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G = C222order 484 = 22·112

Abelian group of type [22,22]

direct product, abelian, monomial

Aliases: C222, SmallGroup(484,12)

Series: Derived Chief Lower central Upper central

C1 — C222
C1C11C112C11×C22 — C222
C1 — C222
C1 — C222

Generators and relations for C222
 G = < a,b | a22=b22=1, ab=ba >

Subgroups: 70, all normal (4 characteristic)
C1, C2 [×3], C22, C11 [×12], C22 [×36], C2×C22 [×12], C112, C11×C22 [×3], C222
Quotients: C1, C2 [×3], C22, C11 [×12], C22 [×36], C2×C22 [×12], C112, C11×C22 [×3], C222

Smallest permutation representation of C222
Regular action on 484 points
Generators in S484
(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)
(1 414 163 446 71 360 286 47 34 210 177 426 351 470 383 128 242 309 289 152 102 247)(2 415 164 447 72 361 265 48 35 211 178 427 352 471 384 129 221 310 290 153 103 248)(3 416 165 448 73 362 266 49 36 212 179 428 331 472 385 130 222 311 291 154 104 249)(4 417 166 449 74 363 267 50 37 213 180 429 332 473 386 131 223 312 292 133 105 250)(5 418 167 450 75 364 268 51 38 214 181 430 333 474 387 132 224 313 293 134 106 251)(6 397 168 451 76 365 269 52 39 215 182 431 334 475 388 111 225 314 294 135 107 252)(7 398 169 452 77 366 270 53 40 216 183 432 335 476 389 112 226 315 295 136 108 253)(8 399 170 453 78 367 271 54 41 217 184 433 336 477 390 113 227 316 296 137 109 254)(9 400 171 454 79 368 272 55 42 218 185 434 337 478 391 114 228 317 297 138 110 255)(10 401 172 455 80 369 273 56 43 219 186 435 338 479 392 115 229 318 298 139 89 256)(11 402 173 456 81 370 274 57 44 220 187 436 339 480 393 116 230 319 299 140 90 257)(12 403 174 457 82 371 275 58 23 199 188 437 340 481 394 117 231 320 300 141 91 258)(13 404 175 458 83 372 276 59 24 200 189 438 341 482 395 118 232 321 301 142 92 259)(14 405 176 459 84 373 277 60 25 201 190 439 342 483 396 119 233 322 302 143 93 260)(15 406 155 460 85 374 278 61 26 202 191 440 343 484 375 120 234 323 303 144 94 261)(16 407 156 461 86 353 279 62 27 203 192 419 344 463 376 121 235 324 304 145 95 262)(17 408 157 462 87 354 280 63 28 204 193 420 345 464 377 122 236 325 305 146 96 263)(18 409 158 441 88 355 281 64 29 205 194 421 346 465 378 123 237 326 306 147 97 264)(19 410 159 442 67 356 282 65 30 206 195 422 347 466 379 124 238 327 307 148 98 243)(20 411 160 443 68 357 283 66 31 207 196 423 348 467 380 125 239 328 308 149 99 244)(21 412 161 444 69 358 284 45 32 208 197 424 349 468 381 126 240 329 287 150 100 245)(22 413 162 445 70 359 285 46 33 209 198 425 350 469 382 127 241 330 288 151 101 246)

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

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

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

484 conjugacy classes

class 1 2A2B2C11A···11DP22A···22MV
order122211···1122···22
size11111···11···1

484 irreducible representations

dim1111
type++
imageC1C2C11C22
kernelC222C11×C22C2×C22C22
# reps13120360

Matrix representation of C222 in GL2(𝔽23) generated by

70
019
,
200
03
G:=sub<GL(2,GF(23))| [7,0,0,19],[20,0,0,3] >;

C222 in GAP, Magma, Sage, TeX

C_{22}^2
% in TeX

G:=Group("C22^2");
// GroupNames label

G:=SmallGroup(484,12);
// by ID

G=gap.SmallGroup(484,12);
# by ID

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

G:=Group<a,b|a^22=b^22=1,a*b=b*a>;
// generators/relations

׿
×
𝔽