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G = C2×C224order 448 = 26·7

Abelian group of type [2,224]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C224, SmallGroup(448,173)

Series: Derived Chief Lower central Upper central

C1 — C2×C224
C1C2C4C8C16C112C224 — C2×C224
C1 — C2×C224
C1 — C2×C224

Generators and relations for C2×C224
 G = < a,b | a2=b224=1, ab=ba >


Smallest permutation representation of C2×C224
Regular action on 448 points
Generators in S448
(1 370)(2 371)(3 372)(4 373)(5 374)(6 375)(7 376)(8 377)(9 378)(10 379)(11 380)(12 381)(13 382)(14 383)(15 384)(16 385)(17 386)(18 387)(19 388)(20 389)(21 390)(22 391)(23 392)(24 393)(25 394)(26 395)(27 396)(28 397)(29 398)(30 399)(31 400)(32 401)(33 402)(34 403)(35 404)(36 405)(37 406)(38 407)(39 408)(40 409)(41 410)(42 411)(43 412)(44 413)(45 414)(46 415)(47 416)(48 417)(49 418)(50 419)(51 420)(52 421)(53 422)(54 423)(55 424)(56 425)(57 426)(58 427)(59 428)(60 429)(61 430)(62 431)(63 432)(64 433)(65 434)(66 435)(67 436)(68 437)(69 438)(70 439)(71 440)(72 441)(73 442)(74 443)(75 444)(76 445)(77 446)(78 447)(79 448)(80 225)(81 226)(82 227)(83 228)(84 229)(85 230)(86 231)(87 232)(88 233)(89 234)(90 235)(91 236)(92 237)(93 238)(94 239)(95 240)(96 241)(97 242)(98 243)(99 244)(100 245)(101 246)(102 247)(103 248)(104 249)(105 250)(106 251)(107 252)(108 253)(109 254)(110 255)(111 256)(112 257)(113 258)(114 259)(115 260)(116 261)(117 262)(118 263)(119 264)(120 265)(121 266)(122 267)(123 268)(124 269)(125 270)(126 271)(127 272)(128 273)(129 274)(130 275)(131 276)(132 277)(133 278)(134 279)(135 280)(136 281)(137 282)(138 283)(139 284)(140 285)(141 286)(142 287)(143 288)(144 289)(145 290)(146 291)(147 292)(148 293)(149 294)(150 295)(151 296)(152 297)(153 298)(154 299)(155 300)(156 301)(157 302)(158 303)(159 304)(160 305)(161 306)(162 307)(163 308)(164 309)(165 310)(166 311)(167 312)(168 313)(169 314)(170 315)(171 316)(172 317)(173 318)(174 319)(175 320)(176 321)(177 322)(178 323)(179 324)(180 325)(181 326)(182 327)(183 328)(184 329)(185 330)(186 331)(187 332)(188 333)(189 334)(190 335)(191 336)(192 337)(193 338)(194 339)(195 340)(196 341)(197 342)(198 343)(199 344)(200 345)(201 346)(202 347)(203 348)(204 349)(205 350)(206 351)(207 352)(208 353)(209 354)(210 355)(211 356)(212 357)(213 358)(214 359)(215 360)(216 361)(217 362)(218 363)(219 364)(220 365)(221 366)(222 367)(223 368)(224 369)
(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)(225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)

G:=sub<Sym(448)| (1,370)(2,371)(3,372)(4,373)(5,374)(6,375)(7,376)(8,377)(9,378)(10,379)(11,380)(12,381)(13,382)(14,383)(15,384)(16,385)(17,386)(18,387)(19,388)(20,389)(21,390)(22,391)(23,392)(24,393)(25,394)(26,395)(27,396)(28,397)(29,398)(30,399)(31,400)(32,401)(33,402)(34,403)(35,404)(36,405)(37,406)(38,407)(39,408)(40,409)(41,410)(42,411)(43,412)(44,413)(45,414)(46,415)(47,416)(48,417)(49,418)(50,419)(51,420)(52,421)(53,422)(54,423)(55,424)(56,425)(57,426)(58,427)(59,428)(60,429)(61,430)(62,431)(63,432)(64,433)(65,434)(66,435)(67,436)(68,437)(69,438)(70,439)(71,440)(72,441)(73,442)(74,443)(75,444)(76,445)(77,446)(78,447)(79,448)(80,225)(81,226)(82,227)(83,228)(84,229)(85,230)(86,231)(87,232)(88,233)(89,234)(90,235)(91,236)(92,237)(93,238)(94,239)(95,240)(96,241)(97,242)(98,243)(99,244)(100,245)(101,246)(102,247)(103,248)(104,249)(105,250)(106,251)(107,252)(108,253)(109,254)(110,255)(111,256)(112,257)(113,258)(114,259)(115,260)(116,261)(117,262)(118,263)(119,264)(120,265)(121,266)(122,267)(123,268)(124,269)(125,270)(126,271)(127,272)(128,273)(129,274)(130,275)(131,276)(132,277)(133,278)(134,279)(135,280)(136,281)(137,282)(138,283)(139,284)(140,285)(141,286)(142,287)(143,288)(144,289)(145,290)(146,291)(147,292)(148,293)(149,294)(150,295)(151,296)(152,297)(153,298)(154,299)(155,300)(156,301)(157,302)(158,303)(159,304)(160,305)(161,306)(162,307)(163,308)(164,309)(165,310)(166,311)(167,312)(168,313)(169,314)(170,315)(171,316)(172,317)(173,318)(174,319)(175,320)(176,321)(177,322)(178,323)(179,324)(180,325)(181,326)(182,327)(183,328)(184,329)(185,330)(186,331)(187,332)(188,333)(189,334)(190,335)(191,336)(192,337)(193,338)(194,339)(195,340)(196,341)(197,342)(198,343)(199,344)(200,345)(201,346)(202,347)(203,348)(204,349)(205,350)(206,351)(207,352)(208,353)(209,354)(210,355)(211,356)(212,357)(213,358)(214,359)(215,360)(216,361)(217,362)(218,363)(219,364)(220,365)(221,366)(222,367)(223,368)(224,369), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)>;

G:=Group( (1,370)(2,371)(3,372)(4,373)(5,374)(6,375)(7,376)(8,377)(9,378)(10,379)(11,380)(12,381)(13,382)(14,383)(15,384)(16,385)(17,386)(18,387)(19,388)(20,389)(21,390)(22,391)(23,392)(24,393)(25,394)(26,395)(27,396)(28,397)(29,398)(30,399)(31,400)(32,401)(33,402)(34,403)(35,404)(36,405)(37,406)(38,407)(39,408)(40,409)(41,410)(42,411)(43,412)(44,413)(45,414)(46,415)(47,416)(48,417)(49,418)(50,419)(51,420)(52,421)(53,422)(54,423)(55,424)(56,425)(57,426)(58,427)(59,428)(60,429)(61,430)(62,431)(63,432)(64,433)(65,434)(66,435)(67,436)(68,437)(69,438)(70,439)(71,440)(72,441)(73,442)(74,443)(75,444)(76,445)(77,446)(78,447)(79,448)(80,225)(81,226)(82,227)(83,228)(84,229)(85,230)(86,231)(87,232)(88,233)(89,234)(90,235)(91,236)(92,237)(93,238)(94,239)(95,240)(96,241)(97,242)(98,243)(99,244)(100,245)(101,246)(102,247)(103,248)(104,249)(105,250)(106,251)(107,252)(108,253)(109,254)(110,255)(111,256)(112,257)(113,258)(114,259)(115,260)(116,261)(117,262)(118,263)(119,264)(120,265)(121,266)(122,267)(123,268)(124,269)(125,270)(126,271)(127,272)(128,273)(129,274)(130,275)(131,276)(132,277)(133,278)(134,279)(135,280)(136,281)(137,282)(138,283)(139,284)(140,285)(141,286)(142,287)(143,288)(144,289)(145,290)(146,291)(147,292)(148,293)(149,294)(150,295)(151,296)(152,297)(153,298)(154,299)(155,300)(156,301)(157,302)(158,303)(159,304)(160,305)(161,306)(162,307)(163,308)(164,309)(165,310)(166,311)(167,312)(168,313)(169,314)(170,315)(171,316)(172,317)(173,318)(174,319)(175,320)(176,321)(177,322)(178,323)(179,324)(180,325)(181,326)(182,327)(183,328)(184,329)(185,330)(186,331)(187,332)(188,333)(189,334)(190,335)(191,336)(192,337)(193,338)(194,339)(195,340)(196,341)(197,342)(198,343)(199,344)(200,345)(201,346)(202,347)(203,348)(204,349)(205,350)(206,351)(207,352)(208,353)(209,354)(210,355)(211,356)(212,357)(213,358)(214,359)(215,360)(216,361)(217,362)(218,363)(219,364)(220,365)(221,366)(222,367)(223,368)(224,369), (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)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448) );

G=PermutationGroup([(1,370),(2,371),(3,372),(4,373),(5,374),(6,375),(7,376),(8,377),(9,378),(10,379),(11,380),(12,381),(13,382),(14,383),(15,384),(16,385),(17,386),(18,387),(19,388),(20,389),(21,390),(22,391),(23,392),(24,393),(25,394),(26,395),(27,396),(28,397),(29,398),(30,399),(31,400),(32,401),(33,402),(34,403),(35,404),(36,405),(37,406),(38,407),(39,408),(40,409),(41,410),(42,411),(43,412),(44,413),(45,414),(46,415),(47,416),(48,417),(49,418),(50,419),(51,420),(52,421),(53,422),(54,423),(55,424),(56,425),(57,426),(58,427),(59,428),(60,429),(61,430),(62,431),(63,432),(64,433),(65,434),(66,435),(67,436),(68,437),(69,438),(70,439),(71,440),(72,441),(73,442),(74,443),(75,444),(76,445),(77,446),(78,447),(79,448),(80,225),(81,226),(82,227),(83,228),(84,229),(85,230),(86,231),(87,232),(88,233),(89,234),(90,235),(91,236),(92,237),(93,238),(94,239),(95,240),(96,241),(97,242),(98,243),(99,244),(100,245),(101,246),(102,247),(103,248),(104,249),(105,250),(106,251),(107,252),(108,253),(109,254),(110,255),(111,256),(112,257),(113,258),(114,259),(115,260),(116,261),(117,262),(118,263),(119,264),(120,265),(121,266),(122,267),(123,268),(124,269),(125,270),(126,271),(127,272),(128,273),(129,274),(130,275),(131,276),(132,277),(133,278),(134,279),(135,280),(136,281),(137,282),(138,283),(139,284),(140,285),(141,286),(142,287),(143,288),(144,289),(145,290),(146,291),(147,292),(148,293),(149,294),(150,295),(151,296),(152,297),(153,298),(154,299),(155,300),(156,301),(157,302),(158,303),(159,304),(160,305),(161,306),(162,307),(163,308),(164,309),(165,310),(166,311),(167,312),(168,313),(169,314),(170,315),(171,316),(172,317),(173,318),(174,319),(175,320),(176,321),(177,322),(178,323),(179,324),(180,325),(181,326),(182,327),(183,328),(184,329),(185,330),(186,331),(187,332),(188,333),(189,334),(190,335),(191,336),(192,337),(193,338),(194,339),(195,340),(196,341),(197,342),(198,343),(199,344),(200,345),(201,346),(202,347),(203,348),(204,349),(205,350),(206,351),(207,352),(208,353),(209,354),(210,355),(211,356),(212,357),(213,358),(214,359),(215,360),(216,361),(217,362),(218,363),(219,364),(220,365),(221,366),(222,367),(223,368),(224,369)], [(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),(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)])

448 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F8A···8H14A···14R16A···16P28A···28X32A···32AF56A···56AV112A···112CR224A···224GJ
order122244447···78···814···1416···1628···2832···3256···56112···112224···224
size111111111···11···11···11···11···11···11···11···11···1

448 irreducible representations

dim11111111111111111111
type+++
imageC1C2C2C4C4C7C8C8C14C14C16C16C28C28C32C56C56C112C112C224
kernelC2×C224C224C2×C112C112C2×C56C2×C32C56C2×C28C32C2×C16C28C2×C14C16C2×C8C14C8C2×C4C4C22C2
# reps121226441268812123224244848192

Matrix representation of C2×C224 in GL2(𝔽449) generated by

10
0448
,
2050
077
G:=sub<GL(2,GF(449))| [1,0,0,448],[205,0,0,77] >;

C2×C224 in GAP, Magma, Sage, TeX

C_2\times C_{224}
% in TeX

G:=Group("C2xC224");
// GroupNames label

G:=SmallGroup(448,173);
// by ID

G=gap.SmallGroup(448,173);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,80,102,124]);
// Polycyclic

G:=Group<a,b|a^2=b^224=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C224 in TeX

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