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G = C2×C216order 432 = 24·33

Abelian group of type [2,216]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C216, SmallGroup(432,23)

Series: Derived Chief Lower central Upper central

C1 — C2×C216
C1C3C6C18C36C108C216 — C2×C216
C1 — C2×C216
C1 — C2×C216

Generators and relations for C2×C216
 G = < a,b | a2=b216=1, ab=ba >


Smallest permutation representation of C2×C216
Regular action on 432 points
Generators in S432
(1 276)(2 277)(3 278)(4 279)(5 280)(6 281)(7 282)(8 283)(9 284)(10 285)(11 286)(12 287)(13 288)(14 289)(15 290)(16 291)(17 292)(18 293)(19 294)(20 295)(21 296)(22 297)(23 298)(24 299)(25 300)(26 301)(27 302)(28 303)(29 304)(30 305)(31 306)(32 307)(33 308)(34 309)(35 310)(36 311)(37 312)(38 313)(39 314)(40 315)(41 316)(42 317)(43 318)(44 319)(45 320)(46 321)(47 322)(48 323)(49 324)(50 325)(51 326)(52 327)(53 328)(54 329)(55 330)(56 331)(57 332)(58 333)(59 334)(60 335)(61 336)(62 337)(63 338)(64 339)(65 340)(66 341)(67 342)(68 343)(69 344)(70 345)(71 346)(72 347)(73 348)(74 349)(75 350)(76 351)(77 352)(78 353)(79 354)(80 355)(81 356)(82 357)(83 358)(84 359)(85 360)(86 361)(87 362)(88 363)(89 364)(90 365)(91 366)(92 367)(93 368)(94 369)(95 370)(96 371)(97 372)(98 373)(99 374)(100 375)(101 376)(102 377)(103 378)(104 379)(105 380)(106 381)(107 382)(108 383)(109 384)(110 385)(111 386)(112 387)(113 388)(114 389)(115 390)(116 391)(117 392)(118 393)(119 394)(120 395)(121 396)(122 397)(123 398)(124 399)(125 400)(126 401)(127 402)(128 403)(129 404)(130 405)(131 406)(132 407)(133 408)(134 409)(135 410)(136 411)(137 412)(138 413)(139 414)(140 415)(141 416)(142 417)(143 418)(144 419)(145 420)(146 421)(147 422)(148 423)(149 424)(150 425)(151 426)(152 427)(153 428)(154 429)(155 430)(156 431)(157 432)(158 217)(159 218)(160 219)(161 220)(162 221)(163 222)(164 223)(165 224)(166 225)(167 226)(168 227)(169 228)(170 229)(171 230)(172 231)(173 232)(174 233)(175 234)(176 235)(177 236)(178 237)(179 238)(180 239)(181 240)(182 241)(183 242)(184 243)(185 244)(186 245)(187 246)(188 247)(189 248)(190 249)(191 250)(192 251)(193 252)(194 253)(195 254)(196 255)(197 256)(198 257)(199 258)(200 259)(201 260)(202 261)(203 262)(204 263)(205 264)(206 265)(207 266)(208 267)(209 268)(210 269)(211 270)(212 271)(213 272)(214 273)(215 274)(216 275)
(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)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)

G:=sub<Sym(432)| (1,276)(2,277)(3,278)(4,279)(5,280)(6,281)(7,282)(8,283)(9,284)(10,285)(11,286)(12,287)(13,288)(14,289)(15,290)(16,291)(17,292)(18,293)(19,294)(20,295)(21,296)(22,297)(23,298)(24,299)(25,300)(26,301)(27,302)(28,303)(29,304)(30,305)(31,306)(32,307)(33,308)(34,309)(35,310)(36,311)(37,312)(38,313)(39,314)(40,315)(41,316)(42,317)(43,318)(44,319)(45,320)(46,321)(47,322)(48,323)(49,324)(50,325)(51,326)(52,327)(53,328)(54,329)(55,330)(56,331)(57,332)(58,333)(59,334)(60,335)(61,336)(62,337)(63,338)(64,339)(65,340)(66,341)(67,342)(68,343)(69,344)(70,345)(71,346)(72,347)(73,348)(74,349)(75,350)(76,351)(77,352)(78,353)(79,354)(80,355)(81,356)(82,357)(83,358)(84,359)(85,360)(86,361)(87,362)(88,363)(89,364)(90,365)(91,366)(92,367)(93,368)(94,369)(95,370)(96,371)(97,372)(98,373)(99,374)(100,375)(101,376)(102,377)(103,378)(104,379)(105,380)(106,381)(107,382)(108,383)(109,384)(110,385)(111,386)(112,387)(113,388)(114,389)(115,390)(116,391)(117,392)(118,393)(119,394)(120,395)(121,396)(122,397)(123,398)(124,399)(125,400)(126,401)(127,402)(128,403)(129,404)(130,405)(131,406)(132,407)(133,408)(134,409)(135,410)(136,411)(137,412)(138,413)(139,414)(140,415)(141,416)(142,417)(143,418)(144,419)(145,420)(146,421)(147,422)(148,423)(149,424)(150,425)(151,426)(152,427)(153,428)(154,429)(155,430)(156,431)(157,432)(158,217)(159,218)(160,219)(161,220)(162,221)(163,222)(164,223)(165,224)(166,225)(167,226)(168,227)(169,228)(170,229)(171,230)(172,231)(173,232)(174,233)(175,234)(176,235)(177,236)(178,237)(179,238)(180,239)(181,240)(182,241)(183,242)(184,243)(185,244)(186,245)(187,246)(188,247)(189,248)(190,249)(191,250)(192,251)(193,252)(194,253)(195,254)(196,255)(197,256)(198,257)(199,258)(200,259)(201,260)(202,261)(203,262)(204,263)(205,264)(206,265)(207,266)(208,267)(209,268)(210,269)(211,270)(212,271)(213,272)(214,273)(215,274)(216,275), (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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)>;

G:=Group( (1,276)(2,277)(3,278)(4,279)(5,280)(6,281)(7,282)(8,283)(9,284)(10,285)(11,286)(12,287)(13,288)(14,289)(15,290)(16,291)(17,292)(18,293)(19,294)(20,295)(21,296)(22,297)(23,298)(24,299)(25,300)(26,301)(27,302)(28,303)(29,304)(30,305)(31,306)(32,307)(33,308)(34,309)(35,310)(36,311)(37,312)(38,313)(39,314)(40,315)(41,316)(42,317)(43,318)(44,319)(45,320)(46,321)(47,322)(48,323)(49,324)(50,325)(51,326)(52,327)(53,328)(54,329)(55,330)(56,331)(57,332)(58,333)(59,334)(60,335)(61,336)(62,337)(63,338)(64,339)(65,340)(66,341)(67,342)(68,343)(69,344)(70,345)(71,346)(72,347)(73,348)(74,349)(75,350)(76,351)(77,352)(78,353)(79,354)(80,355)(81,356)(82,357)(83,358)(84,359)(85,360)(86,361)(87,362)(88,363)(89,364)(90,365)(91,366)(92,367)(93,368)(94,369)(95,370)(96,371)(97,372)(98,373)(99,374)(100,375)(101,376)(102,377)(103,378)(104,379)(105,380)(106,381)(107,382)(108,383)(109,384)(110,385)(111,386)(112,387)(113,388)(114,389)(115,390)(116,391)(117,392)(118,393)(119,394)(120,395)(121,396)(122,397)(123,398)(124,399)(125,400)(126,401)(127,402)(128,403)(129,404)(130,405)(131,406)(132,407)(133,408)(134,409)(135,410)(136,411)(137,412)(138,413)(139,414)(140,415)(141,416)(142,417)(143,418)(144,419)(145,420)(146,421)(147,422)(148,423)(149,424)(150,425)(151,426)(152,427)(153,428)(154,429)(155,430)(156,431)(157,432)(158,217)(159,218)(160,219)(161,220)(162,221)(163,222)(164,223)(165,224)(166,225)(167,226)(168,227)(169,228)(170,229)(171,230)(172,231)(173,232)(174,233)(175,234)(176,235)(177,236)(178,237)(179,238)(180,239)(181,240)(182,241)(183,242)(184,243)(185,244)(186,245)(187,246)(188,247)(189,248)(190,249)(191,250)(192,251)(193,252)(194,253)(195,254)(196,255)(197,256)(198,257)(199,258)(200,259)(201,260)(202,261)(203,262)(204,263)(205,264)(206,265)(207,266)(208,267)(209,268)(210,269)(211,270)(212,271)(213,272)(214,273)(215,274)(216,275), (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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432) );

G=PermutationGroup([(1,276),(2,277),(3,278),(4,279),(5,280),(6,281),(7,282),(8,283),(9,284),(10,285),(11,286),(12,287),(13,288),(14,289),(15,290),(16,291),(17,292),(18,293),(19,294),(20,295),(21,296),(22,297),(23,298),(24,299),(25,300),(26,301),(27,302),(28,303),(29,304),(30,305),(31,306),(32,307),(33,308),(34,309),(35,310),(36,311),(37,312),(38,313),(39,314),(40,315),(41,316),(42,317),(43,318),(44,319),(45,320),(46,321),(47,322),(48,323),(49,324),(50,325),(51,326),(52,327),(53,328),(54,329),(55,330),(56,331),(57,332),(58,333),(59,334),(60,335),(61,336),(62,337),(63,338),(64,339),(65,340),(66,341),(67,342),(68,343),(69,344),(70,345),(71,346),(72,347),(73,348),(74,349),(75,350),(76,351),(77,352),(78,353),(79,354),(80,355),(81,356),(82,357),(83,358),(84,359),(85,360),(86,361),(87,362),(88,363),(89,364),(90,365),(91,366),(92,367),(93,368),(94,369),(95,370),(96,371),(97,372),(98,373),(99,374),(100,375),(101,376),(102,377),(103,378),(104,379),(105,380),(106,381),(107,382),(108,383),(109,384),(110,385),(111,386),(112,387),(113,388),(114,389),(115,390),(116,391),(117,392),(118,393),(119,394),(120,395),(121,396),(122,397),(123,398),(124,399),(125,400),(126,401),(127,402),(128,403),(129,404),(130,405),(131,406),(132,407),(133,408),(134,409),(135,410),(136,411),(137,412),(138,413),(139,414),(140,415),(141,416),(142,417),(143,418),(144,419),(145,420),(146,421),(147,422),(148,423),(149,424),(150,425),(151,426),(152,427),(153,428),(154,429),(155,430),(156,431),(157,432),(158,217),(159,218),(160,219),(161,220),(162,221),(163,222),(164,223),(165,224),(166,225),(167,226),(168,227),(169,228),(170,229),(171,230),(172,231),(173,232),(174,233),(175,234),(176,235),(177,236),(178,237),(179,238),(180,239),(181,240),(182,241),(183,242),(184,243),(185,244),(186,245),(187,246),(188,247),(189,248),(190,249),(191,250),(192,251),(193,252),(194,253),(195,254),(196,255),(197,256),(198,257),(199,258),(200,259),(201,260),(202,261),(203,262),(204,263),(205,264),(206,265),(207,266),(208,267),(209,268),(210,269),(211,270),(212,271),(213,272),(214,273),(215,274),(216,275)], [(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),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)])

432 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F8A···8H9A···9F12A···12H18A···18R24A···24P27A···27R36A···36X54A···54BB72A···72AV108A···108BT216A···216EN
order12223344446···68···89···912···1218···1824···2427···2736···3654···5472···72108···108216···216
size11111111111···11···11···11···11···11···11···11···11···11···11···11···1

432 irreducible representations

dim111111111111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C9C12C12C18C18C24C27C36C36C54C54C72C108C108C216
kernelC2×C216C216C2×C108C2×C72C108C2×C54C72C2×C36C54C2×C24C36C2×C18C24C2×C12C18C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps121222428644126161812123618483636144

Matrix representation of C2×C216 in GL2(𝔽433) generated by

10
0432
,
990
0218
G:=sub<GL(2,GF(433))| [1,0,0,432],[99,0,0,218] >;

C2×C216 in GAP, Magma, Sage, TeX

C_2\times C_{216}
% in TeX

G:=Group("C2xC216");
// GroupNames label

G:=SmallGroup(432,23);
// by ID

G=gap.SmallGroup(432,23);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,142,192,242]);
// Polycyclic

G:=Group<a,b|a^2=b^216=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C216 in TeX

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