Copied to
clipboard

G = C3×C144order 432 = 24·33

Abelian group of type [3,144]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C144, SmallGroup(432,34)

Series: Derived Chief Lower central Upper central

C1 — C3×C144
C1C2C4C12C24C3×C24C3×C72 — C3×C144
C1 — C3×C144
C1 — C3×C144

Generators and relations for C3×C144
 G = < a,b | a3=b144=1, ab=ba >


Smallest permutation representation of C3×C144
Regular action on 432 points
Generators in S432
(1 388 171)(2 389 172)(3 390 173)(4 391 174)(5 392 175)(6 393 176)(7 394 177)(8 395 178)(9 396 179)(10 397 180)(11 398 181)(12 399 182)(13 400 183)(14 401 184)(15 402 185)(16 403 186)(17 404 187)(18 405 188)(19 406 189)(20 407 190)(21 408 191)(22 409 192)(23 410 193)(24 411 194)(25 412 195)(26 413 196)(27 414 197)(28 415 198)(29 416 199)(30 417 200)(31 418 201)(32 419 202)(33 420 203)(34 421 204)(35 422 205)(36 423 206)(37 424 207)(38 425 208)(39 426 209)(40 427 210)(41 428 211)(42 429 212)(43 430 213)(44 431 214)(45 432 215)(46 289 216)(47 290 217)(48 291 218)(49 292 219)(50 293 220)(51 294 221)(52 295 222)(53 296 223)(54 297 224)(55 298 225)(56 299 226)(57 300 227)(58 301 228)(59 302 229)(60 303 230)(61 304 231)(62 305 232)(63 306 233)(64 307 234)(65 308 235)(66 309 236)(67 310 237)(68 311 238)(69 312 239)(70 313 240)(71 314 241)(72 315 242)(73 316 243)(74 317 244)(75 318 245)(76 319 246)(77 320 247)(78 321 248)(79 322 249)(80 323 250)(81 324 251)(82 325 252)(83 326 253)(84 327 254)(85 328 255)(86 329 256)(87 330 257)(88 331 258)(89 332 259)(90 333 260)(91 334 261)(92 335 262)(93 336 263)(94 337 264)(95 338 265)(96 339 266)(97 340 267)(98 341 268)(99 342 269)(100 343 270)(101 344 271)(102 345 272)(103 346 273)(104 347 274)(105 348 275)(106 349 276)(107 350 277)(108 351 278)(109 352 279)(110 353 280)(111 354 281)(112 355 282)(113 356 283)(114 357 284)(115 358 285)(116 359 286)(117 360 287)(118 361 288)(119 362 145)(120 363 146)(121 364 147)(122 365 148)(123 366 149)(124 367 150)(125 368 151)(126 369 152)(127 370 153)(128 371 154)(129 372 155)(130 373 156)(131 374 157)(132 375 158)(133 376 159)(134 377 160)(135 378 161)(136 379 162)(137 380 163)(138 381 164)(139 382 165)(140 383 166)(141 384 167)(142 385 168)(143 386 169)(144 387 170)
(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)| (1,388,171)(2,389,172)(3,390,173)(4,391,174)(5,392,175)(6,393,176)(7,394,177)(8,395,178)(9,396,179)(10,397,180)(11,398,181)(12,399,182)(13,400,183)(14,401,184)(15,402,185)(16,403,186)(17,404,187)(18,405,188)(19,406,189)(20,407,190)(21,408,191)(22,409,192)(23,410,193)(24,411,194)(25,412,195)(26,413,196)(27,414,197)(28,415,198)(29,416,199)(30,417,200)(31,418,201)(32,419,202)(33,420,203)(34,421,204)(35,422,205)(36,423,206)(37,424,207)(38,425,208)(39,426,209)(40,427,210)(41,428,211)(42,429,212)(43,430,213)(44,431,214)(45,432,215)(46,289,216)(47,290,217)(48,291,218)(49,292,219)(50,293,220)(51,294,221)(52,295,222)(53,296,223)(54,297,224)(55,298,225)(56,299,226)(57,300,227)(58,301,228)(59,302,229)(60,303,230)(61,304,231)(62,305,232)(63,306,233)(64,307,234)(65,308,235)(66,309,236)(67,310,237)(68,311,238)(69,312,239)(70,313,240)(71,314,241)(72,315,242)(73,316,243)(74,317,244)(75,318,245)(76,319,246)(77,320,247)(78,321,248)(79,322,249)(80,323,250)(81,324,251)(82,325,252)(83,326,253)(84,327,254)(85,328,255)(86,329,256)(87,330,257)(88,331,258)(89,332,259)(90,333,260)(91,334,261)(92,335,262)(93,336,263)(94,337,264)(95,338,265)(96,339,266)(97,340,267)(98,341,268)(99,342,269)(100,343,270)(101,344,271)(102,345,272)(103,346,273)(104,347,274)(105,348,275)(106,349,276)(107,350,277)(108,351,278)(109,352,279)(110,353,280)(111,354,281)(112,355,282)(113,356,283)(114,357,284)(115,358,285)(116,359,286)(117,360,287)(118,361,288)(119,362,145)(120,363,146)(121,364,147)(122,365,148)(123,366,149)(124,367,150)(125,368,151)(126,369,152)(127,370,153)(128,371,154)(129,372,155)(130,373,156)(131,374,157)(132,375,158)(133,376,159)(134,377,160)(135,378,161)(136,379,162)(137,380,163)(138,381,164)(139,382,165)(140,383,166)(141,384,167)(142,385,168)(143,386,169)(144,387,170), (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( (1,388,171)(2,389,172)(3,390,173)(4,391,174)(5,392,175)(6,393,176)(7,394,177)(8,395,178)(9,396,179)(10,397,180)(11,398,181)(12,399,182)(13,400,183)(14,401,184)(15,402,185)(16,403,186)(17,404,187)(18,405,188)(19,406,189)(20,407,190)(21,408,191)(22,409,192)(23,410,193)(24,411,194)(25,412,195)(26,413,196)(27,414,197)(28,415,198)(29,416,199)(30,417,200)(31,418,201)(32,419,202)(33,420,203)(34,421,204)(35,422,205)(36,423,206)(37,424,207)(38,425,208)(39,426,209)(40,427,210)(41,428,211)(42,429,212)(43,430,213)(44,431,214)(45,432,215)(46,289,216)(47,290,217)(48,291,218)(49,292,219)(50,293,220)(51,294,221)(52,295,222)(53,296,223)(54,297,224)(55,298,225)(56,299,226)(57,300,227)(58,301,228)(59,302,229)(60,303,230)(61,304,231)(62,305,232)(63,306,233)(64,307,234)(65,308,235)(66,309,236)(67,310,237)(68,311,238)(69,312,239)(70,313,240)(71,314,241)(72,315,242)(73,316,243)(74,317,244)(75,318,245)(76,319,246)(77,320,247)(78,321,248)(79,322,249)(80,323,250)(81,324,251)(82,325,252)(83,326,253)(84,327,254)(85,328,255)(86,329,256)(87,330,257)(88,331,258)(89,332,259)(90,333,260)(91,334,261)(92,335,262)(93,336,263)(94,337,264)(95,338,265)(96,339,266)(97,340,267)(98,341,268)(99,342,269)(100,343,270)(101,344,271)(102,345,272)(103,346,273)(104,347,274)(105,348,275)(106,349,276)(107,350,277)(108,351,278)(109,352,279)(110,353,280)(111,354,281)(112,355,282)(113,356,283)(114,357,284)(115,358,285)(116,359,286)(117,360,287)(118,361,288)(119,362,145)(120,363,146)(121,364,147)(122,365,148)(123,366,149)(124,367,150)(125,368,151)(126,369,152)(127,370,153)(128,371,154)(129,372,155)(130,373,156)(131,374,157)(132,375,158)(133,376,159)(134,377,160)(135,378,161)(136,379,162)(137,380,163)(138,381,164)(139,382,165)(140,383,166)(141,384,167)(142,385,168)(143,386,169)(144,387,170), (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([(1,388,171),(2,389,172),(3,390,173),(4,391,174),(5,392,175),(6,393,176),(7,394,177),(8,395,178),(9,396,179),(10,397,180),(11,398,181),(12,399,182),(13,400,183),(14,401,184),(15,402,185),(16,403,186),(17,404,187),(18,405,188),(19,406,189),(20,407,190),(21,408,191),(22,409,192),(23,410,193),(24,411,194),(25,412,195),(26,413,196),(27,414,197),(28,415,198),(29,416,199),(30,417,200),(31,418,201),(32,419,202),(33,420,203),(34,421,204),(35,422,205),(36,423,206),(37,424,207),(38,425,208),(39,426,209),(40,427,210),(41,428,211),(42,429,212),(43,430,213),(44,431,214),(45,432,215),(46,289,216),(47,290,217),(48,291,218),(49,292,219),(50,293,220),(51,294,221),(52,295,222),(53,296,223),(54,297,224),(55,298,225),(56,299,226),(57,300,227),(58,301,228),(59,302,229),(60,303,230),(61,304,231),(62,305,232),(63,306,233),(64,307,234),(65,308,235),(66,309,236),(67,310,237),(68,311,238),(69,312,239),(70,313,240),(71,314,241),(72,315,242),(73,316,243),(74,317,244),(75,318,245),(76,319,246),(77,320,247),(78,321,248),(79,322,249),(80,323,250),(81,324,251),(82,325,252),(83,326,253),(84,327,254),(85,328,255),(86,329,256),(87,330,257),(88,331,258),(89,332,259),(90,333,260),(91,334,261),(92,335,262),(93,336,263),(94,337,264),(95,338,265),(96,339,266),(97,340,267),(98,341,268),(99,342,269),(100,343,270),(101,344,271),(102,345,272),(103,346,273),(104,347,274),(105,348,275),(106,349,276),(107,350,277),(108,351,278),(109,352,279),(110,353,280),(111,354,281),(112,355,282),(113,356,283),(114,357,284),(115,358,285),(116,359,286),(117,360,287),(118,361,288),(119,362,145),(120,363,146),(121,364,147),(122,365,148),(123,366,149),(124,367,150),(125,368,151),(126,369,152),(127,370,153),(128,371,154),(129,372,155),(130,373,156),(131,374,157),(132,375,158),(133,376,159),(134,377,160),(135,378,161),(136,379,162),(137,380,163),(138,381,164),(139,382,165),(140,383,166),(141,384,167),(142,385,168),(143,386,169),(144,387,170)], [(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  2 3A···3H4A4B6A···6H8A8B8C8D9A···9R12A···12P16A···16H18A···18R24A···24AF36A···36AJ48A···48BL72A···72BT144A···144EN
order123···3446···688889···912···1216···1618···1824···2436···3648···4872···72144···144
size111···1111···111111···11···11···11···11···11···11···11···11···1

432 irreducible representations

dim11111111111111111111
type++
imageC1C2C3C3C4C6C6C8C9C12C12C16C18C24C24C36C48C48C72C144
kernelC3×C144C3×C72C144C3×C48C3×C36C72C3×C24C3×C18C48C36C3×C12C3×C9C24C18C3×C6C12C9C32C6C3
# reps116226241812481824836481672144

Matrix representation of C3×C144 in GL2(𝔽433) generated by

1980
0234
,
1940
0300
G:=sub<GL(2,GF(433))| [198,0,0,234],[194,0,0,300] >;

C3×C144 in GAP, Magma, Sage, TeX

C_3\times C_{144}
% in TeX

G:=Group("C3xC144");
// GroupNames label

G:=SmallGroup(432,34);
// by ID

G=gap.SmallGroup(432,34);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-3,-2,-2,126,260,192,124]);
// Polycyclic

G:=Group<a,b|a^3=b^144=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C144 in TeX

׿
×
𝔽