direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C51, C4.C102, C68.3C6, C12.3C34, C204.7C2, C102.24C22, C6.7(C2×C34), C34.7(C2×C6), C2.2(C2×C102), SmallGroup(408,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C51
G = < a,b,c | a51=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C51
►Regular action on 408 points
Generators in S408
(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)
(1 105 191 74)(2 106 192 75)(3 107 193 76)(4 108 194 77)(5 109 195 78)(6 110 196 79)(7 111 197 80)(8 112 198 81)(9 113 199 82)(10 114 200 83)(11 115 201 84)(12 116 202 85)(13 117 203 86)(14 118 204 87)(15 119 154 88)(16 120 155 89)(17 121 156 90)(18 122 157 91)(19 123 158 92)(20 124 159 93)(21 125 160 94)(22 126 161 95)(23 127 162 96)(24 128 163 97)(25 129 164 98)(26 130 165 99)(27 131 166 100)(28 132 167 101)(29 133 168 102)(30 134 169 52)(31 135 170 53)(32 136 171 54)(33 137 172 55)(34 138 173 56)(35 139 174 57)(36 140 175 58)(37 141 176 59)(38 142 177 60)(39 143 178 61)(40 144 179 62)(41 145 180 63)(42 146 181 64)(43 147 182 65)(44 148 183 66)(45 149 184 67)(46 150 185 68)(47 151 186 69)(48 152 187 70)(49 153 188 71)(50 103 189 72)(51 104 190 73)(205 353 273 380)(206 354 274 381)(207 355 275 382)(208 356 276 383)(209 357 277 384)(210 307 278 385)(211 308 279 386)(212 309 280 387)(213 310 281 388)(214 311 282 389)(215 312 283 390)(216 313 284 391)(217 314 285 392)(218 315 286 393)(219 316 287 394)(220 317 288 395)(221 318 289 396)(222 319 290 397)(223 320 291 398)(224 321 292 399)(225 322 293 400)(226 323 294 401)(227 324 295 402)(228 325 296 403)(229 326 297 404)(230 327 298 405)(231 328 299 406)(232 329 300 407)(233 330 301 408)(234 331 302 358)(235 332 303 359)(236 333 304 360)(237 334 305 361)(238 335 306 362)(239 336 256 363)(240 337 257 364)(241 338 258 365)(242 339 259 366)(243 340 260 367)(244 341 261 368)(245 342 262 369)(246 343 263 370)(247 344 264 371)(248 345 265 372)(249 346 266 373)(250 347 267 374)(251 348 268 375)(252 349 269 376)(253 350 270 377)(254 351 271 378)(255 352 272 379)
(1 295 191 227)(2 296 192 228)(3 297 193 229)(4 298 194 230)(5 299 195 231)(6 300 196 232)(7 301 197 233)(8 302 198 234)(9 303 199 235)(10 304 200 236)(11 305 201 237)(12 306 202 238)(13 256 203 239)(14 257 204 240)(15 258 154 241)(16 259 155 242)(17 260 156 243)(18 261 157 244)(19 262 158 245)(20 263 159 246)(21 264 160 247)(22 265 161 248)(23 266 162 249)(24 267 163 250)(25 268 164 251)(26 269 165 252)(27 270 166 253)(28 271 167 254)(29 272 168 255)(30 273 169 205)(31 274 170 206)(32 275 171 207)(33 276 172 208)(34 277 173 209)(35 278 174 210)(36 279 175 211)(37 280 176 212)(38 281 177 213)(39 282 178 214)(40 283 179 215)(41 284 180 216)(42 285 181 217)(43 286 182 218)(44 287 183 219)(45 288 184 220)(46 289 185 221)(47 290 186 222)(48 291 187 223)(49 292 188 224)(50 293 189 225)(51 294 190 226)(52 380 134 353)(53 381 135 354)(54 382 136 355)(55 383 137 356)(56 384 138 357)(57 385 139 307)(58 386 140 308)(59 387 141 309)(60 388 142 310)(61 389 143 311)(62 390 144 312)(63 391 145 313)(64 392 146 314)(65 393 147 315)(66 394 148 316)(67 395 149 317)(68 396 150 318)(69 397 151 319)(70 398 152 320)(71 399 153 321)(72 400 103 322)(73 401 104 323)(74 402 105 324)(75 403 106 325)(76 404 107 326)(77 405 108 327)(78 406 109 328)(79 407 110 329)(80 408 111 330)(81 358 112 331)(82 359 113 332)(83 360 114 333)(84 361 115 334)(85 362 116 335)(86 363 117 336)(87 364 118 337)(88 365 119 338)(89 366 120 339)(90 367 121 340)(91 368 122 341)(92 369 123 342)(93 370 124 343)(94 371 125 344)(95 372 126 345)(96 373 127 346)(97 374 128 347)(98 375 129 348)(99 376 130 349)(100 377 131 350)(101 378 132 351)(102 379 133 352)
G:=sub<Sym(408)| (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), (1,105,191,74)(2,106,192,75)(3,107,193,76)(4,108,194,77)(5,109,195,78)(6,110,196,79)(7,111,197,80)(8,112,198,81)(9,113,199,82)(10,114,200,83)(11,115,201,84)(12,116,202,85)(13,117,203,86)(14,118,204,87)(15,119,154,88)(16,120,155,89)(17,121,156,90)(18,122,157,91)(19,123,158,92)(20,124,159,93)(21,125,160,94)(22,126,161,95)(23,127,162,96)(24,128,163,97)(25,129,164,98)(26,130,165,99)(27,131,166,100)(28,132,167,101)(29,133,168,102)(30,134,169,52)(31,135,170,53)(32,136,171,54)(33,137,172,55)(34,138,173,56)(35,139,174,57)(36,140,175,58)(37,141,176,59)(38,142,177,60)(39,143,178,61)(40,144,179,62)(41,145,180,63)(42,146,181,64)(43,147,182,65)(44,148,183,66)(45,149,184,67)(46,150,185,68)(47,151,186,69)(48,152,187,70)(49,153,188,71)(50,103,189,72)(51,104,190,73)(205,353,273,380)(206,354,274,381)(207,355,275,382)(208,356,276,383)(209,357,277,384)(210,307,278,385)(211,308,279,386)(212,309,280,387)(213,310,281,388)(214,311,282,389)(215,312,283,390)(216,313,284,391)(217,314,285,392)(218,315,286,393)(219,316,287,394)(220,317,288,395)(221,318,289,396)(222,319,290,397)(223,320,291,398)(224,321,292,399)(225,322,293,400)(226,323,294,401)(227,324,295,402)(228,325,296,403)(229,326,297,404)(230,327,298,405)(231,328,299,406)(232,329,300,407)(233,330,301,408)(234,331,302,358)(235,332,303,359)(236,333,304,360)(237,334,305,361)(238,335,306,362)(239,336,256,363)(240,337,257,364)(241,338,258,365)(242,339,259,366)(243,340,260,367)(244,341,261,368)(245,342,262,369)(246,343,263,370)(247,344,264,371)(248,345,265,372)(249,346,266,373)(250,347,267,374)(251,348,268,375)(252,349,269,376)(253,350,270,377)(254,351,271,378)(255,352,272,379), (1,295,191,227)(2,296,192,228)(3,297,193,229)(4,298,194,230)(5,299,195,231)(6,300,196,232)(7,301,197,233)(8,302,198,234)(9,303,199,235)(10,304,200,236)(11,305,201,237)(12,306,202,238)(13,256,203,239)(14,257,204,240)(15,258,154,241)(16,259,155,242)(17,260,156,243)(18,261,157,244)(19,262,158,245)(20,263,159,246)(21,264,160,247)(22,265,161,248)(23,266,162,249)(24,267,163,250)(25,268,164,251)(26,269,165,252)(27,270,166,253)(28,271,167,254)(29,272,168,255)(30,273,169,205)(31,274,170,206)(32,275,171,207)(33,276,172,208)(34,277,173,209)(35,278,174,210)(36,279,175,211)(37,280,176,212)(38,281,177,213)(39,282,178,214)(40,283,179,215)(41,284,180,216)(42,285,181,217)(43,286,182,218)(44,287,183,219)(45,288,184,220)(46,289,185,221)(47,290,186,222)(48,291,187,223)(49,292,188,224)(50,293,189,225)(51,294,190,226)(52,380,134,353)(53,381,135,354)(54,382,136,355)(55,383,137,356)(56,384,138,357)(57,385,139,307)(58,386,140,308)(59,387,141,309)(60,388,142,310)(61,389,143,311)(62,390,144,312)(63,391,145,313)(64,392,146,314)(65,393,147,315)(66,394,148,316)(67,395,149,317)(68,396,150,318)(69,397,151,319)(70,398,152,320)(71,399,153,321)(72,400,103,322)(73,401,104,323)(74,402,105,324)(75,403,106,325)(76,404,107,326)(77,405,108,327)(78,406,109,328)(79,407,110,329)(80,408,111,330)(81,358,112,331)(82,359,113,332)(83,360,114,333)(84,361,115,334)(85,362,116,335)(86,363,117,336)(87,364,118,337)(88,365,119,338)(89,366,120,339)(90,367,121,340)(91,368,122,341)(92,369,123,342)(93,370,124,343)(94,371,125,344)(95,372,126,345)(96,373,127,346)(97,374,128,347)(98,375,129,348)(99,376,130,349)(100,377,131,350)(101,378,132,351)(102,379,133,352)>;
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), (1,105,191,74)(2,106,192,75)(3,107,193,76)(4,108,194,77)(5,109,195,78)(6,110,196,79)(7,111,197,80)(8,112,198,81)(9,113,199,82)(10,114,200,83)(11,115,201,84)(12,116,202,85)(13,117,203,86)(14,118,204,87)(15,119,154,88)(16,120,155,89)(17,121,156,90)(18,122,157,91)(19,123,158,92)(20,124,159,93)(21,125,160,94)(22,126,161,95)(23,127,162,96)(24,128,163,97)(25,129,164,98)(26,130,165,99)(27,131,166,100)(28,132,167,101)(29,133,168,102)(30,134,169,52)(31,135,170,53)(32,136,171,54)(33,137,172,55)(34,138,173,56)(35,139,174,57)(36,140,175,58)(37,141,176,59)(38,142,177,60)(39,143,178,61)(40,144,179,62)(41,145,180,63)(42,146,181,64)(43,147,182,65)(44,148,183,66)(45,149,184,67)(46,150,185,68)(47,151,186,69)(48,152,187,70)(49,153,188,71)(50,103,189,72)(51,104,190,73)(205,353,273,380)(206,354,274,381)(207,355,275,382)(208,356,276,383)(209,357,277,384)(210,307,278,385)(211,308,279,386)(212,309,280,387)(213,310,281,388)(214,311,282,389)(215,312,283,390)(216,313,284,391)(217,314,285,392)(218,315,286,393)(219,316,287,394)(220,317,288,395)(221,318,289,396)(222,319,290,397)(223,320,291,398)(224,321,292,399)(225,322,293,400)(226,323,294,401)(227,324,295,402)(228,325,296,403)(229,326,297,404)(230,327,298,405)(231,328,299,406)(232,329,300,407)(233,330,301,408)(234,331,302,358)(235,332,303,359)(236,333,304,360)(237,334,305,361)(238,335,306,362)(239,336,256,363)(240,337,257,364)(241,338,258,365)(242,339,259,366)(243,340,260,367)(244,341,261,368)(245,342,262,369)(246,343,263,370)(247,344,264,371)(248,345,265,372)(249,346,266,373)(250,347,267,374)(251,348,268,375)(252,349,269,376)(253,350,270,377)(254,351,271,378)(255,352,272,379), (1,295,191,227)(2,296,192,228)(3,297,193,229)(4,298,194,230)(5,299,195,231)(6,300,196,232)(7,301,197,233)(8,302,198,234)(9,303,199,235)(10,304,200,236)(11,305,201,237)(12,306,202,238)(13,256,203,239)(14,257,204,240)(15,258,154,241)(16,259,155,242)(17,260,156,243)(18,261,157,244)(19,262,158,245)(20,263,159,246)(21,264,160,247)(22,265,161,248)(23,266,162,249)(24,267,163,250)(25,268,164,251)(26,269,165,252)(27,270,166,253)(28,271,167,254)(29,272,168,255)(30,273,169,205)(31,274,170,206)(32,275,171,207)(33,276,172,208)(34,277,173,209)(35,278,174,210)(36,279,175,211)(37,280,176,212)(38,281,177,213)(39,282,178,214)(40,283,179,215)(41,284,180,216)(42,285,181,217)(43,286,182,218)(44,287,183,219)(45,288,184,220)(46,289,185,221)(47,290,186,222)(48,291,187,223)(49,292,188,224)(50,293,189,225)(51,294,190,226)(52,380,134,353)(53,381,135,354)(54,382,136,355)(55,383,137,356)(56,384,138,357)(57,385,139,307)(58,386,140,308)(59,387,141,309)(60,388,142,310)(61,389,143,311)(62,390,144,312)(63,391,145,313)(64,392,146,314)(65,393,147,315)(66,394,148,316)(67,395,149,317)(68,396,150,318)(69,397,151,319)(70,398,152,320)(71,399,153,321)(72,400,103,322)(73,401,104,323)(74,402,105,324)(75,403,106,325)(76,404,107,326)(77,405,108,327)(78,406,109,328)(79,407,110,329)(80,408,111,330)(81,358,112,331)(82,359,113,332)(83,360,114,333)(84,361,115,334)(85,362,116,335)(86,363,117,336)(87,364,118,337)(88,365,119,338)(89,366,120,339)(90,367,121,340)(91,368,122,341)(92,369,123,342)(93,370,124,343)(94,371,125,344)(95,372,126,345)(96,373,127,346)(97,374,128,347)(98,375,129,348)(99,376,130,349)(100,377,131,350)(101,378,132,351)(102,379,133,352) );
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)], [(1,105,191,74),(2,106,192,75),(3,107,193,76),(4,108,194,77),(5,109,195,78),(6,110,196,79),(7,111,197,80),(8,112,198,81),(9,113,199,82),(10,114,200,83),(11,115,201,84),(12,116,202,85),(13,117,203,86),(14,118,204,87),(15,119,154,88),(16,120,155,89),(17,121,156,90),(18,122,157,91),(19,123,158,92),(20,124,159,93),(21,125,160,94),(22,126,161,95),(23,127,162,96),(24,128,163,97),(25,129,164,98),(26,130,165,99),(27,131,166,100),(28,132,167,101),(29,133,168,102),(30,134,169,52),(31,135,170,53),(32,136,171,54),(33,137,172,55),(34,138,173,56),(35,139,174,57),(36,140,175,58),(37,141,176,59),(38,142,177,60),(39,143,178,61),(40,144,179,62),(41,145,180,63),(42,146,181,64),(43,147,182,65),(44,148,183,66),(45,149,184,67),(46,150,185,68),(47,151,186,69),(48,152,187,70),(49,153,188,71),(50,103,189,72),(51,104,190,73),(205,353,273,380),(206,354,274,381),(207,355,275,382),(208,356,276,383),(209,357,277,384),(210,307,278,385),(211,308,279,386),(212,309,280,387),(213,310,281,388),(214,311,282,389),(215,312,283,390),(216,313,284,391),(217,314,285,392),(218,315,286,393),(219,316,287,394),(220,317,288,395),(221,318,289,396),(222,319,290,397),(223,320,291,398),(224,321,292,399),(225,322,293,400),(226,323,294,401),(227,324,295,402),(228,325,296,403),(229,326,297,404),(230,327,298,405),(231,328,299,406),(232,329,300,407),(233,330,301,408),(234,331,302,358),(235,332,303,359),(236,333,304,360),(237,334,305,361),(238,335,306,362),(239,336,256,363),(240,337,257,364),(241,338,258,365),(242,339,259,366),(243,340,260,367),(244,341,261,368),(245,342,262,369),(246,343,263,370),(247,344,264,371),(248,345,265,372),(249,346,266,373),(250,347,267,374),(251,348,268,375),(252,349,269,376),(253,350,270,377),(254,351,271,378),(255,352,272,379)], [(1,295,191,227),(2,296,192,228),(3,297,193,229),(4,298,194,230),(5,299,195,231),(6,300,196,232),(7,301,197,233),(8,302,198,234),(9,303,199,235),(10,304,200,236),(11,305,201,237),(12,306,202,238),(13,256,203,239),(14,257,204,240),(15,258,154,241),(16,259,155,242),(17,260,156,243),(18,261,157,244),(19,262,158,245),(20,263,159,246),(21,264,160,247),(22,265,161,248),(23,266,162,249),(24,267,163,250),(25,268,164,251),(26,269,165,252),(27,270,166,253),(28,271,167,254),(29,272,168,255),(30,273,169,205),(31,274,170,206),(32,275,171,207),(33,276,172,208),(34,277,173,209),(35,278,174,210),(36,279,175,211),(37,280,176,212),(38,281,177,213),(39,282,178,214),(40,283,179,215),(41,284,180,216),(42,285,181,217),(43,286,182,218),(44,287,183,219),(45,288,184,220),(46,289,185,221),(47,290,186,222),(48,291,187,223),(49,292,188,224),(50,293,189,225),(51,294,190,226),(52,380,134,353),(53,381,135,354),(54,382,136,355),(55,383,137,356),(56,384,138,357),(57,385,139,307),(58,386,140,308),(59,387,141,309),(60,388,142,310),(61,389,143,311),(62,390,144,312),(63,391,145,313),(64,392,146,314),(65,393,147,315),(66,394,148,316),(67,395,149,317),(68,396,150,318),(69,397,151,319),(70,398,152,320),(71,399,153,321),(72,400,103,322),(73,401,104,323),(74,402,105,324),(75,403,106,325),(76,404,107,326),(77,405,108,327),(78,406,109,328),(79,407,110,329),(80,408,111,330),(81,358,112,331),(82,359,113,332),(83,360,114,333),(84,361,115,334),(85,362,116,335),(86,363,117,336),(87,364,118,337),(88,365,119,338),(89,366,120,339),(90,367,121,340),(91,368,122,341),(92,369,123,342),(93,370,124,343),(94,371,125,344),(95,372,126,345),(96,373,127,346),(97,374,128,347),(98,375,129,348),(99,376,130,349),(100,377,131,350),(101,378,132,351),(102,379,133,352)]])
255 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | ··· | 12F | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51AF | 68A | ··· | 68AV | 102A | ··· | 102AF | 204A | ··· | 204CR |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
255 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C3 | C6 | C17 | C34 | C51 | C102 | Q8 | C3×Q8 | Q8×C17 | Q8×C51 |
| kernel | Q8×C51 | C204 | Q8×C17 | C68 | C3×Q8 | C12 | Q8 | C4 | C51 | C17 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 16 | 48 | 32 | 96 | 1 | 2 | 16 | 32 |
Matrix representation of Q8×C51 ►in GL2(𝔽409) generated by
| 309 | 0 |
| 0 | 309 |
| 0 | 1 |
| 408 | 0 |
| 114 | 136 |
| 136 | 295 |
G:=sub<GL(2,GF(409))| [309,0,0,309],[0,408,1,0],[114,136,136,295] >;
Q8×C51 in GAP, Magma, Sage, TeX
Q_8\times C_{51} % in TeX
G:=Group("Q8xC51"); // GroupNames label
G:=SmallGroup(408,32);
// by ID
G=gap.SmallGroup(408,32);
# by ID
G:=PCGroup([5,-2,-2,-3,-17,-2,1020,2061,1026]);
// Polycyclic
G:=Group<a,b,c|a^51=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export