metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C53⋊2C8, C4.2D53, C2.Dic53, C106.2C4, C212.2C2, SmallGroup(424,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C53⋊2C8 |
Generators and relations for C53⋊2C8
G = < a,b | a53=b8=1, bab-1=a-1 >
Smallest permutation representation of C53⋊2C8
►Regular action on 424 points
Generators in S424
(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)
(1 372 168 266 65 319 155 213)(2 424 169 318 66 371 156 265)(3 423 170 317 67 370 157 264)(4 422 171 316 68 369 158 263)(5 421 172 315 69 368 159 262)(6 420 173 314 70 367 107 261)(7 419 174 313 71 366 108 260)(8 418 175 312 72 365 109 259)(9 417 176 311 73 364 110 258)(10 416 177 310 74 363 111 257)(11 415 178 309 75 362 112 256)(12 414 179 308 76 361 113 255)(13 413 180 307 77 360 114 254)(14 412 181 306 78 359 115 253)(15 411 182 305 79 358 116 252)(16 410 183 304 80 357 117 251)(17 409 184 303 81 356 118 250)(18 408 185 302 82 355 119 249)(19 407 186 301 83 354 120 248)(20 406 187 300 84 353 121 247)(21 405 188 299 85 352 122 246)(22 404 189 298 86 351 123 245)(23 403 190 297 87 350 124 244)(24 402 191 296 88 349 125 243)(25 401 192 295 89 348 126 242)(26 400 193 294 90 347 127 241)(27 399 194 293 91 346 128 240)(28 398 195 292 92 345 129 239)(29 397 196 291 93 344 130 238)(30 396 197 290 94 343 131 237)(31 395 198 289 95 342 132 236)(32 394 199 288 96 341 133 235)(33 393 200 287 97 340 134 234)(34 392 201 286 98 339 135 233)(35 391 202 285 99 338 136 232)(36 390 203 284 100 337 137 231)(37 389 204 283 101 336 138 230)(38 388 205 282 102 335 139 229)(39 387 206 281 103 334 140 228)(40 386 207 280 104 333 141 227)(41 385 208 279 105 332 142 226)(42 384 209 278 106 331 143 225)(43 383 210 277 54 330 144 224)(44 382 211 276 55 329 145 223)(45 381 212 275 56 328 146 222)(46 380 160 274 57 327 147 221)(47 379 161 273 58 326 148 220)(48 378 162 272 59 325 149 219)(49 377 163 271 60 324 150 218)(50 376 164 270 61 323 151 217)(51 375 165 269 62 322 152 216)(52 374 166 268 63 321 153 215)(53 373 167 267 64 320 154 214)
G:=sub<Sym(424)| (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), (1,372,168,266,65,319,155,213)(2,424,169,318,66,371,156,265)(3,423,170,317,67,370,157,264)(4,422,171,316,68,369,158,263)(5,421,172,315,69,368,159,262)(6,420,173,314,70,367,107,261)(7,419,174,313,71,366,108,260)(8,418,175,312,72,365,109,259)(9,417,176,311,73,364,110,258)(10,416,177,310,74,363,111,257)(11,415,178,309,75,362,112,256)(12,414,179,308,76,361,113,255)(13,413,180,307,77,360,114,254)(14,412,181,306,78,359,115,253)(15,411,182,305,79,358,116,252)(16,410,183,304,80,357,117,251)(17,409,184,303,81,356,118,250)(18,408,185,302,82,355,119,249)(19,407,186,301,83,354,120,248)(20,406,187,300,84,353,121,247)(21,405,188,299,85,352,122,246)(22,404,189,298,86,351,123,245)(23,403,190,297,87,350,124,244)(24,402,191,296,88,349,125,243)(25,401,192,295,89,348,126,242)(26,400,193,294,90,347,127,241)(27,399,194,293,91,346,128,240)(28,398,195,292,92,345,129,239)(29,397,196,291,93,344,130,238)(30,396,197,290,94,343,131,237)(31,395,198,289,95,342,132,236)(32,394,199,288,96,341,133,235)(33,393,200,287,97,340,134,234)(34,392,201,286,98,339,135,233)(35,391,202,285,99,338,136,232)(36,390,203,284,100,337,137,231)(37,389,204,283,101,336,138,230)(38,388,205,282,102,335,139,229)(39,387,206,281,103,334,140,228)(40,386,207,280,104,333,141,227)(41,385,208,279,105,332,142,226)(42,384,209,278,106,331,143,225)(43,383,210,277,54,330,144,224)(44,382,211,276,55,329,145,223)(45,381,212,275,56,328,146,222)(46,380,160,274,57,327,147,221)(47,379,161,273,58,326,148,220)(48,378,162,272,59,325,149,219)(49,377,163,271,60,324,150,218)(50,376,164,270,61,323,151,217)(51,375,165,269,62,322,152,216)(52,374,166,268,63,321,153,215)(53,373,167,267,64,320,154,214)>;
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), (1,372,168,266,65,319,155,213)(2,424,169,318,66,371,156,265)(3,423,170,317,67,370,157,264)(4,422,171,316,68,369,158,263)(5,421,172,315,69,368,159,262)(6,420,173,314,70,367,107,261)(7,419,174,313,71,366,108,260)(8,418,175,312,72,365,109,259)(9,417,176,311,73,364,110,258)(10,416,177,310,74,363,111,257)(11,415,178,309,75,362,112,256)(12,414,179,308,76,361,113,255)(13,413,180,307,77,360,114,254)(14,412,181,306,78,359,115,253)(15,411,182,305,79,358,116,252)(16,410,183,304,80,357,117,251)(17,409,184,303,81,356,118,250)(18,408,185,302,82,355,119,249)(19,407,186,301,83,354,120,248)(20,406,187,300,84,353,121,247)(21,405,188,299,85,352,122,246)(22,404,189,298,86,351,123,245)(23,403,190,297,87,350,124,244)(24,402,191,296,88,349,125,243)(25,401,192,295,89,348,126,242)(26,400,193,294,90,347,127,241)(27,399,194,293,91,346,128,240)(28,398,195,292,92,345,129,239)(29,397,196,291,93,344,130,238)(30,396,197,290,94,343,131,237)(31,395,198,289,95,342,132,236)(32,394,199,288,96,341,133,235)(33,393,200,287,97,340,134,234)(34,392,201,286,98,339,135,233)(35,391,202,285,99,338,136,232)(36,390,203,284,100,337,137,231)(37,389,204,283,101,336,138,230)(38,388,205,282,102,335,139,229)(39,387,206,281,103,334,140,228)(40,386,207,280,104,333,141,227)(41,385,208,279,105,332,142,226)(42,384,209,278,106,331,143,225)(43,383,210,277,54,330,144,224)(44,382,211,276,55,329,145,223)(45,381,212,275,56,328,146,222)(46,380,160,274,57,327,147,221)(47,379,161,273,58,326,148,220)(48,378,162,272,59,325,149,219)(49,377,163,271,60,324,150,218)(50,376,164,270,61,323,151,217)(51,375,165,269,62,322,152,216)(52,374,166,268,63,321,153,215)(53,373,167,267,64,320,154,214) );
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)], [(1,372,168,266,65,319,155,213),(2,424,169,318,66,371,156,265),(3,423,170,317,67,370,157,264),(4,422,171,316,68,369,158,263),(5,421,172,315,69,368,159,262),(6,420,173,314,70,367,107,261),(7,419,174,313,71,366,108,260),(8,418,175,312,72,365,109,259),(9,417,176,311,73,364,110,258),(10,416,177,310,74,363,111,257),(11,415,178,309,75,362,112,256),(12,414,179,308,76,361,113,255),(13,413,180,307,77,360,114,254),(14,412,181,306,78,359,115,253),(15,411,182,305,79,358,116,252),(16,410,183,304,80,357,117,251),(17,409,184,303,81,356,118,250),(18,408,185,302,82,355,119,249),(19,407,186,301,83,354,120,248),(20,406,187,300,84,353,121,247),(21,405,188,299,85,352,122,246),(22,404,189,298,86,351,123,245),(23,403,190,297,87,350,124,244),(24,402,191,296,88,349,125,243),(25,401,192,295,89,348,126,242),(26,400,193,294,90,347,127,241),(27,399,194,293,91,346,128,240),(28,398,195,292,92,345,129,239),(29,397,196,291,93,344,130,238),(30,396,197,290,94,343,131,237),(31,395,198,289,95,342,132,236),(32,394,199,288,96,341,133,235),(33,393,200,287,97,340,134,234),(34,392,201,286,98,339,135,233),(35,391,202,285,99,338,136,232),(36,390,203,284,100,337,137,231),(37,389,204,283,101,336,138,230),(38,388,205,282,102,335,139,229),(39,387,206,281,103,334,140,228),(40,386,207,280,104,333,141,227),(41,385,208,279,105,332,142,226),(42,384,209,278,106,331,143,225),(43,383,210,277,54,330,144,224),(44,382,211,276,55,329,145,223),(45,381,212,275,56,328,146,222),(46,380,160,274,57,327,147,221),(47,379,161,273,58,326,148,220),(48,378,162,272,59,325,149,219),(49,377,163,271,60,324,150,218),(50,376,164,270,61,323,151,217),(51,375,165,269,62,322,152,216),(52,374,166,268,63,321,153,215),(53,373,167,267,64,320,154,214)]])
112 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 53A | ··· | 53Z | 106A | ··· | 106Z | 212A | ··· | 212AZ |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 53 | ··· | 53 | 106 | ··· | 106 | 212 | ··· | 212 |
| size | 1 | 1 | 1 | 1 | 53 | 53 | 53 | 53 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D53 | Dic53 | C53⋊2C8 |
| kernel | C53⋊2C8 | C212 | C106 | C53 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 26 | 26 | 52 |
Matrix representation of C53⋊2C8 ►in GL2(𝔽1697) generated by
| 0 | 1 |
| 1696 | 104 |
| 424 | 1437 |
| 1411 | 1273 |
G:=sub<GL(2,GF(1697))| [0,1696,1,104],[424,1411,1437,1273] >;
C53⋊2C8 in GAP, Magma, Sage, TeX
C_{53}\rtimes_2C_8 % in TeX
G:=Group("C53:2C8"); // GroupNames label
G:=SmallGroup(424,1);
// by ID
G=gap.SmallGroup(424,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-53,8,21,6659]);
// Polycyclic
G:=Group<a,b|a^53=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export