direct product, abelian, monomial
Aliases: C212, SmallGroup(441,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C212 |
| C1 — C212 |
| C1 — C212 |
Generators and relations for C212
G = < a,b | a21=b21=1, ab=ba >
Subgroups: 60, all normal (4 characteristic)
C1, C3, C7, C32, C21, C72, C3×C21, C7×C21, C212
Quotients: C1, C3, C7, C32, C21, C72, C3×C21, C7×C21, C212
►Regular action on 441 points
Generators in S441
(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)
(1 189 302 265 353 381 421 51 98 199 418 147 29 108 214 79 245 289 333 157 372)(2 169 303 266 354 382 422 52 99 200 419 127 30 109 215 80 246 290 334 158 373)(3 170 304 267 355 383 423 53 100 201 420 128 31 110 216 81 247 291 335 159 374)(4 171 305 268 356 384 424 54 101 202 400 129 32 111 217 82 248 292 336 160 375)(5 172 306 269 357 385 425 55 102 203 401 130 33 112 218 83 249 293 316 161 376)(6 173 307 270 337 386 426 56 103 204 402 131 34 113 219 84 250 294 317 162 377)(7 174 308 271 338 387 427 57 104 205 403 132 35 114 220 64 251 274 318 163 378)(8 175 309 272 339 388 428 58 105 206 404 133 36 115 221 65 252 275 319 164 358)(9 176 310 273 340 389 429 59 85 207 405 134 37 116 222 66 232 276 320 165 359)(10 177 311 253 341 390 430 60 86 208 406 135 38 117 223 67 233 277 321 166 360)(11 178 312 254 342 391 431 61 87 209 407 136 39 118 224 68 234 278 322 167 361)(12 179 313 255 343 392 432 62 88 210 408 137 40 119 225 69 235 279 323 168 362)(13 180 314 256 344 393 433 63 89 190 409 138 41 120 226 70 236 280 324 148 363)(14 181 315 257 345 394 434 43 90 191 410 139 42 121 227 71 237 281 325 149 364)(15 182 295 258 346 395 435 44 91 192 411 140 22 122 228 72 238 282 326 150 365)(16 183 296 259 347 396 436 45 92 193 412 141 23 123 229 73 239 283 327 151 366)(17 184 297 260 348 397 437 46 93 194 413 142 24 124 230 74 240 284 328 152 367)(18 185 298 261 349 398 438 47 94 195 414 143 25 125 231 75 241 285 329 153 368)(19 186 299 262 350 399 439 48 95 196 415 144 26 126 211 76 242 286 330 154 369)(20 187 300 263 351 379 440 49 96 197 416 145 27 106 212 77 243 287 331 155 370)(21 188 301 264 352 380 441 50 97 198 417 146 28 107 213 78 244 288 332 156 371)
G:=sub<Sym(441)| (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), (1,189,302,265,353,381,421,51,98,199,418,147,29,108,214,79,245,289,333,157,372)(2,169,303,266,354,382,422,52,99,200,419,127,30,109,215,80,246,290,334,158,373)(3,170,304,267,355,383,423,53,100,201,420,128,31,110,216,81,247,291,335,159,374)(4,171,305,268,356,384,424,54,101,202,400,129,32,111,217,82,248,292,336,160,375)(5,172,306,269,357,385,425,55,102,203,401,130,33,112,218,83,249,293,316,161,376)(6,173,307,270,337,386,426,56,103,204,402,131,34,113,219,84,250,294,317,162,377)(7,174,308,271,338,387,427,57,104,205,403,132,35,114,220,64,251,274,318,163,378)(8,175,309,272,339,388,428,58,105,206,404,133,36,115,221,65,252,275,319,164,358)(9,176,310,273,340,389,429,59,85,207,405,134,37,116,222,66,232,276,320,165,359)(10,177,311,253,341,390,430,60,86,208,406,135,38,117,223,67,233,277,321,166,360)(11,178,312,254,342,391,431,61,87,209,407,136,39,118,224,68,234,278,322,167,361)(12,179,313,255,343,392,432,62,88,210,408,137,40,119,225,69,235,279,323,168,362)(13,180,314,256,344,393,433,63,89,190,409,138,41,120,226,70,236,280,324,148,363)(14,181,315,257,345,394,434,43,90,191,410,139,42,121,227,71,237,281,325,149,364)(15,182,295,258,346,395,435,44,91,192,411,140,22,122,228,72,238,282,326,150,365)(16,183,296,259,347,396,436,45,92,193,412,141,23,123,229,73,239,283,327,151,366)(17,184,297,260,348,397,437,46,93,194,413,142,24,124,230,74,240,284,328,152,367)(18,185,298,261,349,398,438,47,94,195,414,143,25,125,231,75,241,285,329,153,368)(19,186,299,262,350,399,439,48,95,196,415,144,26,126,211,76,242,286,330,154,369)(20,187,300,263,351,379,440,49,96,197,416,145,27,106,212,77,243,287,331,155,370)(21,188,301,264,352,380,441,50,97,198,417,146,28,107,213,78,244,288,332,156,371)>;
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), (1,189,302,265,353,381,421,51,98,199,418,147,29,108,214,79,245,289,333,157,372)(2,169,303,266,354,382,422,52,99,200,419,127,30,109,215,80,246,290,334,158,373)(3,170,304,267,355,383,423,53,100,201,420,128,31,110,216,81,247,291,335,159,374)(4,171,305,268,356,384,424,54,101,202,400,129,32,111,217,82,248,292,336,160,375)(5,172,306,269,357,385,425,55,102,203,401,130,33,112,218,83,249,293,316,161,376)(6,173,307,270,337,386,426,56,103,204,402,131,34,113,219,84,250,294,317,162,377)(7,174,308,271,338,387,427,57,104,205,403,132,35,114,220,64,251,274,318,163,378)(8,175,309,272,339,388,428,58,105,206,404,133,36,115,221,65,252,275,319,164,358)(9,176,310,273,340,389,429,59,85,207,405,134,37,116,222,66,232,276,320,165,359)(10,177,311,253,341,390,430,60,86,208,406,135,38,117,223,67,233,277,321,166,360)(11,178,312,254,342,391,431,61,87,209,407,136,39,118,224,68,234,278,322,167,361)(12,179,313,255,343,392,432,62,88,210,408,137,40,119,225,69,235,279,323,168,362)(13,180,314,256,344,393,433,63,89,190,409,138,41,120,226,70,236,280,324,148,363)(14,181,315,257,345,394,434,43,90,191,410,139,42,121,227,71,237,281,325,149,364)(15,182,295,258,346,395,435,44,91,192,411,140,22,122,228,72,238,282,326,150,365)(16,183,296,259,347,396,436,45,92,193,412,141,23,123,229,73,239,283,327,151,366)(17,184,297,260,348,397,437,46,93,194,413,142,24,124,230,74,240,284,328,152,367)(18,185,298,261,349,398,438,47,94,195,414,143,25,125,231,75,241,285,329,153,368)(19,186,299,262,350,399,439,48,95,196,415,144,26,126,211,76,242,286,330,154,369)(20,187,300,263,351,379,440,49,96,197,416,145,27,106,212,77,243,287,331,155,370)(21,188,301,264,352,380,441,50,97,198,417,146,28,107,213,78,244,288,332,156,371) );
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)], [(1,189,302,265,353,381,421,51,98,199,418,147,29,108,214,79,245,289,333,157,372),(2,169,303,266,354,382,422,52,99,200,419,127,30,109,215,80,246,290,334,158,373),(3,170,304,267,355,383,423,53,100,201,420,128,31,110,216,81,247,291,335,159,374),(4,171,305,268,356,384,424,54,101,202,400,129,32,111,217,82,248,292,336,160,375),(5,172,306,269,357,385,425,55,102,203,401,130,33,112,218,83,249,293,316,161,376),(6,173,307,270,337,386,426,56,103,204,402,131,34,113,219,84,250,294,317,162,377),(7,174,308,271,338,387,427,57,104,205,403,132,35,114,220,64,251,274,318,163,378),(8,175,309,272,339,388,428,58,105,206,404,133,36,115,221,65,252,275,319,164,358),(9,176,310,273,340,389,429,59,85,207,405,134,37,116,222,66,232,276,320,165,359),(10,177,311,253,341,390,430,60,86,208,406,135,38,117,223,67,233,277,321,166,360),(11,178,312,254,342,391,431,61,87,209,407,136,39,118,224,68,234,278,322,167,361),(12,179,313,255,343,392,432,62,88,210,408,137,40,119,225,69,235,279,323,168,362),(13,180,314,256,344,393,433,63,89,190,409,138,41,120,226,70,236,280,324,148,363),(14,181,315,257,345,394,434,43,90,191,410,139,42,121,227,71,237,281,325,149,364),(15,182,295,258,346,395,435,44,91,192,411,140,22,122,228,72,238,282,326,150,365),(16,183,296,259,347,396,436,45,92,193,412,141,23,123,229,73,239,283,327,151,366),(17,184,297,260,348,397,437,46,93,194,413,142,24,124,230,74,240,284,328,152,367),(18,185,298,261,349,398,438,47,94,195,414,143,25,125,231,75,241,285,329,153,368),(19,186,299,262,350,399,439,48,95,196,415,144,26,126,211,76,242,286,330,154,369),(20,187,300,263,351,379,440,49,96,197,416,145,27,106,212,77,243,287,331,155,370),(21,188,301,264,352,380,441,50,97,198,417,146,28,107,213,78,244,288,332,156,371)]])
441 conjugacy classes
| class | 1 | 3A | ··· | 3H | 7A | ··· | 7AV | 21A | ··· | 21NT |
| order | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
441 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C7 | C21 |
| kernel | C212 | C7×C21 | C3×C21 | C21 |
| # reps | 1 | 8 | 48 | 384 |
Matrix representation of C212 ►in GL2(𝔽43) generated by
| 13 | 0 |
| 0 | 38 |
| 40 | 0 |
| 0 | 24 |
G:=sub<GL(2,GF(43))| [13,0,0,38],[40,0,0,24] >;
C212 in GAP, Magma, Sage, TeX
C_{21}^2 % in TeX
G:=Group("C21^2"); // GroupNames label
G:=SmallGroup(441,13);
// by ID
G=gap.SmallGroup(441,13);
# by ID
G:=PCGroup([4,-3,-3,-7,-7]);
// Polycyclic
G:=Group<a,b|a^21=b^21=1,a*b=b*a>;
// generators/relations