,88,162,255),(47,89,109,256),(48,90,110,257),(49,91,111,258),(50,92,112,259),(51,93,113,260),(52,94,114,261),(53,95,115,262),(54,96,116,263),(163,301,384,375),(164,302,385,376),(165,303,386,377),(166,304,387,378),(167,305,388,325),(168,306,389,326),(169,307,390,327),(170,308,391,328),(171,309,392,329),(172,310,393,330),(173,311,394,331),(174,312,395,332),(175,313,396,333),(176,314,397,334),(177,315,398,335),(178,316,399,336),(179,317,400,337),(180,318,401,338),(181,319,402,339),(182,320,403,340),(183,321,404,341),(184,322,405,342),(185,323,406,343),(186,324,407,344),(187,271,408,345),(188,272,409,346),(189,273,410,347),(190,274,411,348),(191,275,412,349),(192,276,413,350),(193,277,414,351),(194,278,415,352),(195,279,416,353),(196,280,417,354),(197,281,418,355),(198,282,419,356),(199,283,420,357),(200,284,421,358),(201,285,422,359),(202,286,423,360),(203,287,424,361),(204,288,425,362),(205,289,426,363),(206,290,427,364),(207,291,428,365),(208,292,429,366),(209,293,430,367),(210,294,431,368),(211,295,432,369),(212,296,379,370),(213,297,380,371),(214,298,381,372),(215,299,382,373),(216,300,383,374)], [(1,287,117,361),(2,288,118,362),(3,289,119,363),(4,290,120,364),(5,291,121,365),(6,292,122,366),(7,293,123,367),(8,294,124,368),(9,295,125,369),(10,296,126,370),(11,297,127,371),(12,298,128,372),(13,299,129,373),(14,300,130,374),(15,301,131,375),(16,302,132,376),(17,303,133,377),(18,304,134,378),(19,305,135,325),(20,306,136,326),(21,307,137,327),(22,308,138,328),(23,309,139,329),(24,310,140,330),(25,311,141,331),(26,312,142,332),(27,313,143,333),(28,314,144,334),(29,315,145,335),(30,316,146,336),(31,317,147,337),(32,318,148,338),(33,319,149,339),(34,320,150,340),(35,321,151,341),(36,322,152,342),(37,323,153,343),(38,324,154,344),(39,271,155,345),(40,272,156,346),(41,273,157,347),(42,274,158,348),(43,275,159,349),(44,276,160,350),(45,277,161,351),(46,278,162,352),(47,279,109,353),(48,280,110,354),(49,281,111,355),(50,282,112,356),(51,283,113,357),(52,284,114,358),(53,285,115,359),(54,286,116,360),(55,215,222,382),(56,216,223,383),(57,163,224,384),(58,164,225,385),(59,165,226,386),(60,166,227,387),(61,167,228,388),(62,168,229,389),(63,169,230,390),(64,170,231,391),(65,171,232,392),(66,172,233,393),(67,173,234,394),(68,174,235,395),(69,175,236,396),(70,176,237,397),(71,177,238,398),(72,178,239,399),(73,179,240,400),(74,180,241,401),(75,181,242,402),(76,182,243,403),(77,183,244,404),(78,184,245,405),(79,185,246,406),(80,186,247,407),(81,187,248,408),(82,188,249,409),(83,189,250,410),(84,190,251,411),(85,191,252,412),(86,192,253,413),(87,193,254,414),(88,194,255,415),(89,195,256,416),(90,196,257,417),(91,197,258,418),(92,198,259,419),(93,199,260,420),(94,200,261,421),(95,201,262,422),(96,202,263,423),(97,203,264,424),(98,204,265,425),(99,205,266,426),(100,206,267,427),(101,207,268,428),(102,208,269,429),(103,209,270,430),(104,210,217,431),(105,211,218,432),(106,212,219,379),(107,213,220,380),(108,214,221,381)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12L | 18A | ··· | 18R | 27A | ··· | 27R | 36A | ··· | 36AJ | 54A | ··· | 54BB | 108A | ··· | 108DD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 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 | C6 | C6 | C9 | C18 | C18 | C27 | C54 | C54 | Q8 | C3×Q8 | Q8×C9 | Q8×C27 |
| kernel | Q8×C54 | C2×C108 | Q8×C27 | Q8×C18 | C2×C36 | Q8×C9 | C6×Q8 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C54 | C18 | C6 | C2 |
| # reps | 1 | 3 | 4 | 2 | 6 | 8 | 6 | 18 | 24 | 18 | 54 | 72 | 2 | 4 | 12 | 36 |
Matrix representation of Q8×C54 ►in GL3(𝔽109) generated by
| 108 | 0 | 0 |
| 0 | 97 | 0 |
| 0 | 0 | 97 |
| 108 | 0 | 0 |
| 0 | 0 | 108 |
| 0 | 1 | 0 |
| 108 | 0 | 0 |
| 0 | 66 | 49 |
| 0 | 49 | 43 |
G:=sub<GL(3,GF(109))| [108,0,0,0,97,0,0,0,97],[108,0,0,0,0,1,0,108,0],[108,0,0,0,66,49,0,49,43] >;
Q8×C54 in GAP, Magma, Sage, TeX
Q_8\times C_{54} % in TeX
G:=Group("Q8xC54"); // GroupNames label
G:=SmallGroup(432,55);
// by ID
G=gap.SmallGroup(432,55);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,365,176,192,166]);
// Polycyclic
G:=Group<a,b,c|a^54=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations