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G = C3×C153order 459 = 33·17

Abelian group of type [3,153]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C153, SmallGroup(459,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C153
C1C3C51C153 — C3×C153
C1 — C3×C153
C1 — C3×C153

Generators and relations for C3×C153
 G = < a,b | a3=b153=1, ab=ba >


Smallest permutation representation of C3×C153
Regular action on 459 points
Generators in S459
(1 315 286)(2 316 287)(3 317 288)(4 318 289)(5 319 290)(6 320 291)(7 321 292)(8 322 293)(9 323 294)(10 324 295)(11 325 296)(12 326 297)(13 327 298)(14 328 299)(15 329 300)(16 330 301)(17 331 302)(18 332 303)(19 333 304)(20 334 305)(21 335 306)(22 336 154)(23 337 155)(24 338 156)(25 339 157)(26 340 158)(27 341 159)(28 342 160)(29 343 161)(30 344 162)(31 345 163)(32 346 164)(33 347 165)(34 348 166)(35 349 167)(36 350 168)(37 351 169)(38 352 170)(39 353 171)(40 354 172)(41 355 173)(42 356 174)(43 357 175)(44 358 176)(45 359 177)(46 360 178)(47 361 179)(48 362 180)(49 363 181)(50 364 182)(51 365 183)(52 366 184)(53 367 185)(54 368 186)(55 369 187)(56 370 188)(57 371 189)(58 372 190)(59 373 191)(60 374 192)(61 375 193)(62 376 194)(63 377 195)(64 378 196)(65 379 197)(66 380 198)(67 381 199)(68 382 200)(69 383 201)(70 384 202)(71 385 203)(72 386 204)(73 387 205)(74 388 206)(75 389 207)(76 390 208)(77 391 209)(78 392 210)(79 393 211)(80 394 212)(81 395 213)(82 396 214)(83 397 215)(84 398 216)(85 399 217)(86 400 218)(87 401 219)(88 402 220)(89 403 221)(90 404 222)(91 405 223)(92 406 224)(93 407 225)(94 408 226)(95 409 227)(96 410 228)(97 411 229)(98 412 230)(99 413 231)(100 414 232)(101 415 233)(102 416 234)(103 417 235)(104 418 236)(105 419 237)(106 420 238)(107 421 239)(108 422 240)(109 423 241)(110 424 242)(111 425 243)(112 426 244)(113 427 245)(114 428 246)(115 429 247)(116 430 248)(117 431 249)(118 432 250)(119 433 251)(120 434 252)(121 435 253)(122 436 254)(123 437 255)(124 438 256)(125 439 257)(126 440 258)(127 441 259)(128 442 260)(129 443 261)(130 444 262)(131 445 263)(132 446 264)(133 447 265)(134 448 266)(135 449 267)(136 450 268)(137 451 269)(138 452 270)(139 453 271)(140 454 272)(141 455 273)(142 456 274)(143 457 275)(144 458 276)(145 459 277)(146 307 278)(147 308 279)(148 309 280)(149 310 281)(150 311 282)(151 312 283)(152 313 284)(153 314 285)
(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)(307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459)

G:=sub<Sym(459)| (1,315,286)(2,316,287)(3,317,288)(4,318,289)(5,319,290)(6,320,291)(7,321,292)(8,322,293)(9,323,294)(10,324,295)(11,325,296)(12,326,297)(13,327,298)(14,328,299)(15,329,300)(16,330,301)(17,331,302)(18,332,303)(19,333,304)(20,334,305)(21,335,306)(22,336,154)(23,337,155)(24,338,156)(25,339,157)(26,340,158)(27,341,159)(28,342,160)(29,343,161)(30,344,162)(31,345,163)(32,346,164)(33,347,165)(34,348,166)(35,349,167)(36,350,168)(37,351,169)(38,352,170)(39,353,171)(40,354,172)(41,355,173)(42,356,174)(43,357,175)(44,358,176)(45,359,177)(46,360,178)(47,361,179)(48,362,180)(49,363,181)(50,364,182)(51,365,183)(52,366,184)(53,367,185)(54,368,186)(55,369,187)(56,370,188)(57,371,189)(58,372,190)(59,373,191)(60,374,192)(61,375,193)(62,376,194)(63,377,195)(64,378,196)(65,379,197)(66,380,198)(67,381,199)(68,382,200)(69,383,201)(70,384,202)(71,385,203)(72,386,204)(73,387,205)(74,388,206)(75,389,207)(76,390,208)(77,391,209)(78,392,210)(79,393,211)(80,394,212)(81,395,213)(82,396,214)(83,397,215)(84,398,216)(85,399,217)(86,400,218)(87,401,219)(88,402,220)(89,403,221)(90,404,222)(91,405,223)(92,406,224)(93,407,225)(94,408,226)(95,409,227)(96,410,228)(97,411,229)(98,412,230)(99,413,231)(100,414,232)(101,415,233)(102,416,234)(103,417,235)(104,418,236)(105,419,237)(106,420,238)(107,421,239)(108,422,240)(109,423,241)(110,424,242)(111,425,243)(112,426,244)(113,427,245)(114,428,246)(115,429,247)(116,430,248)(117,431,249)(118,432,250)(119,433,251)(120,434,252)(121,435,253)(122,436,254)(123,437,255)(124,438,256)(125,439,257)(126,440,258)(127,441,259)(128,442,260)(129,443,261)(130,444,262)(131,445,263)(132,446,264)(133,447,265)(134,448,266)(135,449,267)(136,450,268)(137,451,269)(138,452,270)(139,453,271)(140,454,272)(141,455,273)(142,456,274)(143,457,275)(144,458,276)(145,459,277)(146,307,278)(147,308,279)(148,309,280)(149,310,281)(150,311,282)(151,312,283)(152,313,284)(153,314,285), (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)(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459)>;

G:=Group( (1,315,286)(2,316,287)(3,317,288)(4,318,289)(5,319,290)(6,320,291)(7,321,292)(8,322,293)(9,323,294)(10,324,295)(11,325,296)(12,326,297)(13,327,298)(14,328,299)(15,329,300)(16,330,301)(17,331,302)(18,332,303)(19,333,304)(20,334,305)(21,335,306)(22,336,154)(23,337,155)(24,338,156)(25,339,157)(26,340,158)(27,341,159)(28,342,160)(29,343,161)(30,344,162)(31,345,163)(32,346,164)(33,347,165)(34,348,166)(35,349,167)(36,350,168)(37,351,169)(38,352,170)(39,353,171)(40,354,172)(41,355,173)(42,356,174)(43,357,175)(44,358,176)(45,359,177)(46,360,178)(47,361,179)(48,362,180)(49,363,181)(50,364,182)(51,365,183)(52,366,184)(53,367,185)(54,368,186)(55,369,187)(56,370,188)(57,371,189)(58,372,190)(59,373,191)(60,374,192)(61,375,193)(62,376,194)(63,377,195)(64,378,196)(65,379,197)(66,380,198)(67,381,199)(68,382,200)(69,383,201)(70,384,202)(71,385,203)(72,386,204)(73,387,205)(74,388,206)(75,389,207)(76,390,208)(77,391,209)(78,392,210)(79,393,211)(80,394,212)(81,395,213)(82,396,214)(83,397,215)(84,398,216)(85,399,217)(86,400,218)(87,401,219)(88,402,220)(89,403,221)(90,404,222)(91,405,223)(92,406,224)(93,407,225)(94,408,226)(95,409,227)(96,410,228)(97,411,229)(98,412,230)(99,413,231)(100,414,232)(101,415,233)(102,416,234)(103,417,235)(104,418,236)(105,419,237)(106,420,238)(107,421,239)(108,422,240)(109,423,241)(110,424,242)(111,425,243)(112,426,244)(113,427,245)(114,428,246)(115,429,247)(116,430,248)(117,431,249)(118,432,250)(119,433,251)(120,434,252)(121,435,253)(122,436,254)(123,437,255)(124,438,256)(125,439,257)(126,440,258)(127,441,259)(128,442,260)(129,443,261)(130,444,262)(131,445,263)(132,446,264)(133,447,265)(134,448,266)(135,449,267)(136,450,268)(137,451,269)(138,452,270)(139,453,271)(140,454,272)(141,455,273)(142,456,274)(143,457,275)(144,458,276)(145,459,277)(146,307,278)(147,308,279)(148,309,280)(149,310,281)(150,311,282)(151,312,283)(152,313,284)(153,314,285), (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)(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459) );

G=PermutationGroup([(1,315,286),(2,316,287),(3,317,288),(4,318,289),(5,319,290),(6,320,291),(7,321,292),(8,322,293),(9,323,294),(10,324,295),(11,325,296),(12,326,297),(13,327,298),(14,328,299),(15,329,300),(16,330,301),(17,331,302),(18,332,303),(19,333,304),(20,334,305),(21,335,306),(22,336,154),(23,337,155),(24,338,156),(25,339,157),(26,340,158),(27,341,159),(28,342,160),(29,343,161),(30,344,162),(31,345,163),(32,346,164),(33,347,165),(34,348,166),(35,349,167),(36,350,168),(37,351,169),(38,352,170),(39,353,171),(40,354,172),(41,355,173),(42,356,174),(43,357,175),(44,358,176),(45,359,177),(46,360,178),(47,361,179),(48,362,180),(49,363,181),(50,364,182),(51,365,183),(52,366,184),(53,367,185),(54,368,186),(55,369,187),(56,370,188),(57,371,189),(58,372,190),(59,373,191),(60,374,192),(61,375,193),(62,376,194),(63,377,195),(64,378,196),(65,379,197),(66,380,198),(67,381,199),(68,382,200),(69,383,201),(70,384,202),(71,385,203),(72,386,204),(73,387,205),(74,388,206),(75,389,207),(76,390,208),(77,391,209),(78,392,210),(79,393,211),(80,394,212),(81,395,213),(82,396,214),(83,397,215),(84,398,216),(85,399,217),(86,400,218),(87,401,219),(88,402,220),(89,403,221),(90,404,222),(91,405,223),(92,406,224),(93,407,225),(94,408,226),(95,409,227),(96,410,228),(97,411,229),(98,412,230),(99,413,231),(100,414,232),(101,415,233),(102,416,234),(103,417,235),(104,418,236),(105,419,237),(106,420,238),(107,421,239),(108,422,240),(109,423,241),(110,424,242),(111,425,243),(112,426,244),(113,427,245),(114,428,246),(115,429,247),(116,430,248),(117,431,249),(118,432,250),(119,433,251),(120,434,252),(121,435,253),(122,436,254),(123,437,255),(124,438,256),(125,439,257),(126,440,258),(127,441,259),(128,442,260),(129,443,261),(130,444,262),(131,445,263),(132,446,264),(133,447,265),(134,448,266),(135,449,267),(136,450,268),(137,451,269),(138,452,270),(139,453,271),(140,454,272),(141,455,273),(142,456,274),(143,457,275),(144,458,276),(145,459,277),(146,307,278),(147,308,279),(148,309,280),(149,310,281),(150,311,282),(151,312,283),(152,313,284),(153,314,285)], [(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),(307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459)])

459 conjugacy classes

class 1 3A···3H9A···9R17A···17P51A···51DX153A···153KB
order13···39···917···1751···51153···153
size11···11···11···11···11···1

459 irreducible representations

dim11111111
type+
imageC1C3C3C9C17C51C51C153
kernelC3×C153C153C3×C51C51C3×C9C9C32C3
# reps16218169632288

Matrix representation of C3×C153 in GL2(𝔽307) generated by

170
01
,
2620
065
G:=sub<GL(2,GF(307))| [17,0,0,1],[262,0,0,65] >;

C3×C153 in GAP, Magma, Sage, TeX

C_3\times C_{153}
% in TeX

G:=Group("C3xC153");
// GroupNames label

G:=SmallGroup(459,2);
// by ID

G=gap.SmallGroup(459,2);
# by ID

G:=PCGroup([4,-3,-3,-17,-3,612]);
// Polycyclic

G:=Group<a,b|a^3=b^153=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C153 in TeX

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