metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C52⋊1C8, Dic13.4Q8, C26.1M4(2), Dic13.12D4, C4⋊(C13⋊C8), C13⋊1(C4⋊C8), C26.8(C2×C8), (C2×C52).5C4, C26.5(C4⋊C4), C2.1(C52⋊C4), (C2×Dic13).9C4, C2.1(C52.C4), (C4×Dic13).12C2, (C2×Dic13).49C22, C2.4(C2×C13⋊C8), (C2×C13⋊C8).1C2, (C2×C4).6(C13⋊C4), (C2×C26).4(C2×C4), C22.9(C2×C13⋊C4), SmallGroup(416,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊C8
G = < a,b | a52=b8=1, bab-1=a31 >
Smallest permutation representation of C52⋊C8
►Regular action on 416 points
Generators in S416
(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)
(1 159 248 54 293 410 116 355)(2 206 221 85 294 405 141 334)(3 201 246 64 295 400 114 313)(4 196 219 95 296 395 139 344)(5 191 244 74 297 390 112 323)(6 186 217 53 298 385 137 354)(7 181 242 84 299 380 110 333)(8 176 215 63 300 375 135 364)(9 171 240 94 301 370 108 343)(10 166 213 73 302 365 133 322)(11 161 238 104 303 412 106 353)(12 208 211 83 304 407 131 332)(13 203 236 62 305 402 156 363)(14 198 209 93 306 397 129 342)(15 193 234 72 307 392 154 321)(16 188 259 103 308 387 127 352)(17 183 232 82 309 382 152 331)(18 178 257 61 310 377 125 362)(19 173 230 92 311 372 150 341)(20 168 255 71 312 367 123 320)(21 163 228 102 261 414 148 351)(22 158 253 81 262 409 121 330)(23 205 226 60 263 404 146 361)(24 200 251 91 264 399 119 340)(25 195 224 70 265 394 144 319)(26 190 249 101 266 389 117 350)(27 185 222 80 267 384 142 329)(28 180 247 59 268 379 115 360)(29 175 220 90 269 374 140 339)(30 170 245 69 270 369 113 318)(31 165 218 100 271 416 138 349)(32 160 243 79 272 411 111 328)(33 207 216 58 273 406 136 359)(34 202 241 89 274 401 109 338)(35 197 214 68 275 396 134 317)(36 192 239 99 276 391 107 348)(37 187 212 78 277 386 132 327)(38 182 237 57 278 381 105 358)(39 177 210 88 279 376 130 337)(40 172 235 67 280 371 155 316)(41 167 260 98 281 366 128 347)(42 162 233 77 282 413 153 326)(43 157 258 56 283 408 126 357)(44 204 231 87 284 403 151 336)(45 199 256 66 285 398 124 315)(46 194 229 97 286 393 149 346)(47 189 254 76 287 388 122 325)(48 184 227 55 288 383 147 356)(49 179 252 86 289 378 120 335)(50 174 225 65 290 373 145 314)(51 169 250 96 291 368 118 345)(52 164 223 75 292 415 143 324)
G:=sub<Sym(416)| (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), (1,159,248,54,293,410,116,355)(2,206,221,85,294,405,141,334)(3,201,246,64,295,400,114,313)(4,196,219,95,296,395,139,344)(5,191,244,74,297,390,112,323)(6,186,217,53,298,385,137,354)(7,181,242,84,299,380,110,333)(8,176,215,63,300,375,135,364)(9,171,240,94,301,370,108,343)(10,166,213,73,302,365,133,322)(11,161,238,104,303,412,106,353)(12,208,211,83,304,407,131,332)(13,203,236,62,305,402,156,363)(14,198,209,93,306,397,129,342)(15,193,234,72,307,392,154,321)(16,188,259,103,308,387,127,352)(17,183,232,82,309,382,152,331)(18,178,257,61,310,377,125,362)(19,173,230,92,311,372,150,341)(20,168,255,71,312,367,123,320)(21,163,228,102,261,414,148,351)(22,158,253,81,262,409,121,330)(23,205,226,60,263,404,146,361)(24,200,251,91,264,399,119,340)(25,195,224,70,265,394,144,319)(26,190,249,101,266,389,117,350)(27,185,222,80,267,384,142,329)(28,180,247,59,268,379,115,360)(29,175,220,90,269,374,140,339)(30,170,245,69,270,369,113,318)(31,165,218,100,271,416,138,349)(32,160,243,79,272,411,111,328)(33,207,216,58,273,406,136,359)(34,202,241,89,274,401,109,338)(35,197,214,68,275,396,134,317)(36,192,239,99,276,391,107,348)(37,187,212,78,277,386,132,327)(38,182,237,57,278,381,105,358)(39,177,210,88,279,376,130,337)(40,172,235,67,280,371,155,316)(41,167,260,98,281,366,128,347)(42,162,233,77,282,413,153,326)(43,157,258,56,283,408,126,357)(44,204,231,87,284,403,151,336)(45,199,256,66,285,398,124,315)(46,194,229,97,286,393,149,346)(47,189,254,76,287,388,122,325)(48,184,227,55,288,383,147,356)(49,179,252,86,289,378,120,335)(50,174,225,65,290,373,145,314)(51,169,250,96,291,368,118,345)(52,164,223,75,292,415,143,324)>;
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), (1,159,248,54,293,410,116,355)(2,206,221,85,294,405,141,334)(3,201,246,64,295,400,114,313)(4,196,219,95,296,395,139,344)(5,191,244,74,297,390,112,323)(6,186,217,53,298,385,137,354)(7,181,242,84,299,380,110,333)(8,176,215,63,300,375,135,364)(9,171,240,94,301,370,108,343)(10,166,213,73,302,365,133,322)(11,161,238,104,303,412,106,353)(12,208,211,83,304,407,131,332)(13,203,236,62,305,402,156,363)(14,198,209,93,306,397,129,342)(15,193,234,72,307,392,154,321)(16,188,259,103,308,387,127,352)(17,183,232,82,309,382,152,331)(18,178,257,61,310,377,125,362)(19,173,230,92,311,372,150,341)(20,168,255,71,312,367,123,320)(21,163,228,102,261,414,148,351)(22,158,253,81,262,409,121,330)(23,205,226,60,263,404,146,361)(24,200,251,91,264,399,119,340)(25,195,224,70,265,394,144,319)(26,190,249,101,266,389,117,350)(27,185,222,80,267,384,142,329)(28,180,247,59,268,379,115,360)(29,175,220,90,269,374,140,339)(30,170,245,69,270,369,113,318)(31,165,218,100,271,416,138,349)(32,160,243,79,272,411,111,328)(33,207,216,58,273,406,136,359)(34,202,241,89,274,401,109,338)(35,197,214,68,275,396,134,317)(36,192,239,99,276,391,107,348)(37,187,212,78,277,386,132,327)(38,182,237,57,278,381,105,358)(39,177,210,88,279,376,130,337)(40,172,235,67,280,371,155,316)(41,167,260,98,281,366,128,347)(42,162,233,77,282,413,153,326)(43,157,258,56,283,408,126,357)(44,204,231,87,284,403,151,336)(45,199,256,66,285,398,124,315)(46,194,229,97,286,393,149,346)(47,189,254,76,287,388,122,325)(48,184,227,55,288,383,147,356)(49,179,252,86,289,378,120,335)(50,174,225,65,290,373,145,314)(51,169,250,96,291,368,118,345)(52,164,223,75,292,415,143,324) );
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)], [(1,159,248,54,293,410,116,355),(2,206,221,85,294,405,141,334),(3,201,246,64,295,400,114,313),(4,196,219,95,296,395,139,344),(5,191,244,74,297,390,112,323),(6,186,217,53,298,385,137,354),(7,181,242,84,299,380,110,333),(8,176,215,63,300,375,135,364),(9,171,240,94,301,370,108,343),(10,166,213,73,302,365,133,322),(11,161,238,104,303,412,106,353),(12,208,211,83,304,407,131,332),(13,203,236,62,305,402,156,363),(14,198,209,93,306,397,129,342),(15,193,234,72,307,392,154,321),(16,188,259,103,308,387,127,352),(17,183,232,82,309,382,152,331),(18,178,257,61,310,377,125,362),(19,173,230,92,311,372,150,341),(20,168,255,71,312,367,123,320),(21,163,228,102,261,414,148,351),(22,158,253,81,262,409,121,330),(23,205,226,60,263,404,146,361),(24,200,251,91,264,399,119,340),(25,195,224,70,265,394,144,319),(26,190,249,101,266,389,117,350),(27,185,222,80,267,384,142,329),(28,180,247,59,268,379,115,360),(29,175,220,90,269,374,140,339),(30,170,245,69,270,369,113,318),(31,165,218,100,271,416,138,349),(32,160,243,79,272,411,111,328),(33,207,216,58,273,406,136,359),(34,202,241,89,274,401,109,338),(35,197,214,68,275,396,134,317),(36,192,239,99,276,391,107,348),(37,187,212,78,277,386,132,327),(38,182,237,57,278,381,105,358),(39,177,210,88,279,376,130,337),(40,172,235,67,280,371,155,316),(41,167,260,98,281,366,128,347),(42,162,233,77,282,413,153,326),(43,157,258,56,283,408,126,357),(44,204,231,87,284,403,151,336),(45,199,256,66,285,398,124,315),(46,194,229,97,286,393,149,346),(47,189,254,76,287,388,122,325),(48,184,227,55,288,383,147,356),(49,179,252,86,289,378,120,335),(50,174,225,65,290,373,145,314),(51,169,250,96,291,368,118,345),(52,164,223,75,292,415,143,324)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) | C13⋊C4 | C13⋊C8 | C2×C13⋊C4 | C52.C4 | C52⋊C4 |
| kernel | C52⋊C8 | C4×Dic13 | C2×C13⋊C8 | C2×Dic13 | C2×C52 | C52 | Dic13 | Dic13 | C26 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 3 | 6 | 3 | 6 | 6 |
Matrix representation of C52⋊C8 ►in GL6(𝔽313)
| 106 | 199 | 0 | 0 | 0 | 0 |
| 299 | 207 | 0 | 0 | 0 | 0 |
| 0 | 0 | 283 | 21 | 70 | 69 |
| 0 | 0 | 1 | 20 | 62 | 291 |
| 0 | 0 | 275 | 294 | 131 | 78 |
| 0 | 0 | 0 | 43 | 304 | 18 |
| 216 | 251 | 0 | 0 | 0 | 0 |
| 283 | 97 | 0 | 0 | 0 | 0 |
| 0 | 0 | 74 | 63 | 141 | 60 |
| 0 | 0 | 246 | 0 | 197 | 101 |
| 0 | 0 | 71 | 70 | 235 | 150 |
| 0 | 0 | 29 | 258 | 234 | 4 |
G:=sub<GL(6,GF(313))| [106,299,0,0,0,0,199,207,0,0,0,0,0,0,283,1,275,0,0,0,21,20,294,43,0,0,70,62,131,304,0,0,69,291,78,18],[216,283,0,0,0,0,251,97,0,0,0,0,0,0,74,246,71,29,0,0,63,0,70,258,0,0,141,197,235,234,0,0,60,101,150,4] >;
C52⋊C8 in GAP, Magma, Sage, TeX
C_{52}\rtimes C_8 % in TeX
G:=Group("C52:C8"); // GroupNames label
G:=SmallGroup(416,76);
// by ID
G=gap.SmallGroup(416,76);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,9221,3473]);
// Polycyclic
G:=Group<a,b|a^52=b^8=1,b*a*b^-1=a^31>;
// generators/relations
Export