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G = C2×C204order 408 = 23·3·17

Abelian group of type [2,204]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C204, SmallGroup(408,30)

Series: Derived Chief Lower central Upper central

C1 — C2×C204
C1C2C34C102C204 — C2×C204
C1 — C2×C204
C1 — C2×C204

Generators and relations for C2×C204
 G = < a,b | a2=b204=1, ab=ba >


Smallest permutation representation of C2×C204
Regular action on 408 points
Generators in S408
(1 318)(2 319)(3 320)(4 321)(5 322)(6 323)(7 324)(8 325)(9 326)(10 327)(11 328)(12 329)(13 330)(14 331)(15 332)(16 333)(17 334)(18 335)(19 336)(20 337)(21 338)(22 339)(23 340)(24 341)(25 342)(26 343)(27 344)(28 345)(29 346)(30 347)(31 348)(32 349)(33 350)(34 351)(35 352)(36 353)(37 354)(38 355)(39 356)(40 357)(41 358)(42 359)(43 360)(44 361)(45 362)(46 363)(47 364)(48 365)(49 366)(50 367)(51 368)(52 369)(53 370)(54 371)(55 372)(56 373)(57 374)(58 375)(59 376)(60 377)(61 378)(62 379)(63 380)(64 381)(65 382)(66 383)(67 384)(68 385)(69 386)(70 387)(71 388)(72 389)(73 390)(74 391)(75 392)(76 393)(77 394)(78 395)(79 396)(80 397)(81 398)(82 399)(83 400)(84 401)(85 402)(86 403)(87 404)(88 405)(89 406)(90 407)(91 408)(92 205)(93 206)(94 207)(95 208)(96 209)(97 210)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 217)(105 218)(106 219)(107 220)(108 221)(109 222)(110 223)(111 224)(112 225)(113 226)(114 227)(115 228)(116 229)(117 230)(118 231)(119 232)(120 233)(121 234)(122 235)(123 236)(124 237)(125 238)(126 239)(127 240)(128 241)(129 242)(130 243)(131 244)(132 245)(133 246)(134 247)(135 248)(136 249)(137 250)(138 251)(139 252)(140 253)(141 254)(142 255)(143 256)(144 257)(145 258)(146 259)(147 260)(148 261)(149 262)(150 263)(151 264)(152 265)(153 266)(154 267)(155 268)(156 269)(157 270)(158 271)(159 272)(160 273)(161 274)(162 275)(163 276)(164 277)(165 278)(166 279)(167 280)(168 281)(169 282)(170 283)(171 284)(172 285)(173 286)(174 287)(175 288)(176 289)(177 290)(178 291)(179 292)(180 293)(181 294)(182 295)(183 296)(184 297)(185 298)(186 299)(187 300)(188 301)(189 302)(190 303)(191 304)(192 305)(193 306)(194 307)(195 308)(196 309)(197 310)(198 311)(199 312)(200 313)(201 314)(202 315)(203 316)(204 317)
(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)

G:=sub<Sym(408)| (1,318)(2,319)(3,320)(4,321)(5,322)(6,323)(7,324)(8,325)(9,326)(10,327)(11,328)(12,329)(13,330)(14,331)(15,332)(16,333)(17,334)(18,335)(19,336)(20,337)(21,338)(22,339)(23,340)(24,341)(25,342)(26,343)(27,344)(28,345)(29,346)(30,347)(31,348)(32,349)(33,350)(34,351)(35,352)(36,353)(37,354)(38,355)(39,356)(40,357)(41,358)(42,359)(43,360)(44,361)(45,362)(46,363)(47,364)(48,365)(49,366)(50,367)(51,368)(52,369)(53,370)(54,371)(55,372)(56,373)(57,374)(58,375)(59,376)(60,377)(61,378)(62,379)(63,380)(64,381)(65,382)(66,383)(67,384)(68,385)(69,386)(70,387)(71,388)(72,389)(73,390)(74,391)(75,392)(76,393)(77,394)(78,395)(79,396)(80,397)(81,398)(82,399)(83,400)(84,401)(85,402)(86,403)(87,404)(88,405)(89,406)(90,407)(91,408)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,220)(108,221)(109,222)(110,223)(111,224)(112,225)(113,226)(114,227)(115,228)(116,229)(117,230)(118,231)(119,232)(120,233)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,241)(129,242)(130,243)(131,244)(132,245)(133,246)(134,247)(135,248)(136,249)(137,250)(138,251)(139,252)(140,253)(141,254)(142,255)(143,256)(144,257)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,265)(153,266)(154,267)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,281)(169,282)(170,283)(171,284)(172,285)(173,286)(174,287)(175,288)(176,289)(177,290)(178,291)(179,292)(180,293)(181,294)(182,295)(183,296)(184,297)(185,298)(186,299)(187,300)(188,301)(189,302)(190,303)(191,304)(192,305)(193,306)(194,307)(195,308)(196,309)(197,310)(198,311)(199,312)(200,313)(201,314)(202,315)(203,316)(204,317), (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)>;

G:=Group( (1,318)(2,319)(3,320)(4,321)(5,322)(6,323)(7,324)(8,325)(9,326)(10,327)(11,328)(12,329)(13,330)(14,331)(15,332)(16,333)(17,334)(18,335)(19,336)(20,337)(21,338)(22,339)(23,340)(24,341)(25,342)(26,343)(27,344)(28,345)(29,346)(30,347)(31,348)(32,349)(33,350)(34,351)(35,352)(36,353)(37,354)(38,355)(39,356)(40,357)(41,358)(42,359)(43,360)(44,361)(45,362)(46,363)(47,364)(48,365)(49,366)(50,367)(51,368)(52,369)(53,370)(54,371)(55,372)(56,373)(57,374)(58,375)(59,376)(60,377)(61,378)(62,379)(63,380)(64,381)(65,382)(66,383)(67,384)(68,385)(69,386)(70,387)(71,388)(72,389)(73,390)(74,391)(75,392)(76,393)(77,394)(78,395)(79,396)(80,397)(81,398)(82,399)(83,400)(84,401)(85,402)(86,403)(87,404)(88,405)(89,406)(90,407)(91,408)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,220)(108,221)(109,222)(110,223)(111,224)(112,225)(113,226)(114,227)(115,228)(116,229)(117,230)(118,231)(119,232)(120,233)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,241)(129,242)(130,243)(131,244)(132,245)(133,246)(134,247)(135,248)(136,249)(137,250)(138,251)(139,252)(140,253)(141,254)(142,255)(143,256)(144,257)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,265)(153,266)(154,267)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,281)(169,282)(170,283)(171,284)(172,285)(173,286)(174,287)(175,288)(176,289)(177,290)(178,291)(179,292)(180,293)(181,294)(182,295)(183,296)(184,297)(185,298)(186,299)(187,300)(188,301)(189,302)(190,303)(191,304)(192,305)(193,306)(194,307)(195,308)(196,309)(197,310)(198,311)(199,312)(200,313)(201,314)(202,315)(203,316)(204,317), (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) );

G=PermutationGroup([(1,318),(2,319),(3,320),(4,321),(5,322),(6,323),(7,324),(8,325),(9,326),(10,327),(11,328),(12,329),(13,330),(14,331),(15,332),(16,333),(17,334),(18,335),(19,336),(20,337),(21,338),(22,339),(23,340),(24,341),(25,342),(26,343),(27,344),(28,345),(29,346),(30,347),(31,348),(32,349),(33,350),(34,351),(35,352),(36,353),(37,354),(38,355),(39,356),(40,357),(41,358),(42,359),(43,360),(44,361),(45,362),(46,363),(47,364),(48,365),(49,366),(50,367),(51,368),(52,369),(53,370),(54,371),(55,372),(56,373),(57,374),(58,375),(59,376),(60,377),(61,378),(62,379),(63,380),(64,381),(65,382),(66,383),(67,384),(68,385),(69,386),(70,387),(71,388),(72,389),(73,390),(74,391),(75,392),(76,393),(77,394),(78,395),(79,396),(80,397),(81,398),(82,399),(83,400),(84,401),(85,402),(86,403),(87,404),(88,405),(89,406),(90,407),(91,408),(92,205),(93,206),(94,207),(95,208),(96,209),(97,210),(98,211),(99,212),(100,213),(101,214),(102,215),(103,216),(104,217),(105,218),(106,219),(107,220),(108,221),(109,222),(110,223),(111,224),(112,225),(113,226),(114,227),(115,228),(116,229),(117,230),(118,231),(119,232),(120,233),(121,234),(122,235),(123,236),(124,237),(125,238),(126,239),(127,240),(128,241),(129,242),(130,243),(131,244),(132,245),(133,246),(134,247),(135,248),(136,249),(137,250),(138,251),(139,252),(140,253),(141,254),(142,255),(143,256),(144,257),(145,258),(146,259),(147,260),(148,261),(149,262),(150,263),(151,264),(152,265),(153,266),(154,267),(155,268),(156,269),(157,270),(158,271),(159,272),(160,273),(161,274),(162,275),(163,276),(164,277),(165,278),(166,279),(167,280),(168,281),(169,282),(170,283),(171,284),(172,285),(173,286),(174,287),(175,288),(176,289),(177,290),(178,291),(179,292),(180,293),(181,294),(182,295),(183,296),(184,297),(185,298),(186,299),(187,300),(188,301),(189,302),(190,303),(191,304),(192,305),(193,306),(194,307),(195,308),(196,309),(197,310),(198,311),(199,312),(200,313),(201,314),(202,315),(203,316),(204,317)], [(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)])

408 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F12A···12H17A···17P34A···34AV51A···51AF68A···68BL102A···102CR204A···204DX
order12223344446···612···1217···1734···3451···5168···68102···102204···204
size11111111111···11···11···11···11···11···11···11···1

408 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C6C6C12C17C34C34C51C68C102C102C204
kernelC2×C204C204C2×C102C2×C68C102C68C2×C34C34C2×C12C12C2×C6C2×C4C6C4C22C2
# reps1212442816321632646432128

Matrix representation of C2×C204 in GL2(𝔽409) generated by

10
0408
,
2810
096
G:=sub<GL(2,GF(409))| [1,0,0,408],[281,0,0,96] >;

C2×C204 in GAP, Magma, Sage, TeX

C_2\times C_{204}
% in TeX

G:=Group("C2xC204");
// GroupNames label

G:=SmallGroup(408,30);
// by ID

G=gap.SmallGroup(408,30);
# by ID

G:=PCGroup([5,-2,-2,-3,-17,-2,1020]);
// Polycyclic

G:=Group<a,b|a^2=b^204=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C204 in TeX

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