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G = C49⋊C9order 441 = 32·72

The semidirect product of C49 and C9 acting via C9/C3=C3

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C49⋊C9, C147.C3, C7.(C7⋊C9), C3.(C49⋊C3), C21.1(C7⋊C3), SmallGroup(441,1)

Series: Derived Chief Lower central Upper central

C1C49 — C49⋊C9
C1C7C49C147 — C49⋊C9
C49 — C49⋊C9
C1C3

Generators and relations for C49⋊C9
 G = < a,b | a49=b9=1, bab-1=a18 >

49C9
7C7⋊C9

Smallest permutation representation of C49⋊C9
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)
(1 423 294 124 357 242 57 319 151)(2 404 263 125 387 211 58 300 169)(3 434 281 126 368 229 59 330 187)(4 415 250 127 349 198 60 311 156)(5 396 268 128 379 216 61 341 174)(6 426 286 129 360 234 62 322 192)(7 407 255 130 390 203 63 303 161)(8 437 273 131 371 221 64 333 179)(9 418 291 132 352 239 65 314 148)(10 399 260 133 382 208 66 295 166)(11 429 278 134 363 226 67 325 184)(12 410 247 135 344 244 68 306 153)(13 440 265 136 374 213 69 336 171)(14 421 283 137 355 231 70 317 189)(15 402 252 138 385 200 71 298 158)(16 432 270 139 366 218 72 328 176)(17 413 288 140 347 236 73 309 194)(18 394 257 141 377 205 74 339 163)(19 424 275 142 358 223 75 320 181)(20 405 293 143 388 241 76 301 150)(21 435 262 144 369 210 77 331 168)(22 416 280 145 350 228 78 312 186)(23 397 249 146 380 197 79 342 155)(24 427 267 147 361 215 80 323 173)(25 408 285 99 391 233 81 304 191)(26 438 254 100 372 202 82 334 160)(27 419 272 101 353 220 83 315 178)(28 400 290 102 383 238 84 296 196)(29 430 259 103 364 207 85 326 165)(30 411 277 104 345 225 86 307 183)(31 441 246 105 375 243 87 337 152)(32 422 264 106 356 212 88 318 170)(33 403 282 107 386 230 89 299 188)(34 433 251 108 367 199 90 329 157)(35 414 269 109 348 217 91 310 175)(36 395 287 110 378 235 92 340 193)(37 425 256 111 359 204 93 321 162)(38 406 274 112 389 222 94 302 180)(39 436 292 113 370 240 95 332 149)(40 417 261 114 351 209 96 313 167)(41 398 279 115 381 227 97 343 185)(42 428 248 116 362 245 98 324 154)(43 409 266 117 392 214 50 305 172)(44 439 284 118 373 232 51 335 190)(45 420 253 119 354 201 52 316 159)(46 401 271 120 384 219 53 297 177)(47 431 289 121 365 237 54 327 195)(48 412 258 122 346 206 55 308 164)(49 393 276 123 376 224 56 338 182)

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

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

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

57 conjugacy classes

class 1 3A3B7A7B9A···9F21A21B21C21D49A···49N147A···147AB
order133779···92121212149···49147···147
size1113349···4933333···33···3

57 irreducible representations

dim1113333
type+
imageC1C3C9C7⋊C3C7⋊C9C49⋊C3C49⋊C9
kernelC49⋊C9C147C49C21C7C3C1
# reps126241428

Matrix representation of C49⋊C9 in GL4(𝔽883) generated by

1000
02367652
0652286561
0561644600
,
286000
0813837129
048434674
0129712607
G:=sub<GL(4,GF(883))| [1,0,0,0,0,23,652,561,0,67,286,644,0,652,561,600],[286,0,0,0,0,813,484,129,0,837,346,712,0,129,74,607] >;

C49⋊C9 in GAP, Magma, Sage, TeX

C_{49}\rtimes C_9
% in TeX

G:=Group("C49:C9");
// GroupNames label

G:=SmallGroup(441,1);
// by ID

G=gap.SmallGroup(441,1);
# by ID

G:=PCGroup([4,-3,-3,-7,-7,12,974,178,2019]);
// Polycyclic

G:=Group<a,b|a^49=b^9=1,b*a*b^-1=a^18>;
// generators/relations

Export

Subgroup lattice of C49⋊C9 in TeX

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