direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic50, C50⋊Q8, C4.11D50, C20.43D10, C50.1C23, C22.8D50, C10.5Dic10, C100.11C22, Dic25.1C22, C25⋊1(C2×Q8), (C2×C4).4D25, (C2×C20).6D5, C5.(C2×Dic10), (C2×C100).3C2, (C2×C10).23D10, (C2×C50).8C22, C2.3(C22×D25), (C2×Dic25).3C2, C10.19(C22×D5), SmallGroup(400,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic50
G = < a,b,c | a2=b100=1, c2=b50, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 365 in 57 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, Q8, C10, C10, C2×Q8, Dic5, C20, C2×C10, C25, Dic10, C2×Dic5, C2×C20, C50, C50, C2×Dic10, Dic25, C100, C2×C50, Dic50, C2×Dic25, C2×C100, C2×Dic50
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, D10, Dic10, C22×D5, D25, C2×Dic10, D50, Dic50, C22×D25, C2×Dic50
►Regular action on 400 points
Generators in S400
(1 195)(2 196)(3 197)(4 198)(5 199)(6 200)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 161)(68 162)(69 163)(70 164)(71 165)(72 166)(73 167)(74 168)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 177)(84 178)(85 179)(86 180)(87 181)(88 182)(89 183)(90 184)(91 185)(92 186)(93 187)(94 188)(95 189)(96 190)(97 191)(98 192)(99 193)(100 194)(201 392)(202 393)(203 394)(204 395)(205 396)(206 397)(207 398)(208 399)(209 400)(210 301)(211 302)(212 303)(213 304)(214 305)(215 306)(216 307)(217 308)(218 309)(219 310)(220 311)(221 312)(222 313)(223 314)(224 315)(225 316)(226 317)(227 318)(228 319)(229 320)(230 321)(231 322)(232 323)(233 324)(234 325)(235 326)(236 327)(237 328)(238 329)(239 330)(240 331)(241 332)(242 333)(243 334)(244 335)(245 336)(246 337)(247 338)(248 339)(249 340)(250 341)(251 342)(252 343)(253 344)(254 345)(255 346)(256 347)(257 348)(258 349)(259 350)(260 351)(261 352)(262 353)(263 354)(264 355)(265 356)(266 357)(267 358)(268 359)(269 360)(270 361)(271 362)(272 363)(273 364)(274 365)(275 366)(276 367)(277 368)(278 369)(279 370)(280 371)(281 372)(282 373)(283 374)(284 375)(285 376)(286 377)(287 378)(288 379)(289 380)(290 381)(291 382)(292 383)(293 384)(294 385)(295 386)(296 387)(297 388)(298 389)(299 390)(300 391)
(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)
(1 268 51 218)(2 267 52 217)(3 266 53 216)(4 265 54 215)(5 264 55 214)(6 263 56 213)(7 262 57 212)(8 261 58 211)(9 260 59 210)(10 259 60 209)(11 258 61 208)(12 257 62 207)(13 256 63 206)(14 255 64 205)(15 254 65 204)(16 253 66 203)(17 252 67 202)(18 251 68 201)(19 250 69 300)(20 249 70 299)(21 248 71 298)(22 247 72 297)(23 246 73 296)(24 245 74 295)(25 244 75 294)(26 243 76 293)(27 242 77 292)(28 241 78 291)(29 240 79 290)(30 239 80 289)(31 238 81 288)(32 237 82 287)(33 236 83 286)(34 235 84 285)(35 234 85 284)(36 233 86 283)(37 232 87 282)(38 231 88 281)(39 230 89 280)(40 229 90 279)(41 228 91 278)(42 227 92 277)(43 226 93 276)(44 225 94 275)(45 224 95 274)(46 223 96 273)(47 222 97 272)(48 221 98 271)(49 220 99 270)(50 219 100 269)(101 353 151 303)(102 352 152 302)(103 351 153 301)(104 350 154 400)(105 349 155 399)(106 348 156 398)(107 347 157 397)(108 346 158 396)(109 345 159 395)(110 344 160 394)(111 343 161 393)(112 342 162 392)(113 341 163 391)(114 340 164 390)(115 339 165 389)(116 338 166 388)(117 337 167 387)(118 336 168 386)(119 335 169 385)(120 334 170 384)(121 333 171 383)(122 332 172 382)(123 331 173 381)(124 330 174 380)(125 329 175 379)(126 328 176 378)(127 327 177 377)(128 326 178 376)(129 325 179 375)(130 324 180 374)(131 323 181 373)(132 322 182 372)(133 321 183 371)(134 320 184 370)(135 319 185 369)(136 318 186 368)(137 317 187 367)(138 316 188 366)(139 315 189 365)(140 314 190 364)(141 313 191 363)(142 312 192 362)(143 311 193 361)(144 310 194 360)(145 309 195 359)(146 308 196 358)(147 307 197 357)(148 306 198 356)(149 305 199 355)(150 304 200 354)
G:=sub<Sym(400)| (1,195)(2,196)(3,197)(4,198)(5,199)(6,200)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,182)(89,183)(90,184)(91,185)(92,186)(93,187)(94,188)(95,189)(96,190)(97,191)(98,192)(99,193)(100,194)(201,392)(202,393)(203,394)(204,395)(205,396)(206,397)(207,398)(208,399)(209,400)(210,301)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,316)(226,317)(227,318)(228,319)(229,320)(230,321)(231,322)(232,323)(233,324)(234,325)(235,326)(236,327)(237,328)(238,329)(239,330)(240,331)(241,332)(242,333)(243,334)(244,335)(245,336)(246,337)(247,338)(248,339)(249,340)(250,341)(251,342)(252,343)(253,344)(254,345)(255,346)(256,347)(257,348)(258,349)(259,350)(260,351)(261,352)(262,353)(263,354)(264,355)(265,356)(266,357)(267,358)(268,359)(269,360)(270,361)(271,362)(272,363)(273,364)(274,365)(275,366)(276,367)(277,368)(278,369)(279,370)(280,371)(281,372)(282,373)(283,374)(284,375)(285,376)(286,377)(287,378)(288,379)(289,380)(290,381)(291,382)(292,383)(293,384)(294,385)(295,386)(296,387)(297,388)(298,389)(299,390)(300,391), (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), (1,268,51,218)(2,267,52,217)(3,266,53,216)(4,265,54,215)(5,264,55,214)(6,263,56,213)(7,262,57,212)(8,261,58,211)(9,260,59,210)(10,259,60,209)(11,258,61,208)(12,257,62,207)(13,256,63,206)(14,255,64,205)(15,254,65,204)(16,253,66,203)(17,252,67,202)(18,251,68,201)(19,250,69,300)(20,249,70,299)(21,248,71,298)(22,247,72,297)(23,246,73,296)(24,245,74,295)(25,244,75,294)(26,243,76,293)(27,242,77,292)(28,241,78,291)(29,240,79,290)(30,239,80,289)(31,238,81,288)(32,237,82,287)(33,236,83,286)(34,235,84,285)(35,234,85,284)(36,233,86,283)(37,232,87,282)(38,231,88,281)(39,230,89,280)(40,229,90,279)(41,228,91,278)(42,227,92,277)(43,226,93,276)(44,225,94,275)(45,224,95,274)(46,223,96,273)(47,222,97,272)(48,221,98,271)(49,220,99,270)(50,219,100,269)(101,353,151,303)(102,352,152,302)(103,351,153,301)(104,350,154,400)(105,349,155,399)(106,348,156,398)(107,347,157,397)(108,346,158,396)(109,345,159,395)(110,344,160,394)(111,343,161,393)(112,342,162,392)(113,341,163,391)(114,340,164,390)(115,339,165,389)(116,338,166,388)(117,337,167,387)(118,336,168,386)(119,335,169,385)(120,334,170,384)(121,333,171,383)(122,332,172,382)(123,331,173,381)(124,330,174,380)(125,329,175,379)(126,328,176,378)(127,327,177,377)(128,326,178,376)(129,325,179,375)(130,324,180,374)(131,323,181,373)(132,322,182,372)(133,321,183,371)(134,320,184,370)(135,319,185,369)(136,318,186,368)(137,317,187,367)(138,316,188,366)(139,315,189,365)(140,314,190,364)(141,313,191,363)(142,312,192,362)(143,311,193,361)(144,310,194,360)(145,309,195,359)(146,308,196,358)(147,307,197,357)(148,306,198,356)(149,305,199,355)(150,304,200,354)>;
G:=Group( (1,195)(2,196)(3,197)(4,198)(5,199)(6,200)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,182)(89,183)(90,184)(91,185)(92,186)(93,187)(94,188)(95,189)(96,190)(97,191)(98,192)(99,193)(100,194)(201,392)(202,393)(203,394)(204,395)(205,396)(206,397)(207,398)(208,399)(209,400)(210,301)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,316)(226,317)(227,318)(228,319)(229,320)(230,321)(231,322)(232,323)(233,324)(234,325)(235,326)(236,327)(237,328)(238,329)(239,330)(240,331)(241,332)(242,333)(243,334)(244,335)(245,336)(246,337)(247,338)(248,339)(249,340)(250,341)(251,342)(252,343)(253,344)(254,345)(255,346)(256,347)(257,348)(258,349)(259,350)(260,351)(261,352)(262,353)(263,354)(264,355)(265,356)(266,357)(267,358)(268,359)(269,360)(270,361)(271,362)(272,363)(273,364)(274,365)(275,366)(276,367)(277,368)(278,369)(279,370)(280,371)(281,372)(282,373)(283,374)(284,375)(285,376)(286,377)(287,378)(288,379)(289,380)(290,381)(291,382)(292,383)(293,384)(294,385)(295,386)(296,387)(297,388)(298,389)(299,390)(300,391), (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), (1,268,51,218)(2,267,52,217)(3,266,53,216)(4,265,54,215)(5,264,55,214)(6,263,56,213)(7,262,57,212)(8,261,58,211)(9,260,59,210)(10,259,60,209)(11,258,61,208)(12,257,62,207)(13,256,63,206)(14,255,64,205)(15,254,65,204)(16,253,66,203)(17,252,67,202)(18,251,68,201)(19,250,69,300)(20,249,70,299)(21,248,71,298)(22,247,72,297)(23,246,73,296)(24,245,74,295)(25,244,75,294)(26,243,76,293)(27,242,77,292)(28,241,78,291)(29,240,79,290)(30,239,80,289)(31,238,81,288)(32,237,82,287)(33,236,83,286)(34,235,84,285)(35,234,85,284)(36,233,86,283)(37,232,87,282)(38,231,88,281)(39,230,89,280)(40,229,90,279)(41,228,91,278)(42,227,92,277)(43,226,93,276)(44,225,94,275)(45,224,95,274)(46,223,96,273)(47,222,97,272)(48,221,98,271)(49,220,99,270)(50,219,100,269)(101,353,151,303)(102,352,152,302)(103,351,153,301)(104,350,154,400)(105,349,155,399)(106,348,156,398)(107,347,157,397)(108,346,158,396)(109,345,159,395)(110,344,160,394)(111,343,161,393)(112,342,162,392)(113,341,163,391)(114,340,164,390)(115,339,165,389)(116,338,166,388)(117,337,167,387)(118,336,168,386)(119,335,169,385)(120,334,170,384)(121,333,171,383)(122,332,172,382)(123,331,173,381)(124,330,174,380)(125,329,175,379)(126,328,176,378)(127,327,177,377)(128,326,178,376)(129,325,179,375)(130,324,180,374)(131,323,181,373)(132,322,182,372)(133,321,183,371)(134,320,184,370)(135,319,185,369)(136,318,186,368)(137,317,187,367)(138,316,188,366)(139,315,189,365)(140,314,190,364)(141,313,191,363)(142,312,192,362)(143,311,193,361)(144,310,194,360)(145,309,195,359)(146,308,196,358)(147,307,197,357)(148,306,198,356)(149,305,199,355)(150,304,200,354) );
G=PermutationGroup([[(1,195),(2,196),(3,197),(4,198),(5,199),(6,200),(7,101),(8,102),(9,103),(10,104),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,111),(18,112),(19,113),(20,114),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,127),(34,128),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,153),(60,154),(61,155),(62,156),(63,157),(64,158),(65,159),(66,160),(67,161),(68,162),(69,163),(70,164),(71,165),(72,166),(73,167),(74,168),(75,169),(76,170),(77,171),(78,172),(79,173),(80,174),(81,175),(82,176),(83,177),(84,178),(85,179),(86,180),(87,181),(88,182),(89,183),(90,184),(91,185),(92,186),(93,187),(94,188),(95,189),(96,190),(97,191),(98,192),(99,193),(100,194),(201,392),(202,393),(203,394),(204,395),(205,396),(206,397),(207,398),(208,399),(209,400),(210,301),(211,302),(212,303),(213,304),(214,305),(215,306),(216,307),(217,308),(218,309),(219,310),(220,311),(221,312),(222,313),(223,314),(224,315),(225,316),(226,317),(227,318),(228,319),(229,320),(230,321),(231,322),(232,323),(233,324),(234,325),(235,326),(236,327),(237,328),(238,329),(239,330),(240,331),(241,332),(242,333),(243,334),(244,335),(245,336),(246,337),(247,338),(248,339),(249,340),(250,341),(251,342),(252,343),(253,344),(254,345),(255,346),(256,347),(257,348),(258,349),(259,350),(260,351),(261,352),(262,353),(263,354),(264,355),(265,356),(266,357),(267,358),(268,359),(269,360),(270,361),(271,362),(272,363),(273,364),(274,365),(275,366),(276,367),(277,368),(278,369),(279,370),(280,371),(281,372),(282,373),(283,374),(284,375),(285,376),(286,377),(287,378),(288,379),(289,380),(290,381),(291,382),(292,383),(293,384),(294,385),(295,386),(296,387),(297,388),(298,389),(299,390),(300,391)], [(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)], [(1,268,51,218),(2,267,52,217),(3,266,53,216),(4,265,54,215),(5,264,55,214),(6,263,56,213),(7,262,57,212),(8,261,58,211),(9,260,59,210),(10,259,60,209),(11,258,61,208),(12,257,62,207),(13,256,63,206),(14,255,64,205),(15,254,65,204),(16,253,66,203),(17,252,67,202),(18,251,68,201),(19,250,69,300),(20,249,70,299),(21,248,71,298),(22,247,72,297),(23,246,73,296),(24,245,74,295),(25,244,75,294),(26,243,76,293),(27,242,77,292),(28,241,78,291),(29,240,79,290),(30,239,80,289),(31,238,81,288),(32,237,82,287),(33,236,83,286),(34,235,84,285),(35,234,85,284),(36,233,86,283),(37,232,87,282),(38,231,88,281),(39,230,89,280),(40,229,90,279),(41,228,91,278),(42,227,92,277),(43,226,93,276),(44,225,94,275),(45,224,95,274),(46,223,96,273),(47,222,97,272),(48,221,98,271),(49,220,99,270),(50,219,100,269),(101,353,151,303),(102,352,152,302),(103,351,153,301),(104,350,154,400),(105,349,155,399),(106,348,156,398),(107,347,157,397),(108,346,158,396),(109,345,159,395),(110,344,160,394),(111,343,161,393),(112,342,162,392),(113,341,163,391),(114,340,164,390),(115,339,165,389),(116,338,166,388),(117,337,167,387),(118,336,168,386),(119,335,169,385),(120,334,170,384),(121,333,171,383),(122,332,172,382),(123,331,173,381),(124,330,174,380),(125,329,175,379),(126,328,176,378),(127,327,177,377),(128,326,178,376),(129,325,179,375),(130,324,180,374),(131,323,181,373),(132,322,182,372),(133,321,183,371),(134,320,184,370),(135,319,185,369),(136,318,186,368),(137,317,187,367),(138,316,188,366),(139,315,189,365),(140,314,190,364),(141,313,191,363),(142,312,192,362),(143,311,193,361),(144,310,194,360),(145,309,195,359),(146,308,196,358),(147,307,197,357),(148,306,198,356),(149,305,199,355),(150,304,200,354)]])
106 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 25A | ··· | 25J | 50A | ··· | 50AD | 100A | ··· | 100AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 50 | 50 | 50 | 50 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | Dic10 | D25 | D50 | D50 | Dic50 |
| kernel | C2×Dic50 | Dic50 | C2×Dic25 | C2×C100 | C50 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 8 | 10 | 20 | 10 | 40 |
Matrix representation of C2×Dic50 ►in GL3(𝔽101) generated by
| 100 | 0 | 0 |
| 0 | 100 | 0 |
| 0 | 0 | 100 |
| 1 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 51 |
| 100 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 100 | 0 |
G:=sub<GL(3,GF(101))| [100,0,0,0,100,0,0,0,100],[1,0,0,0,2,0,0,0,51],[100,0,0,0,0,100,0,1,0] >;
C2×Dic50 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{50} % in TeX
G:=Group("C2xDic50"); // GroupNames label
G:=SmallGroup(400,35);
// by ID
G=gap.SmallGroup(400,35);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,218,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b,c|a^2=b^100=1,c^2=b^50,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations