direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic51, C6⋊Dic17, C102⋊3C4, C22.D51, C34⋊2Dic3, C6.11D34, C34.11D6, C2.2D102, C102.11C22, (C2×C34).S3, (C2×C6).D17, C51⋊10(C2×C4), C3⋊2(C2×Dic17), C17⋊3(C2×Dic3), (C2×C102).1C2, SmallGroup(408,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C2×Dic51 |
Generators and relations for C2×Dic51
G = < a,b,c | a2=b102=1, c2=b51, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic51
►Regular action on 408 points
Generators in S408
(1 204)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 121)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 129)(29 130)(30 131)(31 132)(32 133)(33 134)(34 135)(35 136)(36 137)(37 138)(38 139)(39 140)(40 141)(41 142)(42 143)(43 144)(44 145)(45 146)(46 147)(47 148)(48 149)(49 150)(50 151)(51 152)(52 153)(53 154)(54 155)(55 156)(56 157)(57 158)(58 159)(59 160)(60 161)(61 162)(62 163)(63 164)(64 165)(65 166)(66 167)(67 168)(68 169)(69 170)(70 171)(71 172)(72 173)(73 174)(74 175)(75 176)(76 177)(77 178)(78 179)(79 180)(80 181)(81 182)(82 183)(83 184)(84 185)(85 186)(86 187)(87 188)(88 189)(89 190)(90 191)(91 192)(92 193)(93 194)(94 195)(95 196)(96 197)(97 198)(98 199)(99 200)(100 201)(101 202)(102 203)(205 358)(206 359)(207 360)(208 361)(209 362)(210 363)(211 364)(212 365)(213 366)(214 367)(215 368)(216 369)(217 370)(218 371)(219 372)(220 373)(221 374)(222 375)(223 376)(224 377)(225 378)(226 379)(227 380)(228 381)(229 382)(230 383)(231 384)(232 385)(233 386)(234 387)(235 388)(236 389)(237 390)(238 391)(239 392)(240 393)(241 394)(242 395)(243 396)(244 397)(245 398)(246 399)(247 400)(248 401)(249 402)(250 403)(251 404)(252 405)(253 406)(254 407)(255 408)(256 307)(257 308)(258 309)(259 310)(260 311)(261 312)(262 313)(263 314)(264 315)(265 316)(266 317)(267 318)(268 319)(269 320)(270 321)(271 322)(272 323)(273 324)(274 325)(275 326)(276 327)(277 328)(278 329)(279 330)(280 331)(281 332)(282 333)(283 334)(284 335)(285 336)(286 337)(287 338)(288 339)(289 340)(290 341)(291 342)(292 343)(293 344)(294 345)(295 346)(296 347)(297 348)(298 349)(299 350)(300 351)(301 352)(302 353)(303 354)(304 355)(305 356)(306 357)
(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)
(1 375 52 324)(2 374 53 323)(3 373 54 322)(4 372 55 321)(5 371 56 320)(6 370 57 319)(7 369 58 318)(8 368 59 317)(9 367 60 316)(10 366 61 315)(11 365 62 314)(12 364 63 313)(13 363 64 312)(14 362 65 311)(15 361 66 310)(16 360 67 309)(17 359 68 308)(18 358 69 307)(19 357 70 408)(20 356 71 407)(21 355 72 406)(22 354 73 405)(23 353 74 404)(24 352 75 403)(25 351 76 402)(26 350 77 401)(27 349 78 400)(28 348 79 399)(29 347 80 398)(30 346 81 397)(31 345 82 396)(32 344 83 395)(33 343 84 394)(34 342 85 393)(35 341 86 392)(36 340 87 391)(37 339 88 390)(38 338 89 389)(39 337 90 388)(40 336 91 387)(41 335 92 386)(42 334 93 385)(43 333 94 384)(44 332 95 383)(45 331 96 382)(46 330 97 381)(47 329 98 380)(48 328 99 379)(49 327 100 378)(50 326 101 377)(51 325 102 376)(103 221 154 272)(104 220 155 271)(105 219 156 270)(106 218 157 269)(107 217 158 268)(108 216 159 267)(109 215 160 266)(110 214 161 265)(111 213 162 264)(112 212 163 263)(113 211 164 262)(114 210 165 261)(115 209 166 260)(116 208 167 259)(117 207 168 258)(118 206 169 257)(119 205 170 256)(120 306 171 255)(121 305 172 254)(122 304 173 253)(123 303 174 252)(124 302 175 251)(125 301 176 250)(126 300 177 249)(127 299 178 248)(128 298 179 247)(129 297 180 246)(130 296 181 245)(131 295 182 244)(132 294 183 243)(133 293 184 242)(134 292 185 241)(135 291 186 240)(136 290 187 239)(137 289 188 238)(138 288 189 237)(139 287 190 236)(140 286 191 235)(141 285 192 234)(142 284 193 233)(143 283 194 232)(144 282 195 231)(145 281 196 230)(146 280 197 229)(147 279 198 228)(148 278 199 227)(149 277 200 226)(150 276 201 225)(151 275 202 224)(152 274 203 223)(153 273 204 222)
G:=sub<Sym(408)| (1,204)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,129)(29,130)(30,131)(31,132)(32,133)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,145)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,171)(71,172)(72,173)(73,174)(74,175)(75,176)(76,177)(77,178)(78,179)(79,180)(80,181)(81,182)(82,183)(83,184)(84,185)(85,186)(86,187)(87,188)(88,189)(89,190)(90,191)(91,192)(92,193)(93,194)(94,195)(95,196)(96,197)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(205,358)(206,359)(207,360)(208,361)(209,362)(210,363)(211,364)(212,365)(213,366)(214,367)(215,368)(216,369)(217,370)(218,371)(219,372)(220,373)(221,374)(222,375)(223,376)(224,377)(225,378)(226,379)(227,380)(228,381)(229,382)(230,383)(231,384)(232,385)(233,386)(234,387)(235,388)(236,389)(237,390)(238,391)(239,392)(240,393)(241,394)(242,395)(243,396)(244,397)(245,398)(246,399)(247,400)(248,401)(249,402)(250,403)(251,404)(252,405)(253,406)(254,407)(255,408)(256,307)(257,308)(258,309)(259,310)(260,311)(261,312)(262,313)(263,314)(264,315)(265,316)(266,317)(267,318)(268,319)(269,320)(270,321)(271,322)(272,323)(273,324)(274,325)(275,326)(276,327)(277,328)(278,329)(279,330)(280,331)(281,332)(282,333)(283,334)(284,335)(285,336)(286,337)(287,338)(288,339)(289,340)(290,341)(291,342)(292,343)(293,344)(294,345)(295,346)(296,347)(297,348)(298,349)(299,350)(300,351)(301,352)(302,353)(303,354)(304,355)(305,356)(306,357), (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), (1,375,52,324)(2,374,53,323)(3,373,54,322)(4,372,55,321)(5,371,56,320)(6,370,57,319)(7,369,58,318)(8,368,59,317)(9,367,60,316)(10,366,61,315)(11,365,62,314)(12,364,63,313)(13,363,64,312)(14,362,65,311)(15,361,66,310)(16,360,67,309)(17,359,68,308)(18,358,69,307)(19,357,70,408)(20,356,71,407)(21,355,72,406)(22,354,73,405)(23,353,74,404)(24,352,75,403)(25,351,76,402)(26,350,77,401)(27,349,78,400)(28,348,79,399)(29,347,80,398)(30,346,81,397)(31,345,82,396)(32,344,83,395)(33,343,84,394)(34,342,85,393)(35,341,86,392)(36,340,87,391)(37,339,88,390)(38,338,89,389)(39,337,90,388)(40,336,91,387)(41,335,92,386)(42,334,93,385)(43,333,94,384)(44,332,95,383)(45,331,96,382)(46,330,97,381)(47,329,98,380)(48,328,99,379)(49,327,100,378)(50,326,101,377)(51,325,102,376)(103,221,154,272)(104,220,155,271)(105,219,156,270)(106,218,157,269)(107,217,158,268)(108,216,159,267)(109,215,160,266)(110,214,161,265)(111,213,162,264)(112,212,163,263)(113,211,164,262)(114,210,165,261)(115,209,166,260)(116,208,167,259)(117,207,168,258)(118,206,169,257)(119,205,170,256)(120,306,171,255)(121,305,172,254)(122,304,173,253)(123,303,174,252)(124,302,175,251)(125,301,176,250)(126,300,177,249)(127,299,178,248)(128,298,179,247)(129,297,180,246)(130,296,181,245)(131,295,182,244)(132,294,183,243)(133,293,184,242)(134,292,185,241)(135,291,186,240)(136,290,187,239)(137,289,188,238)(138,288,189,237)(139,287,190,236)(140,286,191,235)(141,285,192,234)(142,284,193,233)(143,283,194,232)(144,282,195,231)(145,281,196,230)(146,280,197,229)(147,279,198,228)(148,278,199,227)(149,277,200,226)(150,276,201,225)(151,275,202,224)(152,274,203,223)(153,273,204,222)>;
G:=Group( (1,204)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,129)(29,130)(30,131)(31,132)(32,133)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,145)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,171)(71,172)(72,173)(73,174)(74,175)(75,176)(76,177)(77,178)(78,179)(79,180)(80,181)(81,182)(82,183)(83,184)(84,185)(85,186)(86,187)(87,188)(88,189)(89,190)(90,191)(91,192)(92,193)(93,194)(94,195)(95,196)(96,197)(97,198)(98,199)(99,200)(100,201)(101,202)(102,203)(205,358)(206,359)(207,360)(208,361)(209,362)(210,363)(211,364)(212,365)(213,366)(214,367)(215,368)(216,369)(217,370)(218,371)(219,372)(220,373)(221,374)(222,375)(223,376)(224,377)(225,378)(226,379)(227,380)(228,381)(229,382)(230,383)(231,384)(232,385)(233,386)(234,387)(235,388)(236,389)(237,390)(238,391)(239,392)(240,393)(241,394)(242,395)(243,396)(244,397)(245,398)(246,399)(247,400)(248,401)(249,402)(250,403)(251,404)(252,405)(253,406)(254,407)(255,408)(256,307)(257,308)(258,309)(259,310)(260,311)(261,312)(262,313)(263,314)(264,315)(265,316)(266,317)(267,318)(268,319)(269,320)(270,321)(271,322)(272,323)(273,324)(274,325)(275,326)(276,327)(277,328)(278,329)(279,330)(280,331)(281,332)(282,333)(283,334)(284,335)(285,336)(286,337)(287,338)(288,339)(289,340)(290,341)(291,342)(292,343)(293,344)(294,345)(295,346)(296,347)(297,348)(298,349)(299,350)(300,351)(301,352)(302,353)(303,354)(304,355)(305,356)(306,357), (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), (1,375,52,324)(2,374,53,323)(3,373,54,322)(4,372,55,321)(5,371,56,320)(6,370,57,319)(7,369,58,318)(8,368,59,317)(9,367,60,316)(10,366,61,315)(11,365,62,314)(12,364,63,313)(13,363,64,312)(14,362,65,311)(15,361,66,310)(16,360,67,309)(17,359,68,308)(18,358,69,307)(19,357,70,408)(20,356,71,407)(21,355,72,406)(22,354,73,405)(23,353,74,404)(24,352,75,403)(25,351,76,402)(26,350,77,401)(27,349,78,400)(28,348,79,399)(29,347,80,398)(30,346,81,397)(31,345,82,396)(32,344,83,395)(33,343,84,394)(34,342,85,393)(35,341,86,392)(36,340,87,391)(37,339,88,390)(38,338,89,389)(39,337,90,388)(40,336,91,387)(41,335,92,386)(42,334,93,385)(43,333,94,384)(44,332,95,383)(45,331,96,382)(46,330,97,381)(47,329,98,380)(48,328,99,379)(49,327,100,378)(50,326,101,377)(51,325,102,376)(103,221,154,272)(104,220,155,271)(105,219,156,270)(106,218,157,269)(107,217,158,268)(108,216,159,267)(109,215,160,266)(110,214,161,265)(111,213,162,264)(112,212,163,263)(113,211,164,262)(114,210,165,261)(115,209,166,260)(116,208,167,259)(117,207,168,258)(118,206,169,257)(119,205,170,256)(120,306,171,255)(121,305,172,254)(122,304,173,253)(123,303,174,252)(124,302,175,251)(125,301,176,250)(126,300,177,249)(127,299,178,248)(128,298,179,247)(129,297,180,246)(130,296,181,245)(131,295,182,244)(132,294,183,243)(133,293,184,242)(134,292,185,241)(135,291,186,240)(136,290,187,239)(137,289,188,238)(138,288,189,237)(139,287,190,236)(140,286,191,235)(141,285,192,234)(142,284,193,233)(143,283,194,232)(144,282,195,231)(145,281,196,230)(146,280,197,229)(147,279,198,228)(148,278,199,227)(149,277,200,226)(150,276,201,225)(151,275,202,224)(152,274,203,223)(153,273,204,222) );
G=PermutationGroup([[(1,204),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,118),(18,119),(19,120),(20,121),(21,122),(22,123),(23,124),(24,125),(25,126),(26,127),(27,128),(28,129),(29,130),(30,131),(31,132),(32,133),(33,134),(34,135),(35,136),(36,137),(37,138),(38,139),(39,140),(40,141),(41,142),(42,143),(43,144),(44,145),(45,146),(46,147),(47,148),(48,149),(49,150),(50,151),(51,152),(52,153),(53,154),(54,155),(55,156),(56,157),(57,158),(58,159),(59,160),(60,161),(61,162),(62,163),(63,164),(64,165),(65,166),(66,167),(67,168),(68,169),(69,170),(70,171),(71,172),(72,173),(73,174),(74,175),(75,176),(76,177),(77,178),(78,179),(79,180),(80,181),(81,182),(82,183),(83,184),(84,185),(85,186),(86,187),(87,188),(88,189),(89,190),(90,191),(91,192),(92,193),(93,194),(94,195),(95,196),(96,197),(97,198),(98,199),(99,200),(100,201),(101,202),(102,203),(205,358),(206,359),(207,360),(208,361),(209,362),(210,363),(211,364),(212,365),(213,366),(214,367),(215,368),(216,369),(217,370),(218,371),(219,372),(220,373),(221,374),(222,375),(223,376),(224,377),(225,378),(226,379),(227,380),(228,381),(229,382),(230,383),(231,384),(232,385),(233,386),(234,387),(235,388),(236,389),(237,390),(238,391),(239,392),(240,393),(241,394),(242,395),(243,396),(244,397),(245,398),(246,399),(247,400),(248,401),(249,402),(250,403),(251,404),(252,405),(253,406),(254,407),(255,408),(256,307),(257,308),(258,309),(259,310),(260,311),(261,312),(262,313),(263,314),(264,315),(265,316),(266,317),(267,318),(268,319),(269,320),(270,321),(271,322),(272,323),(273,324),(274,325),(275,326),(276,327),(277,328),(278,329),(279,330),(280,331),(281,332),(282,333),(283,334),(284,335),(285,336),(286,337),(287,338),(288,339),(289,340),(290,341),(291,342),(292,343),(293,344),(294,345),(295,346),(296,347),(297,348),(298,349),(299,350),(300,351),(301,352),(302,353),(303,354),(304,355),(305,356),(306,357)], [(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)], [(1,375,52,324),(2,374,53,323),(3,373,54,322),(4,372,55,321),(5,371,56,320),(6,370,57,319),(7,369,58,318),(8,368,59,317),(9,367,60,316),(10,366,61,315),(11,365,62,314),(12,364,63,313),(13,363,64,312),(14,362,65,311),(15,361,66,310),(16,360,67,309),(17,359,68,308),(18,358,69,307),(19,357,70,408),(20,356,71,407),(21,355,72,406),(22,354,73,405),(23,353,74,404),(24,352,75,403),(25,351,76,402),(26,350,77,401),(27,349,78,400),(28,348,79,399),(29,347,80,398),(30,346,81,397),(31,345,82,396),(32,344,83,395),(33,343,84,394),(34,342,85,393),(35,341,86,392),(36,340,87,391),(37,339,88,390),(38,338,89,389),(39,337,90,388),(40,336,91,387),(41,335,92,386),(42,334,93,385),(43,333,94,384),(44,332,95,383),(45,331,96,382),(46,330,97,381),(47,329,98,380),(48,328,99,379),(49,327,100,378),(50,326,101,377),(51,325,102,376),(103,221,154,272),(104,220,155,271),(105,219,156,270),(106,218,157,269),(107,217,158,268),(108,216,159,267),(109,215,160,266),(110,214,161,265),(111,213,162,264),(112,212,163,263),(113,211,164,262),(114,210,165,261),(115,209,166,260),(116,208,167,259),(117,207,168,258),(118,206,169,257),(119,205,170,256),(120,306,171,255),(121,305,172,254),(122,304,173,253),(123,303,174,252),(124,302,175,251),(125,301,176,250),(126,300,177,249),(127,299,178,248),(128,298,179,247),(129,297,180,246),(130,296,181,245),(131,295,182,244),(132,294,183,243),(133,293,184,242),(134,292,185,241),(135,291,186,240),(136,290,187,239),(137,289,188,238),(138,288,189,237),(139,287,190,236),(140,286,191,235),(141,285,192,234),(142,284,193,233),(143,283,194,232),(144,282,195,231),(145,281,196,230),(146,280,197,229),(147,279,198,228),(148,278,199,227),(149,277,200,226),(150,276,201,225),(151,275,202,224),(152,274,203,223),(153,273,204,222)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 17A | ··· | 17H | 34A | ··· | 34X | 51A | ··· | 51P | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 1 | 1 | 2 | 51 | 51 | 51 | 51 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 | D17 | Dic17 | D34 | D51 | Dic51 | D102 |
| kernel | C2×Dic51 | Dic51 | C2×C102 | C102 | C2×C34 | C34 | C34 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 8 | 16 | 8 | 16 | 32 | 16 |
Matrix representation of C2×Dic51 ►in GL3(𝔽409) generated by
| 1 | 0 | 0 |
| 0 | 408 | 0 |
| 0 | 0 | 408 |
| 408 | 0 | 0 |
| 0 | 369 | 226 |
| 0 | 43 | 74 |
| 143 | 0 | 0 |
| 0 | 389 | 75 |
| 0 | 191 | 20 |
G:=sub<GL(3,GF(409))| [1,0,0,0,408,0,0,0,408],[408,0,0,0,369,43,0,226,74],[143,0,0,0,389,191,0,75,20] >;
C2×Dic51 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{51} % in TeX
G:=Group("C2xDic51"); // GroupNames label
G:=SmallGroup(408,28);
// by ID
G=gap.SmallGroup(408,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,323,9604]);
// Polycyclic
G:=Group<a,b,c|a^2=b^102=1,c^2=b^51,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export