metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C53⋊C8, C106.C4, Dic53.2C2, C2.(C53⋊C4), SmallGroup(424,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C53⋊C8 |
Generators and relations for C53⋊C8
G = < a,b | a53=b8=1, bab-1=a23 >
Smallest permutation representation of C53⋊C8
►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)
(1 378 160 305 74 329 107 253)(2 408 212 275 75 359 159 223)(3 385 211 298 76 336 158 246)(4 415 210 268 77 366 157 216)(5 392 209 291 78 343 156 239)(6 422 208 314 79 320 155 262)(7 399 207 284 80 350 154 232)(8 376 206 307 81 327 153 255)(9 406 205 277 82 357 152 225)(10 383 204 300 83 334 151 248)(11 413 203 270 84 364 150 218)(12 390 202 293 85 341 149 241)(13 420 201 316 86 371 148 264)(14 397 200 286 87 348 147 234)(15 374 199 309 88 325 146 257)(16 404 198 279 89 355 145 227)(17 381 197 302 90 332 144 250)(18 411 196 272 91 362 143 220)(19 388 195 295 92 339 142 243)(20 418 194 318 93 369 141 213)(21 395 193 288 94 346 140 236)(22 372 192 311 95 323 139 259)(23 402 191 281 96 353 138 229)(24 379 190 304 97 330 137 252)(25 409 189 274 98 360 136 222)(26 386 188 297 99 337 135 245)(27 416 187 267 100 367 134 215)(28 393 186 290 101 344 133 238)(29 423 185 313 102 321 132 261)(30 400 184 283 103 351 131 231)(31 377 183 306 104 328 130 254)(32 407 182 276 105 358 129 224)(33 384 181 299 106 335 128 247)(34 414 180 269 54 365 127 217)(35 391 179 292 55 342 126 240)(36 421 178 315 56 319 125 263)(37 398 177 285 57 349 124 233)(38 375 176 308 58 326 123 256)(39 405 175 278 59 356 122 226)(40 382 174 301 60 333 121 249)(41 412 173 271 61 363 120 219)(42 389 172 294 62 340 119 242)(43 419 171 317 63 370 118 265)(44 396 170 287 64 347 117 235)(45 373 169 310 65 324 116 258)(46 403 168 280 66 354 115 228)(47 380 167 303 67 331 114 251)(48 410 166 273 68 361 113 221)(49 387 165 296 69 338 112 244)(50 417 164 266 70 368 111 214)(51 394 163 289 71 345 110 237)(52 424 162 312 72 322 109 260)(53 401 161 282 73 352 108 230)
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), (1,378,160,305,74,329,107,253)(2,408,212,275,75,359,159,223)(3,385,211,298,76,336,158,246)(4,415,210,268,77,366,157,216)(5,392,209,291,78,343,156,239)(6,422,208,314,79,320,155,262)(7,399,207,284,80,350,154,232)(8,376,206,307,81,327,153,255)(9,406,205,277,82,357,152,225)(10,383,204,300,83,334,151,248)(11,413,203,270,84,364,150,218)(12,390,202,293,85,341,149,241)(13,420,201,316,86,371,148,264)(14,397,200,286,87,348,147,234)(15,374,199,309,88,325,146,257)(16,404,198,279,89,355,145,227)(17,381,197,302,90,332,144,250)(18,411,196,272,91,362,143,220)(19,388,195,295,92,339,142,243)(20,418,194,318,93,369,141,213)(21,395,193,288,94,346,140,236)(22,372,192,311,95,323,139,259)(23,402,191,281,96,353,138,229)(24,379,190,304,97,330,137,252)(25,409,189,274,98,360,136,222)(26,386,188,297,99,337,135,245)(27,416,187,267,100,367,134,215)(28,393,186,290,101,344,133,238)(29,423,185,313,102,321,132,261)(30,400,184,283,103,351,131,231)(31,377,183,306,104,328,130,254)(32,407,182,276,105,358,129,224)(33,384,181,299,106,335,128,247)(34,414,180,269,54,365,127,217)(35,391,179,292,55,342,126,240)(36,421,178,315,56,319,125,263)(37,398,177,285,57,349,124,233)(38,375,176,308,58,326,123,256)(39,405,175,278,59,356,122,226)(40,382,174,301,60,333,121,249)(41,412,173,271,61,363,120,219)(42,389,172,294,62,340,119,242)(43,419,171,317,63,370,118,265)(44,396,170,287,64,347,117,235)(45,373,169,310,65,324,116,258)(46,403,168,280,66,354,115,228)(47,380,167,303,67,331,114,251)(48,410,166,273,68,361,113,221)(49,387,165,296,69,338,112,244)(50,417,164,266,70,368,111,214)(51,394,163,289,71,345,110,237)(52,424,162,312,72,322,109,260)(53,401,161,282,73,352,108,230)>;
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), (1,378,160,305,74,329,107,253)(2,408,212,275,75,359,159,223)(3,385,211,298,76,336,158,246)(4,415,210,268,77,366,157,216)(5,392,209,291,78,343,156,239)(6,422,208,314,79,320,155,262)(7,399,207,284,80,350,154,232)(8,376,206,307,81,327,153,255)(9,406,205,277,82,357,152,225)(10,383,204,300,83,334,151,248)(11,413,203,270,84,364,150,218)(12,390,202,293,85,341,149,241)(13,420,201,316,86,371,148,264)(14,397,200,286,87,348,147,234)(15,374,199,309,88,325,146,257)(16,404,198,279,89,355,145,227)(17,381,197,302,90,332,144,250)(18,411,196,272,91,362,143,220)(19,388,195,295,92,339,142,243)(20,418,194,318,93,369,141,213)(21,395,193,288,94,346,140,236)(22,372,192,311,95,323,139,259)(23,402,191,281,96,353,138,229)(24,379,190,304,97,330,137,252)(25,409,189,274,98,360,136,222)(26,386,188,297,99,337,135,245)(27,416,187,267,100,367,134,215)(28,393,186,290,101,344,133,238)(29,423,185,313,102,321,132,261)(30,400,184,283,103,351,131,231)(31,377,183,306,104,328,130,254)(32,407,182,276,105,358,129,224)(33,384,181,299,106,335,128,247)(34,414,180,269,54,365,127,217)(35,391,179,292,55,342,126,240)(36,421,178,315,56,319,125,263)(37,398,177,285,57,349,124,233)(38,375,176,308,58,326,123,256)(39,405,175,278,59,356,122,226)(40,382,174,301,60,333,121,249)(41,412,173,271,61,363,120,219)(42,389,172,294,62,340,119,242)(43,419,171,317,63,370,118,265)(44,396,170,287,64,347,117,235)(45,373,169,310,65,324,116,258)(46,403,168,280,66,354,115,228)(47,380,167,303,67,331,114,251)(48,410,166,273,68,361,113,221)(49,387,165,296,69,338,112,244)(50,417,164,266,70,368,111,214)(51,394,163,289,71,345,110,237)(52,424,162,312,72,322,109,260)(53,401,161,282,73,352,108,230) );
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)], [(1,378,160,305,74,329,107,253),(2,408,212,275,75,359,159,223),(3,385,211,298,76,336,158,246),(4,415,210,268,77,366,157,216),(5,392,209,291,78,343,156,239),(6,422,208,314,79,320,155,262),(7,399,207,284,80,350,154,232),(8,376,206,307,81,327,153,255),(9,406,205,277,82,357,152,225),(10,383,204,300,83,334,151,248),(11,413,203,270,84,364,150,218),(12,390,202,293,85,341,149,241),(13,420,201,316,86,371,148,264),(14,397,200,286,87,348,147,234),(15,374,199,309,88,325,146,257),(16,404,198,279,89,355,145,227),(17,381,197,302,90,332,144,250),(18,411,196,272,91,362,143,220),(19,388,195,295,92,339,142,243),(20,418,194,318,93,369,141,213),(21,395,193,288,94,346,140,236),(22,372,192,311,95,323,139,259),(23,402,191,281,96,353,138,229),(24,379,190,304,97,330,137,252),(25,409,189,274,98,360,136,222),(26,386,188,297,99,337,135,245),(27,416,187,267,100,367,134,215),(28,393,186,290,101,344,133,238),(29,423,185,313,102,321,132,261),(30,400,184,283,103,351,131,231),(31,377,183,306,104,328,130,254),(32,407,182,276,105,358,129,224),(33,384,181,299,106,335,128,247),(34,414,180,269,54,365,127,217),(35,391,179,292,55,342,126,240),(36,421,178,315,56,319,125,263),(37,398,177,285,57,349,124,233),(38,375,176,308,58,326,123,256),(39,405,175,278,59,356,122,226),(40,382,174,301,60,333,121,249),(41,412,173,271,61,363,120,219),(42,389,172,294,62,340,119,242),(43,419,171,317,63,370,118,265),(44,396,170,287,64,347,117,235),(45,373,169,310,65,324,116,258),(46,403,168,280,66,354,115,228),(47,380,167,303,67,331,114,251),(48,410,166,273,68,361,113,221),(49,387,165,296,69,338,112,244),(50,417,164,266,70,368,111,214),(51,394,163,289,71,345,110,237),(52,424,162,312,72,322,109,260),(53,401,161,282,73,352,108,230)]])
34 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 53A | ··· | 53M | 106A | ··· | 106M |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 53 | ··· | 53 | 106 | ··· | 106 |
| size | 1 | 1 | 53 | 53 | 53 | 53 | 53 | 53 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | - | ||
| image | C1 | C2 | C4 | C8 | C53⋊C4 | C53⋊C8 |
| kernel | C53⋊C8 | Dic53 | C106 | C53 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 13 | 13 |
Matrix representation of C53⋊C8 ►in GL4(𝔽1697) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1696 | 710 | 1460 | 710 |
| 1298 | 1528 | 1461 | 396 |
| 277 | 1249 | 312 | 316 |
| 1163 | 1343 | 1684 | 1266 |
| 1343 | 381 | 110 | 860 |
G:=sub<GL(4,GF(1697))| [0,0,0,1696,1,0,0,710,0,1,0,1460,0,0,1,710],[1298,277,1163,1343,1528,1249,1343,381,1461,312,1684,110,396,316,1266,860] >;
C53⋊C8 in GAP, Magma, Sage, TeX
C_{53}\rtimes C_8 % in TeX
G:=Group("C53:C8"); // GroupNames label
G:=SmallGroup(424,3);
// by ID
G=gap.SmallGroup(424,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-53,8,21,3843,3335]);
// Polycyclic
G:=Group<a,b|a^53=b^8=1,b*a*b^-1=a^23>;
// generators/relations
Export