metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C51⋊Q8, C34.7D6, C6.7D34, C3⋊1Dic34, C17⋊1Dic6, Dic3.D17, C102.7C22, Dic51.2C2, Dic17.1S3, C2.7(S3×D17), (C3×Dic17).1C2, (Dic3×C17).1C2, SmallGroup(408,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C51⋊Q8
G = < a,b,c | a51=b4=1, c2=b2, bab-1=a35, cac-1=a16, cbc-1=b-1 >
Smallest permutation representation of C51⋊Q8
►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)
(1 182 61 149)(2 166 62 133)(3 201 63 117)(4 185 64 152)(5 169 65 136)(6 204 66 120)(7 188 67 104)(8 172 68 139)(9 156 69 123)(10 191 70 107)(11 175 71 142)(12 159 72 126)(13 194 73 110)(14 178 74 145)(15 162 75 129)(16 197 76 113)(17 181 77 148)(18 165 78 132)(19 200 79 116)(20 184 80 151)(21 168 81 135)(22 203 82 119)(23 187 83 103)(24 171 84 138)(25 155 85 122)(26 190 86 106)(27 174 87 141)(28 158 88 125)(29 193 89 109)(30 177 90 144)(31 161 91 128)(32 196 92 112)(33 180 93 147)(34 164 94 131)(35 199 95 115)(36 183 96 150)(37 167 97 134)(38 202 98 118)(39 186 99 153)(40 170 100 137)(41 154 101 121)(42 189 102 105)(43 173 52 140)(44 157 53 124)(45 192 54 108)(46 176 55 143)(47 160 56 127)(48 195 57 111)(49 179 58 146)(50 163 59 130)(51 198 60 114)(205 313 276 405)(206 348 277 389)(207 332 278 373)(208 316 279 408)(209 351 280 392)(210 335 281 376)(211 319 282 360)(212 354 283 395)(213 338 284 379)(214 322 285 363)(215 357 286 398)(216 341 287 382)(217 325 288 366)(218 309 289 401)(219 344 290 385)(220 328 291 369)(221 312 292 404)(222 347 293 388)(223 331 294 372)(224 315 295 407)(225 350 296 391)(226 334 297 375)(227 318 298 359)(228 353 299 394)(229 337 300 378)(230 321 301 362)(231 356 302 397)(232 340 303 381)(233 324 304 365)(234 308 305 400)(235 343 306 384)(236 327 256 368)(237 311 257 403)(238 346 258 387)(239 330 259 371)(240 314 260 406)(241 349 261 390)(242 333 262 374)(243 317 263 358)(244 352 264 393)(245 336 265 377)(246 320 266 361)(247 355 267 396)(248 339 268 380)(249 323 269 364)(250 307 270 399)(251 342 271 383)(252 326 272 367)(253 310 273 402)(254 345 274 386)(255 329 275 370)
(1 303 61 232)(2 268 62 248)(3 284 63 213)(4 300 64 229)(5 265 65 245)(6 281 66 210)(7 297 67 226)(8 262 68 242)(9 278 69 207)(10 294 70 223)(11 259 71 239)(12 275 72 255)(13 291 73 220)(14 256 74 236)(15 272 75 252)(16 288 76 217)(17 304 77 233)(18 269 78 249)(19 285 79 214)(20 301 80 230)(21 266 81 246)(22 282 82 211)(23 298 83 227)(24 263 84 243)(25 279 85 208)(26 295 86 224)(27 260 87 240)(28 276 88 205)(29 292 89 221)(30 257 90 237)(31 273 91 253)(32 289 92 218)(33 305 93 234)(34 270 94 250)(35 286 95 215)(36 302 96 231)(37 267 97 247)(38 283 98 212)(39 299 99 228)(40 264 100 244)(41 280 101 209)(42 296 102 225)(43 261 52 241)(44 277 53 206)(45 293 54 222)(46 258 55 238)(47 274 56 254)(48 290 57 219)(49 306 58 235)(50 271 59 251)(51 287 60 216)(103 359 187 318)(104 375 188 334)(105 391 189 350)(106 407 190 315)(107 372 191 331)(108 388 192 347)(109 404 193 312)(110 369 194 328)(111 385 195 344)(112 401 196 309)(113 366 197 325)(114 382 198 341)(115 398 199 357)(116 363 200 322)(117 379 201 338)(118 395 202 354)(119 360 203 319)(120 376 204 335)(121 392 154 351)(122 408 155 316)(123 373 156 332)(124 389 157 348)(125 405 158 313)(126 370 159 329)(127 386 160 345)(128 402 161 310)(129 367 162 326)(130 383 163 342)(131 399 164 307)(132 364 165 323)(133 380 166 339)(134 396 167 355)(135 361 168 320)(136 377 169 336)(137 393 170 352)(138 358 171 317)(139 374 172 333)(140 390 173 349)(141 406 174 314)(142 371 175 330)(143 387 176 346)(144 403 177 311)(145 368 178 327)(146 384 179 343)(147 400 180 308)(148 365 181 324)(149 381 182 340)(150 397 183 356)(151 362 184 321)(152 378 185 337)(153 394 186 353)
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), (1,182,61,149)(2,166,62,133)(3,201,63,117)(4,185,64,152)(5,169,65,136)(6,204,66,120)(7,188,67,104)(8,172,68,139)(9,156,69,123)(10,191,70,107)(11,175,71,142)(12,159,72,126)(13,194,73,110)(14,178,74,145)(15,162,75,129)(16,197,76,113)(17,181,77,148)(18,165,78,132)(19,200,79,116)(20,184,80,151)(21,168,81,135)(22,203,82,119)(23,187,83,103)(24,171,84,138)(25,155,85,122)(26,190,86,106)(27,174,87,141)(28,158,88,125)(29,193,89,109)(30,177,90,144)(31,161,91,128)(32,196,92,112)(33,180,93,147)(34,164,94,131)(35,199,95,115)(36,183,96,150)(37,167,97,134)(38,202,98,118)(39,186,99,153)(40,170,100,137)(41,154,101,121)(42,189,102,105)(43,173,52,140)(44,157,53,124)(45,192,54,108)(46,176,55,143)(47,160,56,127)(48,195,57,111)(49,179,58,146)(50,163,59,130)(51,198,60,114)(205,313,276,405)(206,348,277,389)(207,332,278,373)(208,316,279,408)(209,351,280,392)(210,335,281,376)(211,319,282,360)(212,354,283,395)(213,338,284,379)(214,322,285,363)(215,357,286,398)(216,341,287,382)(217,325,288,366)(218,309,289,401)(219,344,290,385)(220,328,291,369)(221,312,292,404)(222,347,293,388)(223,331,294,372)(224,315,295,407)(225,350,296,391)(226,334,297,375)(227,318,298,359)(228,353,299,394)(229,337,300,378)(230,321,301,362)(231,356,302,397)(232,340,303,381)(233,324,304,365)(234,308,305,400)(235,343,306,384)(236,327,256,368)(237,311,257,403)(238,346,258,387)(239,330,259,371)(240,314,260,406)(241,349,261,390)(242,333,262,374)(243,317,263,358)(244,352,264,393)(245,336,265,377)(246,320,266,361)(247,355,267,396)(248,339,268,380)(249,323,269,364)(250,307,270,399)(251,342,271,383)(252,326,272,367)(253,310,273,402)(254,345,274,386)(255,329,275,370), (1,303,61,232)(2,268,62,248)(3,284,63,213)(4,300,64,229)(5,265,65,245)(6,281,66,210)(7,297,67,226)(8,262,68,242)(9,278,69,207)(10,294,70,223)(11,259,71,239)(12,275,72,255)(13,291,73,220)(14,256,74,236)(15,272,75,252)(16,288,76,217)(17,304,77,233)(18,269,78,249)(19,285,79,214)(20,301,80,230)(21,266,81,246)(22,282,82,211)(23,298,83,227)(24,263,84,243)(25,279,85,208)(26,295,86,224)(27,260,87,240)(28,276,88,205)(29,292,89,221)(30,257,90,237)(31,273,91,253)(32,289,92,218)(33,305,93,234)(34,270,94,250)(35,286,95,215)(36,302,96,231)(37,267,97,247)(38,283,98,212)(39,299,99,228)(40,264,100,244)(41,280,101,209)(42,296,102,225)(43,261,52,241)(44,277,53,206)(45,293,54,222)(46,258,55,238)(47,274,56,254)(48,290,57,219)(49,306,58,235)(50,271,59,251)(51,287,60,216)(103,359,187,318)(104,375,188,334)(105,391,189,350)(106,407,190,315)(107,372,191,331)(108,388,192,347)(109,404,193,312)(110,369,194,328)(111,385,195,344)(112,401,196,309)(113,366,197,325)(114,382,198,341)(115,398,199,357)(116,363,200,322)(117,379,201,338)(118,395,202,354)(119,360,203,319)(120,376,204,335)(121,392,154,351)(122,408,155,316)(123,373,156,332)(124,389,157,348)(125,405,158,313)(126,370,159,329)(127,386,160,345)(128,402,161,310)(129,367,162,326)(130,383,163,342)(131,399,164,307)(132,364,165,323)(133,380,166,339)(134,396,167,355)(135,361,168,320)(136,377,169,336)(137,393,170,352)(138,358,171,317)(139,374,172,333)(140,390,173,349)(141,406,174,314)(142,371,175,330)(143,387,176,346)(144,403,177,311)(145,368,178,327)(146,384,179,343)(147,400,180,308)(148,365,181,324)(149,381,182,340)(150,397,183,356)(151,362,184,321)(152,378,185,337)(153,394,186,353)>;
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), (1,182,61,149)(2,166,62,133)(3,201,63,117)(4,185,64,152)(5,169,65,136)(6,204,66,120)(7,188,67,104)(8,172,68,139)(9,156,69,123)(10,191,70,107)(11,175,71,142)(12,159,72,126)(13,194,73,110)(14,178,74,145)(15,162,75,129)(16,197,76,113)(17,181,77,148)(18,165,78,132)(19,200,79,116)(20,184,80,151)(21,168,81,135)(22,203,82,119)(23,187,83,103)(24,171,84,138)(25,155,85,122)(26,190,86,106)(27,174,87,141)(28,158,88,125)(29,193,89,109)(30,177,90,144)(31,161,91,128)(32,196,92,112)(33,180,93,147)(34,164,94,131)(35,199,95,115)(36,183,96,150)(37,167,97,134)(38,202,98,118)(39,186,99,153)(40,170,100,137)(41,154,101,121)(42,189,102,105)(43,173,52,140)(44,157,53,124)(45,192,54,108)(46,176,55,143)(47,160,56,127)(48,195,57,111)(49,179,58,146)(50,163,59,130)(51,198,60,114)(205,313,276,405)(206,348,277,389)(207,332,278,373)(208,316,279,408)(209,351,280,392)(210,335,281,376)(211,319,282,360)(212,354,283,395)(213,338,284,379)(214,322,285,363)(215,357,286,398)(216,341,287,382)(217,325,288,366)(218,309,289,401)(219,344,290,385)(220,328,291,369)(221,312,292,404)(222,347,293,388)(223,331,294,372)(224,315,295,407)(225,350,296,391)(226,334,297,375)(227,318,298,359)(228,353,299,394)(229,337,300,378)(230,321,301,362)(231,356,302,397)(232,340,303,381)(233,324,304,365)(234,308,305,400)(235,343,306,384)(236,327,256,368)(237,311,257,403)(238,346,258,387)(239,330,259,371)(240,314,260,406)(241,349,261,390)(242,333,262,374)(243,317,263,358)(244,352,264,393)(245,336,265,377)(246,320,266,361)(247,355,267,396)(248,339,268,380)(249,323,269,364)(250,307,270,399)(251,342,271,383)(252,326,272,367)(253,310,273,402)(254,345,274,386)(255,329,275,370), (1,303,61,232)(2,268,62,248)(3,284,63,213)(4,300,64,229)(5,265,65,245)(6,281,66,210)(7,297,67,226)(8,262,68,242)(9,278,69,207)(10,294,70,223)(11,259,71,239)(12,275,72,255)(13,291,73,220)(14,256,74,236)(15,272,75,252)(16,288,76,217)(17,304,77,233)(18,269,78,249)(19,285,79,214)(20,301,80,230)(21,266,81,246)(22,282,82,211)(23,298,83,227)(24,263,84,243)(25,279,85,208)(26,295,86,224)(27,260,87,240)(28,276,88,205)(29,292,89,221)(30,257,90,237)(31,273,91,253)(32,289,92,218)(33,305,93,234)(34,270,94,250)(35,286,95,215)(36,302,96,231)(37,267,97,247)(38,283,98,212)(39,299,99,228)(40,264,100,244)(41,280,101,209)(42,296,102,225)(43,261,52,241)(44,277,53,206)(45,293,54,222)(46,258,55,238)(47,274,56,254)(48,290,57,219)(49,306,58,235)(50,271,59,251)(51,287,60,216)(103,359,187,318)(104,375,188,334)(105,391,189,350)(106,407,190,315)(107,372,191,331)(108,388,192,347)(109,404,193,312)(110,369,194,328)(111,385,195,344)(112,401,196,309)(113,366,197,325)(114,382,198,341)(115,398,199,357)(116,363,200,322)(117,379,201,338)(118,395,202,354)(119,360,203,319)(120,376,204,335)(121,392,154,351)(122,408,155,316)(123,373,156,332)(124,389,157,348)(125,405,158,313)(126,370,159,329)(127,386,160,345)(128,402,161,310)(129,367,162,326)(130,383,163,342)(131,399,164,307)(132,364,165,323)(133,380,166,339)(134,396,167,355)(135,361,168,320)(136,377,169,336)(137,393,170,352)(138,358,171,317)(139,374,172,333)(140,390,173,349)(141,406,174,314)(142,371,175,330)(143,387,176,346)(144,403,177,311)(145,368,178,327)(146,384,179,343)(147,400,180,308)(148,365,181,324)(149,381,182,340)(150,397,183,356)(151,362,184,321)(152,378,185,337)(153,394,186,353) );
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)], [(1,182,61,149),(2,166,62,133),(3,201,63,117),(4,185,64,152),(5,169,65,136),(6,204,66,120),(7,188,67,104),(8,172,68,139),(9,156,69,123),(10,191,70,107),(11,175,71,142),(12,159,72,126),(13,194,73,110),(14,178,74,145),(15,162,75,129),(16,197,76,113),(17,181,77,148),(18,165,78,132),(19,200,79,116),(20,184,80,151),(21,168,81,135),(22,203,82,119),(23,187,83,103),(24,171,84,138),(25,155,85,122),(26,190,86,106),(27,174,87,141),(28,158,88,125),(29,193,89,109),(30,177,90,144),(31,161,91,128),(32,196,92,112),(33,180,93,147),(34,164,94,131),(35,199,95,115),(36,183,96,150),(37,167,97,134),(38,202,98,118),(39,186,99,153),(40,170,100,137),(41,154,101,121),(42,189,102,105),(43,173,52,140),(44,157,53,124),(45,192,54,108),(46,176,55,143),(47,160,56,127),(48,195,57,111),(49,179,58,146),(50,163,59,130),(51,198,60,114),(205,313,276,405),(206,348,277,389),(207,332,278,373),(208,316,279,408),(209,351,280,392),(210,335,281,376),(211,319,282,360),(212,354,283,395),(213,338,284,379),(214,322,285,363),(215,357,286,398),(216,341,287,382),(217,325,288,366),(218,309,289,401),(219,344,290,385),(220,328,291,369),(221,312,292,404),(222,347,293,388),(223,331,294,372),(224,315,295,407),(225,350,296,391),(226,334,297,375),(227,318,298,359),(228,353,299,394),(229,337,300,378),(230,321,301,362),(231,356,302,397),(232,340,303,381),(233,324,304,365),(234,308,305,400),(235,343,306,384),(236,327,256,368),(237,311,257,403),(238,346,258,387),(239,330,259,371),(240,314,260,406),(241,349,261,390),(242,333,262,374),(243,317,263,358),(244,352,264,393),(245,336,265,377),(246,320,266,361),(247,355,267,396),(248,339,268,380),(249,323,269,364),(250,307,270,399),(251,342,271,383),(252,326,272,367),(253,310,273,402),(254,345,274,386),(255,329,275,370)], [(1,303,61,232),(2,268,62,248),(3,284,63,213),(4,300,64,229),(5,265,65,245),(6,281,66,210),(7,297,67,226),(8,262,68,242),(9,278,69,207),(10,294,70,223),(11,259,71,239),(12,275,72,255),(13,291,73,220),(14,256,74,236),(15,272,75,252),(16,288,76,217),(17,304,77,233),(18,269,78,249),(19,285,79,214),(20,301,80,230),(21,266,81,246),(22,282,82,211),(23,298,83,227),(24,263,84,243),(25,279,85,208),(26,295,86,224),(27,260,87,240),(28,276,88,205),(29,292,89,221),(30,257,90,237),(31,273,91,253),(32,289,92,218),(33,305,93,234),(34,270,94,250),(35,286,95,215),(36,302,96,231),(37,267,97,247),(38,283,98,212),(39,299,99,228),(40,264,100,244),(41,280,101,209),(42,296,102,225),(43,261,52,241),(44,277,53,206),(45,293,54,222),(46,258,55,238),(47,274,56,254),(48,290,57,219),(49,306,58,235),(50,271,59,251),(51,287,60,216),(103,359,187,318),(104,375,188,334),(105,391,189,350),(106,407,190,315),(107,372,191,331),(108,388,192,347),(109,404,193,312),(110,369,194,328),(111,385,195,344),(112,401,196,309),(113,366,197,325),(114,382,198,341),(115,398,199,357),(116,363,200,322),(117,379,201,338),(118,395,202,354),(119,360,203,319),(120,376,204,335),(121,392,154,351),(122,408,155,316),(123,373,156,332),(124,389,157,348),(125,405,158,313),(126,370,159,329),(127,386,160,345),(128,402,161,310),(129,367,162,326),(130,383,163,342),(131,399,164,307),(132,364,165,323),(133,380,166,339),(134,396,167,355),(135,361,168,320),(136,377,169,336),(137,393,170,352),(138,358,171,317),(139,374,172,333),(140,390,173,349),(141,406,174,314),(142,371,175,330),(143,387,176,346),(144,403,177,311),(145,368,178,327),(146,384,179,343),(147,400,180,308),(148,365,181,324),(149,381,182,340),(150,397,183,356),(151,362,184,321),(152,378,185,337),(153,394,186,353)]])
57 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51H | 68A | ··· | 68P | 102A | ··· | 102H |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 2 | 6 | 34 | 102 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | Dic6 | D17 | D34 | Dic34 | S3×D17 | C51⋊Q8 |
| kernel | C51⋊Q8 | Dic3×C17 | C3×Dic17 | Dic51 | Dic17 | C51 | C34 | C17 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 8 | 8 |
Matrix representation of C51⋊Q8 ►in GL4(𝔽409) generated by
| 65 | 270 | 0 | 0 |
| 370 | 96 | 0 | 0 |
| 0 | 0 | 407 | 64 |
| 0 | 0 | 377 | 1 |
| 350 | 91 | 0 | 0 |
| 155 | 59 | 0 | 0 |
| 0 | 0 | 241 | 312 |
| 0 | 0 | 97 | 168 |
| 286 | 206 | 0 | 0 |
| 2 | 123 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(409))| [65,370,0,0,270,96,0,0,0,0,407,377,0,0,64,1],[350,155,0,0,91,59,0,0,0,0,241,97,0,0,312,168],[286,2,0,0,206,123,0,0,0,0,1,0,0,0,0,1] >;
C51⋊Q8 in GAP, Magma, Sage, TeX
C_{51}\rtimes Q_8 % in TeX
G:=Group("C51:Q8"); // GroupNames label
G:=SmallGroup(408,13);
// by ID
G=gap.SmallGroup(408,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,61,26,168,9604]);
// Polycyclic
G:=Group<a,b,c|a^51=b^4=1,c^2=b^2,b*a*b^-1=a^35,c*a*c^-1=a^16,c*b*c^-1=b^-1>;
// generators/relations
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