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G = C212order 441 = 32·72

Abelian group of type [21,21]

direct product, abelian, monomial

Aliases: C212, SmallGroup(441,13)

Series: Derived Chief Lower central Upper central

C1 — C212
C1C7C72C7×C21 — 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 [×4], C7 [×8], C32, C21 [×32], C72, C3×C21 [×8], C7×C21 [×4], C212
Quotients: C1, C3 [×4], C7 [×8], C32, C21 [×32], C72, C3×C21 [×8], C7×C21 [×4], C212

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

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

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

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

441 conjugacy classes

class 1 3A···3H7A···7AV21A···21NT
order13···37···721···21
size11···11···11···1

441 irreducible representations

dim1111
type+
imageC1C3C7C21
kernelC212C7×C21C3×C21C21
# reps1848384

Matrix representation of C212 in GL2(𝔽43) generated by

130
038
,
400
024
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

׿
×
𝔽