direct product, abelian, monomial
Aliases: C222, SmallGroup(484,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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, C22, C11, C22, C2×C22, C112, C11×C22, C222
Quotients: C1, C2, C22, C11, C22, C2×C22, C112, C11×C22, 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)
(1 408 271 438 92 159 257 138 371 294 53 80 196 391 221 117 316 465 24 350 200 458)(2 409 272 439 93 160 258 139 372 295 54 81 197 392 222 118 317 466 25 351 201 459)(3 410 273 440 94 161 259 140 373 296 55 82 198 393 223 119 318 467 26 352 202 460)(4 411 274 419 95 162 260 141 374 297 56 83 177 394 224 120 319 468 27 331 203 461)(5 412 275 420 96 163 261 142 353 298 57 84 178 395 225 121 320 469 28 332 204 462)(6 413 276 421 97 164 262 143 354 299 58 85 179 396 226 122 321 470 29 333 205 441)(7 414 277 422 98 165 263 144 355 300 59 86 180 375 227 123 322 471 30 334 206 442)(8 415 278 423 99 166 264 145 356 301 60 87 181 376 228 124 323 472 31 335 207 443)(9 416 279 424 100 167 243 146 357 302 61 88 182 377 229 125 324 473 32 336 208 444)(10 417 280 425 101 168 244 147 358 303 62 67 183 378 230 126 325 474 33 337 209 445)(11 418 281 426 102 169 245 148 359 304 63 68 184 379 231 127 326 475 34 338 210 446)(12 397 282 427 103 170 246 149 360 305 64 69 185 380 232 128 327 476 35 339 211 447)(13 398 283 428 104 171 247 150 361 306 65 70 186 381 233 129 328 477 36 340 212 448)(14 399 284 429 105 172 248 151 362 307 66 71 187 382 234 130 329 478 37 341 213 449)(15 400 285 430 106 173 249 152 363 308 45 72 188 383 235 131 330 479 38 342 214 450)(16 401 286 431 107 174 250 153 364 287 46 73 189 384 236 132 309 480 39 343 215 451)(17 402 265 432 108 175 251 154 365 288 47 74 190 385 237 111 310 481 40 344 216 452)(18 403 266 433 109 176 252 133 366 289 48 75 191 386 238 112 311 482 41 345 217 453)(19 404 267 434 110 155 253 134 367 290 49 76 192 387 239 113 312 483 42 346 218 454)(20 405 268 435 89 156 254 135 368 291 50 77 193 388 240 114 313 484 43 347 219 455)(21 406 269 436 90 157 255 136 369 292 51 78 194 389 241 115 314 463 44 348 220 456)(22 407 270 437 91 158 256 137 370 293 52 79 195 390 242 116 315 464 23 349 199 457)
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), (1,408,271,438,92,159,257,138,371,294,53,80,196,391,221,117,316,465,24,350,200,458)(2,409,272,439,93,160,258,139,372,295,54,81,197,392,222,118,317,466,25,351,201,459)(3,410,273,440,94,161,259,140,373,296,55,82,198,393,223,119,318,467,26,352,202,460)(4,411,274,419,95,162,260,141,374,297,56,83,177,394,224,120,319,468,27,331,203,461)(5,412,275,420,96,163,261,142,353,298,57,84,178,395,225,121,320,469,28,332,204,462)(6,413,276,421,97,164,262,143,354,299,58,85,179,396,226,122,321,470,29,333,205,441)(7,414,277,422,98,165,263,144,355,300,59,86,180,375,227,123,322,471,30,334,206,442)(8,415,278,423,99,166,264,145,356,301,60,87,181,376,228,124,323,472,31,335,207,443)(9,416,279,424,100,167,243,146,357,302,61,88,182,377,229,125,324,473,32,336,208,444)(10,417,280,425,101,168,244,147,358,303,62,67,183,378,230,126,325,474,33,337,209,445)(11,418,281,426,102,169,245,148,359,304,63,68,184,379,231,127,326,475,34,338,210,446)(12,397,282,427,103,170,246,149,360,305,64,69,185,380,232,128,327,476,35,339,211,447)(13,398,283,428,104,171,247,150,361,306,65,70,186,381,233,129,328,477,36,340,212,448)(14,399,284,429,105,172,248,151,362,307,66,71,187,382,234,130,329,478,37,341,213,449)(15,400,285,430,106,173,249,152,363,308,45,72,188,383,235,131,330,479,38,342,214,450)(16,401,286,431,107,174,250,153,364,287,46,73,189,384,236,132,309,480,39,343,215,451)(17,402,265,432,108,175,251,154,365,288,47,74,190,385,237,111,310,481,40,344,216,452)(18,403,266,433,109,176,252,133,366,289,48,75,191,386,238,112,311,482,41,345,217,453)(19,404,267,434,110,155,253,134,367,290,49,76,192,387,239,113,312,483,42,346,218,454)(20,405,268,435,89,156,254,135,368,291,50,77,193,388,240,114,313,484,43,347,219,455)(21,406,269,436,90,157,255,136,369,292,51,78,194,389,241,115,314,463,44,348,220,456)(22,407,270,437,91,158,256,137,370,293,52,79,195,390,242,116,315,464,23,349,199,457)>;
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), (1,408,271,438,92,159,257,138,371,294,53,80,196,391,221,117,316,465,24,350,200,458)(2,409,272,439,93,160,258,139,372,295,54,81,197,392,222,118,317,466,25,351,201,459)(3,410,273,440,94,161,259,140,373,296,55,82,198,393,223,119,318,467,26,352,202,460)(4,411,274,419,95,162,260,141,374,297,56,83,177,394,224,120,319,468,27,331,203,461)(5,412,275,420,96,163,261,142,353,298,57,84,178,395,225,121,320,469,28,332,204,462)(6,413,276,421,97,164,262,143,354,299,58,85,179,396,226,122,321,470,29,333,205,441)(7,414,277,422,98,165,263,144,355,300,59,86,180,375,227,123,322,471,30,334,206,442)(8,415,278,423,99,166,264,145,356,301,60,87,181,376,228,124,323,472,31,335,207,443)(9,416,279,424,100,167,243,146,357,302,61,88,182,377,229,125,324,473,32,336,208,444)(10,417,280,425,101,168,244,147,358,303,62,67,183,378,230,126,325,474,33,337,209,445)(11,418,281,426,102,169,245,148,359,304,63,68,184,379,231,127,326,475,34,338,210,446)(12,397,282,427,103,170,246,149,360,305,64,69,185,380,232,128,327,476,35,339,211,447)(13,398,283,428,104,171,247,150,361,306,65,70,186,381,233,129,328,477,36,340,212,448)(14,399,284,429,105,172,248,151,362,307,66,71,187,382,234,130,329,478,37,341,213,449)(15,400,285,430,106,173,249,152,363,308,45,72,188,383,235,131,330,479,38,342,214,450)(16,401,286,431,107,174,250,153,364,287,46,73,189,384,236,132,309,480,39,343,215,451)(17,402,265,432,108,175,251,154,365,288,47,74,190,385,237,111,310,481,40,344,216,452)(18,403,266,433,109,176,252,133,366,289,48,75,191,386,238,112,311,482,41,345,217,453)(19,404,267,434,110,155,253,134,367,290,49,76,192,387,239,113,312,483,42,346,218,454)(20,405,268,435,89,156,254,135,368,291,50,77,193,388,240,114,313,484,43,347,219,455)(21,406,269,436,90,157,255,136,369,292,51,78,194,389,241,115,314,463,44,348,220,456)(22,407,270,437,91,158,256,137,370,293,52,79,195,390,242,116,315,464,23,349,199,457) );
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)], [(1,408,271,438,92,159,257,138,371,294,53,80,196,391,221,117,316,465,24,350,200,458),(2,409,272,439,93,160,258,139,372,295,54,81,197,392,222,118,317,466,25,351,201,459),(3,410,273,440,94,161,259,140,373,296,55,82,198,393,223,119,318,467,26,352,202,460),(4,411,274,419,95,162,260,141,374,297,56,83,177,394,224,120,319,468,27,331,203,461),(5,412,275,420,96,163,261,142,353,298,57,84,178,395,225,121,320,469,28,332,204,462),(6,413,276,421,97,164,262,143,354,299,58,85,179,396,226,122,321,470,29,333,205,441),(7,414,277,422,98,165,263,144,355,300,59,86,180,375,227,123,322,471,30,334,206,442),(8,415,278,423,99,166,264,145,356,301,60,87,181,376,228,124,323,472,31,335,207,443),(9,416,279,424,100,167,243,146,357,302,61,88,182,377,229,125,324,473,32,336,208,444),(10,417,280,425,101,168,244,147,358,303,62,67,183,378,230,126,325,474,33,337,209,445),(11,418,281,426,102,169,245,148,359,304,63,68,184,379,231,127,326,475,34,338,210,446),(12,397,282,427,103,170,246,149,360,305,64,69,185,380,232,128,327,476,35,339,211,447),(13,398,283,428,104,171,247,150,361,306,65,70,186,381,233,129,328,477,36,340,212,448),(14,399,284,429,105,172,248,151,362,307,66,71,187,382,234,130,329,478,37,341,213,449),(15,400,285,430,106,173,249,152,363,308,45,72,188,383,235,131,330,479,38,342,214,450),(16,401,286,431,107,174,250,153,364,287,46,73,189,384,236,132,309,480,39,343,215,451),(17,402,265,432,108,175,251,154,365,288,47,74,190,385,237,111,310,481,40,344,216,452),(18,403,266,433,109,176,252,133,366,289,48,75,191,386,238,112,311,482,41,345,217,453),(19,404,267,434,110,155,253,134,367,290,49,76,192,387,239,113,312,483,42,346,218,454),(20,405,268,435,89,156,254,135,368,291,50,77,193,388,240,114,313,484,43,347,219,455),(21,406,269,436,90,157,255,136,369,292,51,78,194,389,241,115,314,463,44,348,220,456),(22,407,270,437,91,158,256,137,370,293,52,79,195,390,242,116,315,464,23,349,199,457)]])
484 conjugacy classes
| class | 1 | 2A | 2B | 2C | 11A | ··· | 11DP | 22A | ··· | 22MV |
| order | 1 | 2 | 2 | 2 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
484 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C11 | C22 |
| kernel | C222 | C11×C22 | C2×C22 | C22 |
| # reps | 1 | 3 | 120 | 360 |
Matrix representation of C222 ►in GL2(𝔽23) generated by
| 7 | 0 |
| 0 | 19 |
| 20 | 0 |
| 0 | 3 |
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