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G = C11×C44order 484 = 22·112

Abelian group of type [11,44]

direct product, abelian, monomial, 11-elementary

Aliases: C11×C44, SmallGroup(484,7)

Series: Derived Chief Lower central Upper central

C1 — C11×C44
C1C2C22C11×C22 — C11×C44
C1 — C11×C44
C1 — C11×C44

Generators and relations for C11×C44
 G = < a,b | a11=b44=1, ab=ba >


Smallest permutation representation of C11×C44
Regular action on 484 points
Generators in S484
(1 147 413 460 304 224 369 195 320 68 99)(2 148 414 461 305 225 370 196 321 69 100)(3 149 415 462 306 226 371 197 322 70 101)(4 150 416 463 307 227 372 198 323 71 102)(5 151 417 464 308 228 373 199 324 72 103)(6 152 418 465 265 229 374 200 325 73 104)(7 153 419 466 266 230 375 201 326 74 105)(8 154 420 467 267 231 376 202 327 75 106)(9 155 421 468 268 232 377 203 328 76 107)(10 156 422 469 269 233 378 204 329 77 108)(11 157 423 470 270 234 379 205 330 78 109)(12 158 424 471 271 235 380 206 331 79 110)(13 159 425 472 272 236 381 207 332 80 111)(14 160 426 473 273 237 382 208 333 81 112)(15 161 427 474 274 238 383 209 334 82 113)(16 162 428 475 275 239 384 210 335 83 114)(17 163 429 476 276 240 385 211 336 84 115)(18 164 430 477 277 241 386 212 337 85 116)(19 165 431 478 278 242 387 213 338 86 117)(20 166 432 479 279 243 388 214 339 87 118)(21 167 433 480 280 244 389 215 340 88 119)(22 168 434 481 281 245 390 216 341 45 120)(23 169 435 482 282 246 391 217 342 46 121)(24 170 436 483 283 247 392 218 343 47 122)(25 171 437 484 284 248 393 219 344 48 123)(26 172 438 441 285 249 394 220 345 49 124)(27 173 439 442 286 250 395 177 346 50 125)(28 174 440 443 287 251 396 178 347 51 126)(29 175 397 444 288 252 353 179 348 52 127)(30 176 398 445 289 253 354 180 349 53 128)(31 133 399 446 290 254 355 181 350 54 129)(32 134 400 447 291 255 356 182 351 55 130)(33 135 401 448 292 256 357 183 352 56 131)(34 136 402 449 293 257 358 184 309 57 132)(35 137 403 450 294 258 359 185 310 58 89)(36 138 404 451 295 259 360 186 311 59 90)(37 139 405 452 296 260 361 187 312 60 91)(38 140 406 453 297 261 362 188 313 61 92)(39 141 407 454 298 262 363 189 314 62 93)(40 142 408 455 299 263 364 190 315 63 94)(41 143 409 456 300 264 365 191 316 64 95)(42 144 410 457 301 221 366 192 317 65 96)(43 145 411 458 302 222 367 193 318 66 97)(44 146 412 459 303 223 368 194 319 67 98)
(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)(441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484)

G:=sub<Sym(484)| (1,147,413,460,304,224,369,195,320,68,99)(2,148,414,461,305,225,370,196,321,69,100)(3,149,415,462,306,226,371,197,322,70,101)(4,150,416,463,307,227,372,198,323,71,102)(5,151,417,464,308,228,373,199,324,72,103)(6,152,418,465,265,229,374,200,325,73,104)(7,153,419,466,266,230,375,201,326,74,105)(8,154,420,467,267,231,376,202,327,75,106)(9,155,421,468,268,232,377,203,328,76,107)(10,156,422,469,269,233,378,204,329,77,108)(11,157,423,470,270,234,379,205,330,78,109)(12,158,424,471,271,235,380,206,331,79,110)(13,159,425,472,272,236,381,207,332,80,111)(14,160,426,473,273,237,382,208,333,81,112)(15,161,427,474,274,238,383,209,334,82,113)(16,162,428,475,275,239,384,210,335,83,114)(17,163,429,476,276,240,385,211,336,84,115)(18,164,430,477,277,241,386,212,337,85,116)(19,165,431,478,278,242,387,213,338,86,117)(20,166,432,479,279,243,388,214,339,87,118)(21,167,433,480,280,244,389,215,340,88,119)(22,168,434,481,281,245,390,216,341,45,120)(23,169,435,482,282,246,391,217,342,46,121)(24,170,436,483,283,247,392,218,343,47,122)(25,171,437,484,284,248,393,219,344,48,123)(26,172,438,441,285,249,394,220,345,49,124)(27,173,439,442,286,250,395,177,346,50,125)(28,174,440,443,287,251,396,178,347,51,126)(29,175,397,444,288,252,353,179,348,52,127)(30,176,398,445,289,253,354,180,349,53,128)(31,133,399,446,290,254,355,181,350,54,129)(32,134,400,447,291,255,356,182,351,55,130)(33,135,401,448,292,256,357,183,352,56,131)(34,136,402,449,293,257,358,184,309,57,132)(35,137,403,450,294,258,359,185,310,58,89)(36,138,404,451,295,259,360,186,311,59,90)(37,139,405,452,296,260,361,187,312,60,91)(38,140,406,453,297,261,362,188,313,61,92)(39,141,407,454,298,262,363,189,314,62,93)(40,142,408,455,299,263,364,190,315,63,94)(41,143,409,456,300,264,365,191,316,64,95)(42,144,410,457,301,221,366,192,317,65,96)(43,145,411,458,302,222,367,193,318,66,97)(44,146,412,459,303,223,368,194,319,67,98), (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)(441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484)>;

G:=Group( (1,147,413,460,304,224,369,195,320,68,99)(2,148,414,461,305,225,370,196,321,69,100)(3,149,415,462,306,226,371,197,322,70,101)(4,150,416,463,307,227,372,198,323,71,102)(5,151,417,464,308,228,373,199,324,72,103)(6,152,418,465,265,229,374,200,325,73,104)(7,153,419,466,266,230,375,201,326,74,105)(8,154,420,467,267,231,376,202,327,75,106)(9,155,421,468,268,232,377,203,328,76,107)(10,156,422,469,269,233,378,204,329,77,108)(11,157,423,470,270,234,379,205,330,78,109)(12,158,424,471,271,235,380,206,331,79,110)(13,159,425,472,272,236,381,207,332,80,111)(14,160,426,473,273,237,382,208,333,81,112)(15,161,427,474,274,238,383,209,334,82,113)(16,162,428,475,275,239,384,210,335,83,114)(17,163,429,476,276,240,385,211,336,84,115)(18,164,430,477,277,241,386,212,337,85,116)(19,165,431,478,278,242,387,213,338,86,117)(20,166,432,479,279,243,388,214,339,87,118)(21,167,433,480,280,244,389,215,340,88,119)(22,168,434,481,281,245,390,216,341,45,120)(23,169,435,482,282,246,391,217,342,46,121)(24,170,436,483,283,247,392,218,343,47,122)(25,171,437,484,284,248,393,219,344,48,123)(26,172,438,441,285,249,394,220,345,49,124)(27,173,439,442,286,250,395,177,346,50,125)(28,174,440,443,287,251,396,178,347,51,126)(29,175,397,444,288,252,353,179,348,52,127)(30,176,398,445,289,253,354,180,349,53,128)(31,133,399,446,290,254,355,181,350,54,129)(32,134,400,447,291,255,356,182,351,55,130)(33,135,401,448,292,256,357,183,352,56,131)(34,136,402,449,293,257,358,184,309,57,132)(35,137,403,450,294,258,359,185,310,58,89)(36,138,404,451,295,259,360,186,311,59,90)(37,139,405,452,296,260,361,187,312,60,91)(38,140,406,453,297,261,362,188,313,61,92)(39,141,407,454,298,262,363,189,314,62,93)(40,142,408,455,299,263,364,190,315,63,94)(41,143,409,456,300,264,365,191,316,64,95)(42,144,410,457,301,221,366,192,317,65,96)(43,145,411,458,302,222,367,193,318,66,97)(44,146,412,459,303,223,368,194,319,67,98), (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)(441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484) );

G=PermutationGroup([(1,147,413,460,304,224,369,195,320,68,99),(2,148,414,461,305,225,370,196,321,69,100),(3,149,415,462,306,226,371,197,322,70,101),(4,150,416,463,307,227,372,198,323,71,102),(5,151,417,464,308,228,373,199,324,72,103),(6,152,418,465,265,229,374,200,325,73,104),(7,153,419,466,266,230,375,201,326,74,105),(8,154,420,467,267,231,376,202,327,75,106),(9,155,421,468,268,232,377,203,328,76,107),(10,156,422,469,269,233,378,204,329,77,108),(11,157,423,470,270,234,379,205,330,78,109),(12,158,424,471,271,235,380,206,331,79,110),(13,159,425,472,272,236,381,207,332,80,111),(14,160,426,473,273,237,382,208,333,81,112),(15,161,427,474,274,238,383,209,334,82,113),(16,162,428,475,275,239,384,210,335,83,114),(17,163,429,476,276,240,385,211,336,84,115),(18,164,430,477,277,241,386,212,337,85,116),(19,165,431,478,278,242,387,213,338,86,117),(20,166,432,479,279,243,388,214,339,87,118),(21,167,433,480,280,244,389,215,340,88,119),(22,168,434,481,281,245,390,216,341,45,120),(23,169,435,482,282,246,391,217,342,46,121),(24,170,436,483,283,247,392,218,343,47,122),(25,171,437,484,284,248,393,219,344,48,123),(26,172,438,441,285,249,394,220,345,49,124),(27,173,439,442,286,250,395,177,346,50,125),(28,174,440,443,287,251,396,178,347,51,126),(29,175,397,444,288,252,353,179,348,52,127),(30,176,398,445,289,253,354,180,349,53,128),(31,133,399,446,290,254,355,181,350,54,129),(32,134,400,447,291,255,356,182,351,55,130),(33,135,401,448,292,256,357,183,352,56,131),(34,136,402,449,293,257,358,184,309,57,132),(35,137,403,450,294,258,359,185,310,58,89),(36,138,404,451,295,259,360,186,311,59,90),(37,139,405,452,296,260,361,187,312,60,91),(38,140,406,453,297,261,362,188,313,61,92),(39,141,407,454,298,262,363,189,314,62,93),(40,142,408,455,299,263,364,190,315,63,94),(41,143,409,456,300,264,365,191,316,64,95),(42,144,410,457,301,221,366,192,317,65,96),(43,145,411,458,302,222,367,193,318,66,97),(44,146,412,459,303,223,368,194,319,67,98)], [(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),(441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484)])

484 conjugacy classes

class 1  2 4A4B11A···11DP22A···22DP44A···44IF
order124411···1122···2244···44
size11111···11···11···1

484 irreducible representations

dim111111
type++
imageC1C2C4C11C22C44
kernelC11×C44C11×C22C112C44C22C11
# reps112120120240

Matrix representation of C11×C44 in GL2(𝔽89) generated by

780
016
,
160
018
G:=sub<GL(2,GF(89))| [78,0,0,16],[16,0,0,18] >;

C11×C44 in GAP, Magma, Sage, TeX

C_{11}\times C_{44}
% in TeX

G:=Group("C11xC44");
// GroupNames label

G:=SmallGroup(484,7);
// by ID

G=gap.SmallGroup(484,7);
# by ID

G:=PCGroup([4,-2,-11,-11,-2,968]);
// Polycyclic

G:=Group<a,b|a^11=b^44=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C11×C44 in TeX

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