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G = C2×C202order 404 = 22·101

Abelian group of type [2,202]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C202, SmallGroup(404,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C202
C1C101C202 — C2×C202
C1 — C2×C202
C1 — C2×C202

Generators and relations for C2×C202
 G = < a,b | a2=b202=1, ab=ba >


Smallest permutation representation of C2×C202
Regular action on 404 points
Generators in S404
(1 355)(2 356)(3 357)(4 358)(5 359)(6 360)(7 361)(8 362)(9 363)(10 364)(11 365)(12 366)(13 367)(14 368)(15 369)(16 370)(17 371)(18 372)(19 373)(20 374)(21 375)(22 376)(23 377)(24 378)(25 379)(26 380)(27 381)(28 382)(29 383)(30 384)(31 385)(32 386)(33 387)(34 388)(35 389)(36 390)(37 391)(38 392)(39 393)(40 394)(41 395)(42 396)(43 397)(44 398)(45 399)(46 400)(47 401)(48 402)(49 403)(50 404)(51 203)(52 204)(53 205)(54 206)(55 207)(56 208)(57 209)(58 210)(59 211)(60 212)(61 213)(62 214)(63 215)(64 216)(65 217)(66 218)(67 219)(68 220)(69 221)(70 222)(71 223)(72 224)(73 225)(74 226)(75 227)(76 228)(77 229)(78 230)(79 231)(80 232)(81 233)(82 234)(83 235)(84 236)(85 237)(86 238)(87 239)(88 240)(89 241)(90 242)(91 243)(92 244)(93 245)(94 246)(95 247)(96 248)(97 249)(98 250)(99 251)(100 252)(101 253)(102 254)(103 255)(104 256)(105 257)(106 258)(107 259)(108 260)(109 261)(110 262)(111 263)(112 264)(113 265)(114 266)(115 267)(116 268)(117 269)(118 270)(119 271)(120 272)(121 273)(122 274)(123 275)(124 276)(125 277)(126 278)(127 279)(128 280)(129 281)(130 282)(131 283)(132 284)(133 285)(134 286)(135 287)(136 288)(137 289)(138 290)(139 291)(140 292)(141 293)(142 294)(143 295)(144 296)(145 297)(146 298)(147 299)(148 300)(149 301)(150 302)(151 303)(152 304)(153 305)(154 306)(155 307)(156 308)(157 309)(158 310)(159 311)(160 312)(161 313)(162 314)(163 315)(164 316)(165 317)(166 318)(167 319)(168 320)(169 321)(170 322)(171 323)(172 324)(173 325)(174 326)(175 327)(176 328)(177 329)(178 330)(179 331)(180 332)(181 333)(182 334)(183 335)(184 336)(185 337)(186 338)(187 339)(188 340)(189 341)(190 342)(191 343)(192 344)(193 345)(194 346)(195 347)(196 348)(197 349)(198 350)(199 351)(200 352)(201 353)(202 354)
(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)

G:=sub<Sym(404)| (1,355)(2,356)(3,357)(4,358)(5,359)(6,360)(7,361)(8,362)(9,363)(10,364)(11,365)(12,366)(13,367)(14,368)(15,369)(16,370)(17,371)(18,372)(19,373)(20,374)(21,375)(22,376)(23,377)(24,378)(25,379)(26,380)(27,381)(28,382)(29,383)(30,384)(31,385)(32,386)(33,387)(34,388)(35,389)(36,390)(37,391)(38,392)(39,393)(40,394)(41,395)(42,396)(43,397)(44,398)(45,399)(46,400)(47,401)(48,402)(49,403)(50,404)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,209)(58,210)(59,211)(60,212)(61,213)(62,214)(63,215)(64,216)(65,217)(66,218)(67,219)(68,220)(69,221)(70,222)(71,223)(72,224)(73,225)(74,226)(75,227)(76,228)(77,229)(78,230)(79,231)(80,232)(81,233)(82,234)(83,235)(84,236)(85,237)(86,238)(87,239)(88,240)(89,241)(90,242)(91,243)(92,244)(93,245)(94,246)(95,247)(96,248)(97,249)(98,250)(99,251)(100,252)(101,253)(102,254)(103,255)(104,256)(105,257)(106,258)(107,259)(108,260)(109,261)(110,262)(111,263)(112,264)(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)(120,272)(121,273)(122,274)(123,275)(124,276)(125,277)(126,278)(127,279)(128,280)(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)(136,288)(137,289)(138,290)(139,291)(140,292)(141,293)(142,294)(143,295)(144,296)(145,297)(146,298)(147,299)(148,300)(149,301)(150,302)(151,303)(152,304)(153,305)(154,306)(155,307)(156,308)(157,309)(158,310)(159,311)(160,312)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(169,321)(170,322)(171,323)(172,324)(173,325)(174,326)(175,327)(176,328)(177,329)(178,330)(179,331)(180,332)(181,333)(182,334)(183,335)(184,336)(185,337)(186,338)(187,339)(188,340)(189,341)(190,342)(191,343)(192,344)(193,345)(194,346)(195,347)(196,348)(197,349)(198,350)(199,351)(200,352)(201,353)(202,354), (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)>;

G:=Group( (1,355)(2,356)(3,357)(4,358)(5,359)(6,360)(7,361)(8,362)(9,363)(10,364)(11,365)(12,366)(13,367)(14,368)(15,369)(16,370)(17,371)(18,372)(19,373)(20,374)(21,375)(22,376)(23,377)(24,378)(25,379)(26,380)(27,381)(28,382)(29,383)(30,384)(31,385)(32,386)(33,387)(34,388)(35,389)(36,390)(37,391)(38,392)(39,393)(40,394)(41,395)(42,396)(43,397)(44,398)(45,399)(46,400)(47,401)(48,402)(49,403)(50,404)(51,203)(52,204)(53,205)(54,206)(55,207)(56,208)(57,209)(58,210)(59,211)(60,212)(61,213)(62,214)(63,215)(64,216)(65,217)(66,218)(67,219)(68,220)(69,221)(70,222)(71,223)(72,224)(73,225)(74,226)(75,227)(76,228)(77,229)(78,230)(79,231)(80,232)(81,233)(82,234)(83,235)(84,236)(85,237)(86,238)(87,239)(88,240)(89,241)(90,242)(91,243)(92,244)(93,245)(94,246)(95,247)(96,248)(97,249)(98,250)(99,251)(100,252)(101,253)(102,254)(103,255)(104,256)(105,257)(106,258)(107,259)(108,260)(109,261)(110,262)(111,263)(112,264)(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)(120,272)(121,273)(122,274)(123,275)(124,276)(125,277)(126,278)(127,279)(128,280)(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)(136,288)(137,289)(138,290)(139,291)(140,292)(141,293)(142,294)(143,295)(144,296)(145,297)(146,298)(147,299)(148,300)(149,301)(150,302)(151,303)(152,304)(153,305)(154,306)(155,307)(156,308)(157,309)(158,310)(159,311)(160,312)(161,313)(162,314)(163,315)(164,316)(165,317)(166,318)(167,319)(168,320)(169,321)(170,322)(171,323)(172,324)(173,325)(174,326)(175,327)(176,328)(177,329)(178,330)(179,331)(180,332)(181,333)(182,334)(183,335)(184,336)(185,337)(186,338)(187,339)(188,340)(189,341)(190,342)(191,343)(192,344)(193,345)(194,346)(195,347)(196,348)(197,349)(198,350)(199,351)(200,352)(201,353)(202,354), (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) );

G=PermutationGroup([(1,355),(2,356),(3,357),(4,358),(5,359),(6,360),(7,361),(8,362),(9,363),(10,364),(11,365),(12,366),(13,367),(14,368),(15,369),(16,370),(17,371),(18,372),(19,373),(20,374),(21,375),(22,376),(23,377),(24,378),(25,379),(26,380),(27,381),(28,382),(29,383),(30,384),(31,385),(32,386),(33,387),(34,388),(35,389),(36,390),(37,391),(38,392),(39,393),(40,394),(41,395),(42,396),(43,397),(44,398),(45,399),(46,400),(47,401),(48,402),(49,403),(50,404),(51,203),(52,204),(53,205),(54,206),(55,207),(56,208),(57,209),(58,210),(59,211),(60,212),(61,213),(62,214),(63,215),(64,216),(65,217),(66,218),(67,219),(68,220),(69,221),(70,222),(71,223),(72,224),(73,225),(74,226),(75,227),(76,228),(77,229),(78,230),(79,231),(80,232),(81,233),(82,234),(83,235),(84,236),(85,237),(86,238),(87,239),(88,240),(89,241),(90,242),(91,243),(92,244),(93,245),(94,246),(95,247),(96,248),(97,249),(98,250),(99,251),(100,252),(101,253),(102,254),(103,255),(104,256),(105,257),(106,258),(107,259),(108,260),(109,261),(110,262),(111,263),(112,264),(113,265),(114,266),(115,267),(116,268),(117,269),(118,270),(119,271),(120,272),(121,273),(122,274),(123,275),(124,276),(125,277),(126,278),(127,279),(128,280),(129,281),(130,282),(131,283),(132,284),(133,285),(134,286),(135,287),(136,288),(137,289),(138,290),(139,291),(140,292),(141,293),(142,294),(143,295),(144,296),(145,297),(146,298),(147,299),(148,300),(149,301),(150,302),(151,303),(152,304),(153,305),(154,306),(155,307),(156,308),(157,309),(158,310),(159,311),(160,312),(161,313),(162,314),(163,315),(164,316),(165,317),(166,318),(167,319),(168,320),(169,321),(170,322),(171,323),(172,324),(173,325),(174,326),(175,327),(176,328),(177,329),(178,330),(179,331),(180,332),(181,333),(182,334),(183,335),(184,336),(185,337),(186,338),(187,339),(188,340),(189,341),(190,342),(191,343),(192,344),(193,345),(194,346),(195,347),(196,348),(197,349),(198,350),(199,351),(200,352),(201,353),(202,354)], [(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)])

404 conjugacy classes

class 1 2A2B2C101A···101CV202A···202KN
order1222101···101202···202
size11111···11···1

404 irreducible representations

dim1111
type++
imageC1C2C101C202
kernelC2×C202C202C22C2
# reps13100300

Matrix representation of C2×C202 in GL2(𝔽607) generated by

6060
01
,
4520
033
G:=sub<GL(2,GF(607))| [606,0,0,1],[452,0,0,33] >;

C2×C202 in GAP, Magma, Sage, TeX

C_2\times C_{202}
% in TeX

G:=Group("C2xC202");
// GroupNames label

G:=SmallGroup(404,5);
// by ID

G=gap.SmallGroup(404,5);
# by ID

G:=PCGroup([3,-2,-2,-101]);
// Polycyclic

G:=Group<a,b|a^2=b^202=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C202 in TeX

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