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## G = Q8×C51order 408 = 23·3·17

### Direct product of C51 and Q8

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

 Derived series C1 — C2 — Q8×C51
 Chief series C1 — C2 — C34 — C102 — C204 — Q8×C51
 Lower central C1 — C2 — Q8×C51
 Upper central C1 — C102 — Q8×C51

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

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

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

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

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

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