direct product, abelian, monomial, 2-elementary
Aliases: C2×C200, SmallGroup(400,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C200 |
| C1 — C2×C200 |
| C1 — C2×C200 |
Generators and relations for C2×C200
G = < a,b | a2=b200=1, ab=ba >
Smallest permutation representation of C2×C200
►Regular action on 400 points
Generators in S400
(1 336)(2 337)(3 338)(4 339)(5 340)(6 341)(7 342)(8 343)(9 344)(10 345)(11 346)(12 347)(13 348)(14 349)(15 350)(16 351)(17 352)(18 353)(19 354)(20 355)(21 356)(22 357)(23 358)(24 359)(25 360)(26 361)(27 362)(28 363)(29 364)(30 365)(31 366)(32 367)(33 368)(34 369)(35 370)(36 371)(37 372)(38 373)(39 374)(40 375)(41 376)(42 377)(43 378)(44 379)(45 380)(46 381)(47 382)(48 383)(49 384)(50 385)(51 386)(52 387)(53 388)(54 389)(55 390)(56 391)(57 392)(58 393)(59 394)(60 395)(61 396)(62 397)(63 398)(64 399)(65 400)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 209)(75 210)(76 211)(77 212)(78 213)(79 214)(80 215)(81 216)(82 217)(83 218)(84 219)(85 220)(86 221)(87 222)(88 223)(89 224)(90 225)(91 226)(92 227)(93 228)(94 229)(95 230)(96 231)(97 232)(98 233)(99 234)(100 235)(101 236)(102 237)(103 238)(104 239)(105 240)(106 241)(107 242)(108 243)(109 244)(110 245)(111 246)(112 247)(113 248)(114 249)(115 250)(116 251)(117 252)(118 253)(119 254)(120 255)(121 256)(122 257)(123 258)(124 259)(125 260)(126 261)(127 262)(128 263)(129 264)(130 265)(131 266)(132 267)(133 268)(134 269)(135 270)(136 271)(137 272)(138 273)(139 274)(140 275)(141 276)(142 277)(143 278)(144 279)(145 280)(146 281)(147 282)(148 283)(149 284)(150 285)(151 286)(152 287)(153 288)(154 289)(155 290)(156 291)(157 292)(158 293)(159 294)(160 295)(161 296)(162 297)(163 298)(164 299)(165 300)(166 301)(167 302)(168 303)(169 304)(170 305)(171 306)(172 307)(173 308)(174 309)(175 310)(176 311)(177 312)(178 313)(179 314)(180 315)(181 316)(182 317)(183 318)(184 319)(185 320)(186 321)(187 322)(188 323)(189 324)(190 325)(191 326)(192 327)(193 328)(194 329)(195 330)(196 331)(197 332)(198 333)(199 334)(200 335)
(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)
G:=sub<Sym(400)| (1,336)(2,337)(3,338)(4,339)(5,340)(6,341)(7,342)(8,343)(9,344)(10,345)(11,346)(12,347)(13,348)(14,349)(15,350)(16,351)(17,352)(18,353)(19,354)(20,355)(21,356)(22,357)(23,358)(24,359)(25,360)(26,361)(27,362)(28,363)(29,364)(30,365)(31,366)(32,367)(33,368)(34,369)(35,370)(36,371)(37,372)(38,373)(39,374)(40,375)(41,376)(42,377)(43,378)(44,379)(45,380)(46,381)(47,382)(48,383)(49,384)(50,385)(51,386)(52,387)(53,388)(54,389)(55,390)(56,391)(57,392)(58,393)(59,394)(60,395)(61,396)(62,397)(63,398)(64,399)(65,400)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,220)(86,221)(87,222)(88,223)(89,224)(90,225)(91,226)(92,227)(93,228)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)(112,247)(113,248)(114,249)(115,250)(116,251)(117,252)(118,253)(119,254)(120,255)(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,262)(128,263)(129,264)(130,265)(131,266)(132,267)(133,268)(134,269)(135,270)(136,271)(137,272)(138,273)(139,274)(140,275)(141,276)(142,277)(143,278)(144,279)(145,280)(146,281)(147,282)(148,283)(149,284)(150,285)(151,286)(152,287)(153,288)(154,289)(155,290)(156,291)(157,292)(158,293)(159,294)(160,295)(161,296)(162,297)(163,298)(164,299)(165,300)(166,301)(167,302)(168,303)(169,304)(170,305)(171,306)(172,307)(173,308)(174,309)(175,310)(176,311)(177,312)(178,313)(179,314)(180,315)(181,316)(182,317)(183,318)(184,319)(185,320)(186,321)(187,322)(188,323)(189,324)(190,325)(191,326)(192,327)(193,328)(194,329)(195,330)(196,331)(197,332)(198,333)(199,334)(200,335), (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)>;
G:=Group( (1,336)(2,337)(3,338)(4,339)(5,340)(6,341)(7,342)(8,343)(9,344)(10,345)(11,346)(12,347)(13,348)(14,349)(15,350)(16,351)(17,352)(18,353)(19,354)(20,355)(21,356)(22,357)(23,358)(24,359)(25,360)(26,361)(27,362)(28,363)(29,364)(30,365)(31,366)(32,367)(33,368)(34,369)(35,370)(36,371)(37,372)(38,373)(39,374)(40,375)(41,376)(42,377)(43,378)(44,379)(45,380)(46,381)(47,382)(48,383)(49,384)(50,385)(51,386)(52,387)(53,388)(54,389)(55,390)(56,391)(57,392)(58,393)(59,394)(60,395)(61,396)(62,397)(63,398)(64,399)(65,400)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,220)(86,221)(87,222)(88,223)(89,224)(90,225)(91,226)(92,227)(93,228)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,241)(107,242)(108,243)(109,244)(110,245)(111,246)(112,247)(113,248)(114,249)(115,250)(116,251)(117,252)(118,253)(119,254)(120,255)(121,256)(122,257)(123,258)(124,259)(125,260)(126,261)(127,262)(128,263)(129,264)(130,265)(131,266)(132,267)(133,268)(134,269)(135,270)(136,271)(137,272)(138,273)(139,274)(140,275)(141,276)(142,277)(143,278)(144,279)(145,280)(146,281)(147,282)(148,283)(149,284)(150,285)(151,286)(152,287)(153,288)(154,289)(155,290)(156,291)(157,292)(158,293)(159,294)(160,295)(161,296)(162,297)(163,298)(164,299)(165,300)(166,301)(167,302)(168,303)(169,304)(170,305)(171,306)(172,307)(173,308)(174,309)(175,310)(176,311)(177,312)(178,313)(179,314)(180,315)(181,316)(182,317)(183,318)(184,319)(185,320)(186,321)(187,322)(188,323)(189,324)(190,325)(191,326)(192,327)(193,328)(194,329)(195,330)(196,331)(197,332)(198,333)(199,334)(200,335), (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) );
G=PermutationGroup([[(1,336),(2,337),(3,338),(4,339),(5,340),(6,341),(7,342),(8,343),(9,344),(10,345),(11,346),(12,347),(13,348),(14,349),(15,350),(16,351),(17,352),(18,353),(19,354),(20,355),(21,356),(22,357),(23,358),(24,359),(25,360),(26,361),(27,362),(28,363),(29,364),(30,365),(31,366),(32,367),(33,368),(34,369),(35,370),(36,371),(37,372),(38,373),(39,374),(40,375),(41,376),(42,377),(43,378),(44,379),(45,380),(46,381),(47,382),(48,383),(49,384),(50,385),(51,386),(52,387),(53,388),(54,389),(55,390),(56,391),(57,392),(58,393),(59,394),(60,395),(61,396),(62,397),(63,398),(64,399),(65,400),(66,201),(67,202),(68,203),(69,204),(70,205),(71,206),(72,207),(73,208),(74,209),(75,210),(76,211),(77,212),(78,213),(79,214),(80,215),(81,216),(82,217),(83,218),(84,219),(85,220),(86,221),(87,222),(88,223),(89,224),(90,225),(91,226),(92,227),(93,228),(94,229),(95,230),(96,231),(97,232),(98,233),(99,234),(100,235),(101,236),(102,237),(103,238),(104,239),(105,240),(106,241),(107,242),(108,243),(109,244),(110,245),(111,246),(112,247),(113,248),(114,249),(115,250),(116,251),(117,252),(118,253),(119,254),(120,255),(121,256),(122,257),(123,258),(124,259),(125,260),(126,261),(127,262),(128,263),(129,264),(130,265),(131,266),(132,267),(133,268),(134,269),(135,270),(136,271),(137,272),(138,273),(139,274),(140,275),(141,276),(142,277),(143,278),(144,279),(145,280),(146,281),(147,282),(148,283),(149,284),(150,285),(151,286),(152,287),(153,288),(154,289),(155,290),(156,291),(157,292),(158,293),(159,294),(160,295),(161,296),(162,297),(163,298),(164,299),(165,300),(166,301),(167,302),(168,303),(169,304),(170,305),(171,306),(172,307),(173,308),(174,309),(175,310),(176,311),(177,312),(178,313),(179,314),(180,315),(181,316),(182,317),(183,318),(184,319),(185,320),(186,321),(187,322),(188,323),(189,324),(190,325),(191,326),(192,327),(193,328),(194,329),(195,330),(196,331),(197,332),(198,333),(199,334),(200,335)], [(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)]])
400 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 20A | ··· | 20P | 25A | ··· | 25T | 40A | ··· | 40AF | 50A | ··· | 50BH | 100A | ··· | 100CB | 200A | ··· | 200FD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 40 | ··· | 40 | 50 | ··· | 50 | 100 | ··· | 100 | 200 | ··· | 200 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
400 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C25 | C40 | C50 | C50 | C100 | C100 | C200 |
| kernel | C2×C200 | C200 | C2×C100 | C100 | C2×C50 | C2×C40 | C50 | C40 | C2×C20 | C20 | C2×C10 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 8 | 20 | 32 | 40 | 20 | 40 | 40 | 160 |
Matrix representation of C2×C200 ►in GL2(𝔽401) generated by
| 400 | 0 |
| 0 | 1 |
| 256 | 0 |
| 0 | 390 |
G:=sub<GL(2,GF(401))| [400,0,0,1],[256,0,0,390] >;
C2×C200 in GAP, Magma, Sage, TeX
C_2\times C_{200} % in TeX
G:=Group("C2xC200"); // GroupNames label
G:=SmallGroup(400,23);
// by ID
G=gap.SmallGroup(400,23);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-5,-2,120,194,261]);
// Polycyclic
G:=Group<a,b|a^2=b^200=1,a*b=b*a>;
// generators/relations
Export