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## G = Q8×C53order 424 = 23·53

### Direct product of C53 and Q8

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

 Derived series C1 — C2 — Q8×C53
 Chief series C1 — C2 — C106 — C212 — Q8×C53
 Lower central C1 — C2 — Q8×C53
 Upper central C1 — C106 — Q8×C53

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

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

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

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

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

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