Copied to
clipboard

G = C2×C200order 400 = 24·52

Abelian group of type [2,200]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C200, SmallGroup(400,23)

Series: Derived Chief Lower central Upper central

C1 — C2×C200
C1C2C10C20C100C200 — 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
(1 347)(2 348)(3 349)(4 350)(5 351)(6 352)(7 353)(8 354)(9 355)(10 356)(11 357)(12 358)(13 359)(14 360)(15 361)(16 362)(17 363)(18 364)(19 365)(20 366)(21 367)(22 368)(23 369)(24 370)(25 371)(26 372)(27 373)(28 374)(29 375)(30 376)(31 377)(32 378)(33 379)(34 380)(35 381)(36 382)(37 383)(38 384)(39 385)(40 386)(41 387)(42 388)(43 389)(44 390)(45 391)(46 392)(47 393)(48 394)(49 395)(50 396)(51 397)(52 398)(53 399)(54 400)(55 201)(56 202)(57 203)(58 204)(59 205)(60 206)(61 207)(62 208)(63 209)(64 210)(65 211)(66 212)(67 213)(68 214)(69 215)(70 216)(71 217)(72 218)(73 219)(74 220)(75 221)(76 222)(77 223)(78 224)(79 225)(80 226)(81 227)(82 228)(83 229)(84 230)(85 231)(86 232)(87 233)(88 234)(89 235)(90 236)(91 237)(92 238)(93 239)(94 240)(95 241)(96 242)(97 243)(98 244)(99 245)(100 246)(101 247)(102 248)(103 249)(104 250)(105 251)(106 252)(107 253)(108 254)(109 255)(110 256)(111 257)(112 258)(113 259)(114 260)(115 261)(116 262)(117 263)(118 264)(119 265)(120 266)(121 267)(122 268)(123 269)(124 270)(125 271)(126 272)(127 273)(128 274)(129 275)(130 276)(131 277)(132 278)(133 279)(134 280)(135 281)(136 282)(137 283)(138 284)(139 285)(140 286)(141 287)(142 288)(143 289)(144 290)(145 291)(146 292)(147 293)(148 294)(149 295)(150 296)(151 297)(152 298)(153 299)(154 300)(155 301)(156 302)(157 303)(158 304)(159 305)(160 306)(161 307)(162 308)(163 309)(164 310)(165 311)(166 312)(167 313)(168 314)(169 315)(170 316)(171 317)(172 318)(173 319)(174 320)(175 321)(176 322)(177 323)(178 324)(179 325)(180 326)(181 327)(182 328)(183 329)(184 330)(185 331)(186 332)(187 333)(188 334)(189 335)(190 336)(191 337)(192 338)(193 339)(194 340)(195 341)(196 342)(197 343)(198 344)(199 345)(200 346)
(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)| (1,347)(2,348)(3,349)(4,350)(5,351)(6,352)(7,353)(8,354)(9,355)(10,356)(11,357)(12,358)(13,359)(14,360)(15,361)(16,362)(17,363)(18,364)(19,365)(20,366)(21,367)(22,368)(23,369)(24,370)(25,371)(26,372)(27,373)(28,374)(29,375)(30,376)(31,377)(32,378)(33,379)(34,380)(35,381)(36,382)(37,383)(38,384)(39,385)(40,386)(41,387)(42,388)(43,389)(44,390)(45,391)(46,392)(47,393)(48,394)(49,395)(50,396)(51,397)(52,398)(53,399)(54,400)(55,201)(56,202)(57,203)(58,204)(59,205)(60,206)(61,207)(62,208)(63,209)(64,210)(65,211)(66,212)(67,213)(68,214)(69,215)(70,216)(71,217)(72,218)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,253)(108,254)(109,255)(110,256)(111,257)(112,258)(113,259)(114,260)(115,261)(116,262)(117,263)(118,264)(119,265)(120,266)(121,267)(122,268)(123,269)(124,270)(125,271)(126,272)(127,273)(128,274)(129,275)(130,276)(131,277)(132,278)(133,279)(134,280)(135,281)(136,282)(137,283)(138,284)(139,285)(140,286)(141,287)(142,288)(143,289)(144,290)(145,291)(146,292)(147,293)(148,294)(149,295)(150,296)(151,297)(152,298)(153,299)(154,300)(155,301)(156,302)(157,303)(158,304)(159,305)(160,306)(161,307)(162,308)(163,309)(164,310)(165,311)(166,312)(167,313)(168,314)(169,315)(170,316)(171,317)(172,318)(173,319)(174,320)(175,321)(176,322)(177,323)(178,324)(179,325)(180,326)(181,327)(182,328)(183,329)(184,330)(185,331)(186,332)(187,333)(188,334)(189,335)(190,336)(191,337)(192,338)(193,339)(194,340)(195,341)(196,342)(197,343)(198,344)(199,345)(200,346), (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( (1,347)(2,348)(3,349)(4,350)(5,351)(6,352)(7,353)(8,354)(9,355)(10,356)(11,357)(12,358)(13,359)(14,360)(15,361)(16,362)(17,363)(18,364)(19,365)(20,366)(21,367)(22,368)(23,369)(24,370)(25,371)(26,372)(27,373)(28,374)(29,375)(30,376)(31,377)(32,378)(33,379)(34,380)(35,381)(36,382)(37,383)(38,384)(39,385)(40,386)(41,387)(42,388)(43,389)(44,390)(45,391)(46,392)(47,393)(48,394)(49,395)(50,396)(51,397)(52,398)(53,399)(54,400)(55,201)(56,202)(57,203)(58,204)(59,205)(60,206)(61,207)(62,208)(63,209)(64,210)(65,211)(66,212)(67,213)(68,214)(69,215)(70,216)(71,217)(72,218)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,253)(108,254)(109,255)(110,256)(111,257)(112,258)(113,259)(114,260)(115,261)(116,262)(117,263)(118,264)(119,265)(120,266)(121,267)(122,268)(123,269)(124,270)(125,271)(126,272)(127,273)(128,274)(129,275)(130,276)(131,277)(132,278)(133,279)(134,280)(135,281)(136,282)(137,283)(138,284)(139,285)(140,286)(141,287)(142,288)(143,289)(144,290)(145,291)(146,292)(147,293)(148,294)(149,295)(150,296)(151,297)(152,298)(153,299)(154,300)(155,301)(156,302)(157,303)(158,304)(159,305)(160,306)(161,307)(162,308)(163,309)(164,310)(165,311)(166,312)(167,313)(168,314)(169,315)(170,316)(171,317)(172,318)(173,319)(174,320)(175,321)(176,322)(177,323)(178,324)(179,325)(180,326)(181,327)(182,328)(183,329)(184,330)(185,331)(186,332)(187,333)(188,334)(189,335)(190,336)(191,337)(192,338)(193,339)(194,340)(195,341)(196,342)(197,343)(198,344)(199,345)(200,346), (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([(1,347),(2,348),(3,349),(4,350),(5,351),(6,352),(7,353),(8,354),(9,355),(10,356),(11,357),(12,358),(13,359),(14,360),(15,361),(16,362),(17,363),(18,364),(19,365),(20,366),(21,367),(22,368),(23,369),(24,370),(25,371),(26,372),(27,373),(28,374),(29,375),(30,376),(31,377),(32,378),(33,379),(34,380),(35,381),(36,382),(37,383),(38,384),(39,385),(40,386),(41,387),(42,388),(43,389),(44,390),(45,391),(46,392),(47,393),(48,394),(49,395),(50,396),(51,397),(52,398),(53,399),(54,400),(55,201),(56,202),(57,203),(58,204),(59,205),(60,206),(61,207),(62,208),(63,209),(64,210),(65,211),(66,212),(67,213),(68,214),(69,215),(70,216),(71,217),(72,218),(73,219),(74,220),(75,221),(76,222),(77,223),(78,224),(79,225),(80,226),(81,227),(82,228),(83,229),(84,230),(85,231),(86,232),(87,233),(88,234),(89,235),(90,236),(91,237),(92,238),(93,239),(94,240),(95,241),(96,242),(97,243),(98,244),(99,245),(100,246),(101,247),(102,248),(103,249),(104,250),(105,251),(106,252),(107,253),(108,254),(109,255),(110,256),(111,257),(112,258),(113,259),(114,260),(115,261),(116,262),(117,263),(118,264),(119,265),(120,266),(121,267),(122,268),(123,269),(124,270),(125,271),(126,272),(127,273),(128,274),(129,275),(130,276),(131,277),(132,278),(133,279),(134,280),(135,281),(136,282),(137,283),(138,284),(139,285),(140,286),(141,287),(142,288),(143,289),(144,290),(145,291),(146,292),(147,293),(148,294),(149,295),(150,296),(151,297),(152,298),(153,299),(154,300),(155,301),(156,302),(157,303),(158,304),(159,305),(160,306),(161,307),(162,308),(163,309),(164,310),(165,311),(166,312),(167,313),(168,314),(169,315),(170,316),(171,317),(172,318),(173,319),(174,320),(175,321),(176,322),(177,323),(178,324),(179,325),(180,326),(181,327),(182,328),(183,329),(184,330),(185,331),(186,332),(187,333),(188,334),(189,335),(190,336),(191,337),(192,338),(193,339),(194,340),(195,341),(196,342),(197,343),(198,344),(199,345),(200,346)], [(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 2A2B2C4A4B4C4D5A5B5C5D8A···8H10A···10L20A···20P25A···25T40A···40AF50A···50BH100A···100CB200A···200FD
order1222444455558···810···1020···2025···2540···4050···50100···100200···200
size1111111111111···11···11···11···11···11···11···11···1

400 irreducible representations

dim111111111111111111
type+++
imageC1C2C2C4C4C5C8C10C10C20C20C25C40C50C50C100C100C200
kernelC2×C200C200C2×C100C100C2×C50C2×C40C50C40C2×C20C20C2×C10C2×C8C10C8C2×C4C4C22C2
# reps12122488488203240204040160

Matrix representation of C2×C200 in GL2(𝔽401) generated by

4000
01
,
2560
0390
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

Subgroup lattice of C2×C200 in TeX

׿
×
𝔽