direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C53, C4.C106, C212.3C2, C106.7C22, C2.2(C2×C106), SmallGroup(424,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C53
G = < a,b,c | a53=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C53
►Regular action on 424 points
Generators in S424
(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)
(1 106 122 179)(2 54 123 180)(3 55 124 181)(4 56 125 182)(5 57 126 183)(6 58 127 184)(7 59 128 185)(8 60 129 186)(9 61 130 187)(10 62 131 188)(11 63 132 189)(12 64 133 190)(13 65 134 191)(14 66 135 192)(15 67 136 193)(16 68 137 194)(17 69 138 195)(18 70 139 196)(19 71 140 197)(20 72 141 198)(21 73 142 199)(22 74 143 200)(23 75 144 201)(24 76 145 202)(25 77 146 203)(26 78 147 204)(27 79 148 205)(28 80 149 206)(29 81 150 207)(30 82 151 208)(31 83 152 209)(32 84 153 210)(33 85 154 211)(34 86 155 212)(35 87 156 160)(36 88 157 161)(37 89 158 162)(38 90 159 163)(39 91 107 164)(40 92 108 165)(41 93 109 166)(42 94 110 167)(43 95 111 168)(44 96 112 169)(45 97 113 170)(46 98 114 171)(47 99 115 172)(48 100 116 173)(49 101 117 174)(50 102 118 175)(51 103 119 176)(52 104 120 177)(53 105 121 178)(213 292 382 360)(214 293 383 361)(215 294 384 362)(216 295 385 363)(217 296 386 364)(218 297 387 365)(219 298 388 366)(220 299 389 367)(221 300 390 368)(222 301 391 369)(223 302 392 370)(224 303 393 371)(225 304 394 319)(226 305 395 320)(227 306 396 321)(228 307 397 322)(229 308 398 323)(230 309 399 324)(231 310 400 325)(232 311 401 326)(233 312 402 327)(234 313 403 328)(235 314 404 329)(236 315 405 330)(237 316 406 331)(238 317 407 332)(239 318 408 333)(240 266 409 334)(241 267 410 335)(242 268 411 336)(243 269 412 337)(244 270 413 338)(245 271 414 339)(246 272 415 340)(247 273 416 341)(248 274 417 342)(249 275 418 343)(250 276 419 344)(251 277 420 345)(252 278 421 346)(253 279 422 347)(254 280 423 348)(255 281 424 349)(256 282 372 350)(257 283 373 351)(258 284 374 352)(259 285 375 353)(260 286 376 354)(261 287 377 355)(262 288 378 356)(263 289 379 357)(264 290 380 358)(265 291 381 359)
(1 317 122 332)(2 318 123 333)(3 266 124 334)(4 267 125 335)(5 268 126 336)(6 269 127 337)(7 270 128 338)(8 271 129 339)(9 272 130 340)(10 273 131 341)(11 274 132 342)(12 275 133 343)(13 276 134 344)(14 277 135 345)(15 278 136 346)(16 279 137 347)(17 280 138 348)(18 281 139 349)(19 282 140 350)(20 283 141 351)(21 284 142 352)(22 285 143 353)(23 286 144 354)(24 287 145 355)(25 288 146 356)(26 289 147 357)(27 290 148 358)(28 291 149 359)(29 292 150 360)(30 293 151 361)(31 294 152 362)(32 295 153 363)(33 296 154 364)(34 297 155 365)(35 298 156 366)(36 299 157 367)(37 300 158 368)(38 301 159 369)(39 302 107 370)(40 303 108 371)(41 304 109 319)(42 305 110 320)(43 306 111 321)(44 307 112 322)(45 308 113 323)(46 309 114 324)(47 310 115 325)(48 311 116 326)(49 312 117 327)(50 313 118 328)(51 314 119 329)(52 315 120 330)(53 316 121 331)(54 239 180 408)(55 240 181 409)(56 241 182 410)(57 242 183 411)(58 243 184 412)(59 244 185 413)(60 245 186 414)(61 246 187 415)(62 247 188 416)(63 248 189 417)(64 249 190 418)(65 250 191 419)(66 251 192 420)(67 252 193 421)(68 253 194 422)(69 254 195 423)(70 255 196 424)(71 256 197 372)(72 257 198 373)(73 258 199 374)(74 259 200 375)(75 260 201 376)(76 261 202 377)(77 262 203 378)(78 263 204 379)(79 264 205 380)(80 265 206 381)(81 213 207 382)(82 214 208 383)(83 215 209 384)(84 216 210 385)(85 217 211 386)(86 218 212 387)(87 219 160 388)(88 220 161 389)(89 221 162 390)(90 222 163 391)(91 223 164 392)(92 224 165 393)(93 225 166 394)(94 226 167 395)(95 227 168 396)(96 228 169 397)(97 229 170 398)(98 230 171 399)(99 231 172 400)(100 232 173 401)(101 233 174 402)(102 234 175 403)(103 235 176 404)(104 236 177 405)(105 237 178 406)(106 238 179 407)
G:=sub<Sym(424)| (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), (1,106,122,179)(2,54,123,180)(3,55,124,181)(4,56,125,182)(5,57,126,183)(6,58,127,184)(7,59,128,185)(8,60,129,186)(9,61,130,187)(10,62,131,188)(11,63,132,189)(12,64,133,190)(13,65,134,191)(14,66,135,192)(15,67,136,193)(16,68,137,194)(17,69,138,195)(18,70,139,196)(19,71,140,197)(20,72,141,198)(21,73,142,199)(22,74,143,200)(23,75,144,201)(24,76,145,202)(25,77,146,203)(26,78,147,204)(27,79,148,205)(28,80,149,206)(29,81,150,207)(30,82,151,208)(31,83,152,209)(32,84,153,210)(33,85,154,211)(34,86,155,212)(35,87,156,160)(36,88,157,161)(37,89,158,162)(38,90,159,163)(39,91,107,164)(40,92,108,165)(41,93,109,166)(42,94,110,167)(43,95,111,168)(44,96,112,169)(45,97,113,170)(46,98,114,171)(47,99,115,172)(48,100,116,173)(49,101,117,174)(50,102,118,175)(51,103,119,176)(52,104,120,177)(53,105,121,178)(213,292,382,360)(214,293,383,361)(215,294,384,362)(216,295,385,363)(217,296,386,364)(218,297,387,365)(219,298,388,366)(220,299,389,367)(221,300,390,368)(222,301,391,369)(223,302,392,370)(224,303,393,371)(225,304,394,319)(226,305,395,320)(227,306,396,321)(228,307,397,322)(229,308,398,323)(230,309,399,324)(231,310,400,325)(232,311,401,326)(233,312,402,327)(234,313,403,328)(235,314,404,329)(236,315,405,330)(237,316,406,331)(238,317,407,332)(239,318,408,333)(240,266,409,334)(241,267,410,335)(242,268,411,336)(243,269,412,337)(244,270,413,338)(245,271,414,339)(246,272,415,340)(247,273,416,341)(248,274,417,342)(249,275,418,343)(250,276,419,344)(251,277,420,345)(252,278,421,346)(253,279,422,347)(254,280,423,348)(255,281,424,349)(256,282,372,350)(257,283,373,351)(258,284,374,352)(259,285,375,353)(260,286,376,354)(261,287,377,355)(262,288,378,356)(263,289,379,357)(264,290,380,358)(265,291,381,359), (1,317,122,332)(2,318,123,333)(3,266,124,334)(4,267,125,335)(5,268,126,336)(6,269,127,337)(7,270,128,338)(8,271,129,339)(9,272,130,340)(10,273,131,341)(11,274,132,342)(12,275,133,343)(13,276,134,344)(14,277,135,345)(15,278,136,346)(16,279,137,347)(17,280,138,348)(18,281,139,349)(19,282,140,350)(20,283,141,351)(21,284,142,352)(22,285,143,353)(23,286,144,354)(24,287,145,355)(25,288,146,356)(26,289,147,357)(27,290,148,358)(28,291,149,359)(29,292,150,360)(30,293,151,361)(31,294,152,362)(32,295,153,363)(33,296,154,364)(34,297,155,365)(35,298,156,366)(36,299,157,367)(37,300,158,368)(38,301,159,369)(39,302,107,370)(40,303,108,371)(41,304,109,319)(42,305,110,320)(43,306,111,321)(44,307,112,322)(45,308,113,323)(46,309,114,324)(47,310,115,325)(48,311,116,326)(49,312,117,327)(50,313,118,328)(51,314,119,329)(52,315,120,330)(53,316,121,331)(54,239,180,408)(55,240,181,409)(56,241,182,410)(57,242,183,411)(58,243,184,412)(59,244,185,413)(60,245,186,414)(61,246,187,415)(62,247,188,416)(63,248,189,417)(64,249,190,418)(65,250,191,419)(66,251,192,420)(67,252,193,421)(68,253,194,422)(69,254,195,423)(70,255,196,424)(71,256,197,372)(72,257,198,373)(73,258,199,374)(74,259,200,375)(75,260,201,376)(76,261,202,377)(77,262,203,378)(78,263,204,379)(79,264,205,380)(80,265,206,381)(81,213,207,382)(82,214,208,383)(83,215,209,384)(84,216,210,385)(85,217,211,386)(86,218,212,387)(87,219,160,388)(88,220,161,389)(89,221,162,390)(90,222,163,391)(91,223,164,392)(92,224,165,393)(93,225,166,394)(94,226,167,395)(95,227,168,396)(96,228,169,397)(97,229,170,398)(98,230,171,399)(99,231,172,400)(100,232,173,401)(101,233,174,402)(102,234,175,403)(103,235,176,404)(104,236,177,405)(105,237,178,406)(106,238,179,407)>;
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), (1,106,122,179)(2,54,123,180)(3,55,124,181)(4,56,125,182)(5,57,126,183)(6,58,127,184)(7,59,128,185)(8,60,129,186)(9,61,130,187)(10,62,131,188)(11,63,132,189)(12,64,133,190)(13,65,134,191)(14,66,135,192)(15,67,136,193)(16,68,137,194)(17,69,138,195)(18,70,139,196)(19,71,140,197)(20,72,141,198)(21,73,142,199)(22,74,143,200)(23,75,144,201)(24,76,145,202)(25,77,146,203)(26,78,147,204)(27,79,148,205)(28,80,149,206)(29,81,150,207)(30,82,151,208)(31,83,152,209)(32,84,153,210)(33,85,154,211)(34,86,155,212)(35,87,156,160)(36,88,157,161)(37,89,158,162)(38,90,159,163)(39,91,107,164)(40,92,108,165)(41,93,109,166)(42,94,110,167)(43,95,111,168)(44,96,112,169)(45,97,113,170)(46,98,114,171)(47,99,115,172)(48,100,116,173)(49,101,117,174)(50,102,118,175)(51,103,119,176)(52,104,120,177)(53,105,121,178)(213,292,382,360)(214,293,383,361)(215,294,384,362)(216,295,385,363)(217,296,386,364)(218,297,387,365)(219,298,388,366)(220,299,389,367)(221,300,390,368)(222,301,391,369)(223,302,392,370)(224,303,393,371)(225,304,394,319)(226,305,395,320)(227,306,396,321)(228,307,397,322)(229,308,398,323)(230,309,399,324)(231,310,400,325)(232,311,401,326)(233,312,402,327)(234,313,403,328)(235,314,404,329)(236,315,405,330)(237,316,406,331)(238,317,407,332)(239,318,408,333)(240,266,409,334)(241,267,410,335)(242,268,411,336)(243,269,412,337)(244,270,413,338)(245,271,414,339)(246,272,415,340)(247,273,416,341)(248,274,417,342)(249,275,418,343)(250,276,419,344)(251,277,420,345)(252,278,421,346)(253,279,422,347)(254,280,423,348)(255,281,424,349)(256,282,372,350)(257,283,373,351)(258,284,374,352)(259,285,375,353)(260,286,376,354)(261,287,377,355)(262,288,378,356)(263,289,379,357)(264,290,380,358)(265,291,381,359), (1,317,122,332)(2,318,123,333)(3,266,124,334)(4,267,125,335)(5,268,126,336)(6,269,127,337)(7,270,128,338)(8,271,129,339)(9,272,130,340)(10,273,131,341)(11,274,132,342)(12,275,133,343)(13,276,134,344)(14,277,135,345)(15,278,136,346)(16,279,137,347)(17,280,138,348)(18,281,139,349)(19,282,140,350)(20,283,141,351)(21,284,142,352)(22,285,143,353)(23,286,144,354)(24,287,145,355)(25,288,146,356)(26,289,147,357)(27,290,148,358)(28,291,149,359)(29,292,150,360)(30,293,151,361)(31,294,152,362)(32,295,153,363)(33,296,154,364)(34,297,155,365)(35,298,156,366)(36,299,157,367)(37,300,158,368)(38,301,159,369)(39,302,107,370)(40,303,108,371)(41,304,109,319)(42,305,110,320)(43,306,111,321)(44,307,112,322)(45,308,113,323)(46,309,114,324)(47,310,115,325)(48,311,116,326)(49,312,117,327)(50,313,118,328)(51,314,119,329)(52,315,120,330)(53,316,121,331)(54,239,180,408)(55,240,181,409)(56,241,182,410)(57,242,183,411)(58,243,184,412)(59,244,185,413)(60,245,186,414)(61,246,187,415)(62,247,188,416)(63,248,189,417)(64,249,190,418)(65,250,191,419)(66,251,192,420)(67,252,193,421)(68,253,194,422)(69,254,195,423)(70,255,196,424)(71,256,197,372)(72,257,198,373)(73,258,199,374)(74,259,200,375)(75,260,201,376)(76,261,202,377)(77,262,203,378)(78,263,204,379)(79,264,205,380)(80,265,206,381)(81,213,207,382)(82,214,208,383)(83,215,209,384)(84,216,210,385)(85,217,211,386)(86,218,212,387)(87,219,160,388)(88,220,161,389)(89,221,162,390)(90,222,163,391)(91,223,164,392)(92,224,165,393)(93,225,166,394)(94,226,167,395)(95,227,168,396)(96,228,169,397)(97,229,170,398)(98,230,171,399)(99,231,172,400)(100,232,173,401)(101,233,174,402)(102,234,175,403)(103,235,176,404)(104,236,177,405)(105,237,178,406)(106,238,179,407) );
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)], [(1,106,122,179),(2,54,123,180),(3,55,124,181),(4,56,125,182),(5,57,126,183),(6,58,127,184),(7,59,128,185),(8,60,129,186),(9,61,130,187),(10,62,131,188),(11,63,132,189),(12,64,133,190),(13,65,134,191),(14,66,135,192),(15,67,136,193),(16,68,137,194),(17,69,138,195),(18,70,139,196),(19,71,140,197),(20,72,141,198),(21,73,142,199),(22,74,143,200),(23,75,144,201),(24,76,145,202),(25,77,146,203),(26,78,147,204),(27,79,148,205),(28,80,149,206),(29,81,150,207),(30,82,151,208),(31,83,152,209),(32,84,153,210),(33,85,154,211),(34,86,155,212),(35,87,156,160),(36,88,157,161),(37,89,158,162),(38,90,159,163),(39,91,107,164),(40,92,108,165),(41,93,109,166),(42,94,110,167),(43,95,111,168),(44,96,112,169),(45,97,113,170),(46,98,114,171),(47,99,115,172),(48,100,116,173),(49,101,117,174),(50,102,118,175),(51,103,119,176),(52,104,120,177),(53,105,121,178),(213,292,382,360),(214,293,383,361),(215,294,384,362),(216,295,385,363),(217,296,386,364),(218,297,387,365),(219,298,388,366),(220,299,389,367),(221,300,390,368),(222,301,391,369),(223,302,392,370),(224,303,393,371),(225,304,394,319),(226,305,395,320),(227,306,396,321),(228,307,397,322),(229,308,398,323),(230,309,399,324),(231,310,400,325),(232,311,401,326),(233,312,402,327),(234,313,403,328),(235,314,404,329),(236,315,405,330),(237,316,406,331),(238,317,407,332),(239,318,408,333),(240,266,409,334),(241,267,410,335),(242,268,411,336),(243,269,412,337),(244,270,413,338),(245,271,414,339),(246,272,415,340),(247,273,416,341),(248,274,417,342),(249,275,418,343),(250,276,419,344),(251,277,420,345),(252,278,421,346),(253,279,422,347),(254,280,423,348),(255,281,424,349),(256,282,372,350),(257,283,373,351),(258,284,374,352),(259,285,375,353),(260,286,376,354),(261,287,377,355),(262,288,378,356),(263,289,379,357),(264,290,380,358),(265,291,381,359)], [(1,317,122,332),(2,318,123,333),(3,266,124,334),(4,267,125,335),(5,268,126,336),(6,269,127,337),(7,270,128,338),(8,271,129,339),(9,272,130,340),(10,273,131,341),(11,274,132,342),(12,275,133,343),(13,276,134,344),(14,277,135,345),(15,278,136,346),(16,279,137,347),(17,280,138,348),(18,281,139,349),(19,282,140,350),(20,283,141,351),(21,284,142,352),(22,285,143,353),(23,286,144,354),(24,287,145,355),(25,288,146,356),(26,289,147,357),(27,290,148,358),(28,291,149,359),(29,292,150,360),(30,293,151,361),(31,294,152,362),(32,295,153,363),(33,296,154,364),(34,297,155,365),(35,298,156,366),(36,299,157,367),(37,300,158,368),(38,301,159,369),(39,302,107,370),(40,303,108,371),(41,304,109,319),(42,305,110,320),(43,306,111,321),(44,307,112,322),(45,308,113,323),(46,309,114,324),(47,310,115,325),(48,311,116,326),(49,312,117,327),(50,313,118,328),(51,314,119,329),(52,315,120,330),(53,316,121,331),(54,239,180,408),(55,240,181,409),(56,241,182,410),(57,242,183,411),(58,243,184,412),(59,244,185,413),(60,245,186,414),(61,246,187,415),(62,247,188,416),(63,248,189,417),(64,249,190,418),(65,250,191,419),(66,251,192,420),(67,252,193,421),(68,253,194,422),(69,254,195,423),(70,255,196,424),(71,256,197,372),(72,257,198,373),(73,258,199,374),(74,259,200,375),(75,260,201,376),(76,261,202,377),(77,262,203,378),(78,263,204,379),(79,264,205,380),(80,265,206,381),(81,213,207,382),(82,214,208,383),(83,215,209,384),(84,216,210,385),(85,217,211,386),(86,218,212,387),(87,219,160,388),(88,220,161,389),(89,221,162,390),(90,222,163,391),(91,223,164,392),(92,224,165,393),(93,225,166,394),(94,226,167,395),(95,227,168,396),(96,228,169,397),(97,229,170,398),(98,230,171,399),(99,231,172,400),(100,232,173,401),(101,233,174,402),(102,234,175,403),(103,235,176,404),(104,236,177,405),(105,237,178,406),(106,238,179,407)]])
265 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 53A | ··· | 53AZ | 106A | ··· | 106AZ | 212A | ··· | 212EZ |
| order | 1 | 2 | 4 | 4 | 4 | 53 | ··· | 53 | 106 | ··· | 106 | 212 | ··· | 212 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
265 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C53 | C106 | Q8 | Q8×C53 |
| kernel | Q8×C53 | C212 | Q8 | C4 | C53 | C1 |
| # reps | 1 | 3 | 52 | 156 | 1 | 52 |
Matrix representation of Q8×C53 ►in GL2(𝔽1061) generated by
| 451 | 0 |
| 0 | 451 |
| 925 | 1059 |
| 230 | 136 |
| 527 | 944 |
| 814 | 534 |
G:=sub<GL(2,GF(1061))| [451,0,0,451],[925,230,1059,136],[527,814,944,534] >;
Q8×C53 in GAP, Magma, Sage, TeX
Q_8\times C_{53} % in TeX
G:=Group("Q8xC53"); // GroupNames label
G:=SmallGroup(424,11);
// by ID
G=gap.SmallGroup(424,11);
# by ID
G:=PCGroup([4,-2,-2,-53,-2,848,1713,853]);
// Polycyclic
G:=Group<a,b,c|a^53=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export