metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C52⋊3C8, C52.7Q8, C52.32D4, C4.16D52, C4.7Dic26, C42.2D13, C26.10M4(2), C4⋊(C13⋊2C8), C13⋊3(C4⋊C8), (C4×C52).4C2, C26.8(C4⋊C4), C26.16(C2×C8), (C2×C52).18C4, (C2×C4).89D26, (C2×C4).3Dic13, C2.1(C52⋊3C4), C2.2(C52.4C4), (C2×C52).103C22, C22.8(C2×Dic13), C2.3(C2×C13⋊2C8), (C2×C13⋊2C8).8C2, (C2×C26).46(C2×C4), SmallGroup(416,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊3C8
G = < a,b | a52=b8=1, bab-1=a-1 >
Smallest permutation representation of C52⋊3C8
►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)
(1 117 299 160 374 255 75 331)(2 116 300 159 375 254 76 330)(3 115 301 158 376 253 77 329)(4 114 302 157 377 252 78 328)(5 113 303 208 378 251 79 327)(6 112 304 207 379 250 80 326)(7 111 305 206 380 249 81 325)(8 110 306 205 381 248 82 324)(9 109 307 204 382 247 83 323)(10 108 308 203 383 246 84 322)(11 107 309 202 384 245 85 321)(12 106 310 201 385 244 86 320)(13 105 311 200 386 243 87 319)(14 156 312 199 387 242 88 318)(15 155 261 198 388 241 89 317)(16 154 262 197 389 240 90 316)(17 153 263 196 390 239 91 315)(18 152 264 195 391 238 92 314)(19 151 265 194 392 237 93 313)(20 150 266 193 393 236 94 364)(21 149 267 192 394 235 95 363)(22 148 268 191 395 234 96 362)(23 147 269 190 396 233 97 361)(24 146 270 189 397 232 98 360)(25 145 271 188 398 231 99 359)(26 144 272 187 399 230 100 358)(27 143 273 186 400 229 101 357)(28 142 274 185 401 228 102 356)(29 141 275 184 402 227 103 355)(30 140 276 183 403 226 104 354)(31 139 277 182 404 225 53 353)(32 138 278 181 405 224 54 352)(33 137 279 180 406 223 55 351)(34 136 280 179 407 222 56 350)(35 135 281 178 408 221 57 349)(36 134 282 177 409 220 58 348)(37 133 283 176 410 219 59 347)(38 132 284 175 411 218 60 346)(39 131 285 174 412 217 61 345)(40 130 286 173 413 216 62 344)(41 129 287 172 414 215 63 343)(42 128 288 171 415 214 64 342)(43 127 289 170 416 213 65 341)(44 126 290 169 365 212 66 340)(45 125 291 168 366 211 67 339)(46 124 292 167 367 210 68 338)(47 123 293 166 368 209 69 337)(48 122 294 165 369 260 70 336)(49 121 295 164 370 259 71 335)(50 120 296 163 371 258 72 334)(51 119 297 162 372 257 73 333)(52 118 298 161 373 256 74 332)
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), (1,117,299,160,374,255,75,331)(2,116,300,159,375,254,76,330)(3,115,301,158,376,253,77,329)(4,114,302,157,377,252,78,328)(5,113,303,208,378,251,79,327)(6,112,304,207,379,250,80,326)(7,111,305,206,380,249,81,325)(8,110,306,205,381,248,82,324)(9,109,307,204,382,247,83,323)(10,108,308,203,383,246,84,322)(11,107,309,202,384,245,85,321)(12,106,310,201,385,244,86,320)(13,105,311,200,386,243,87,319)(14,156,312,199,387,242,88,318)(15,155,261,198,388,241,89,317)(16,154,262,197,389,240,90,316)(17,153,263,196,390,239,91,315)(18,152,264,195,391,238,92,314)(19,151,265,194,392,237,93,313)(20,150,266,193,393,236,94,364)(21,149,267,192,394,235,95,363)(22,148,268,191,395,234,96,362)(23,147,269,190,396,233,97,361)(24,146,270,189,397,232,98,360)(25,145,271,188,398,231,99,359)(26,144,272,187,399,230,100,358)(27,143,273,186,400,229,101,357)(28,142,274,185,401,228,102,356)(29,141,275,184,402,227,103,355)(30,140,276,183,403,226,104,354)(31,139,277,182,404,225,53,353)(32,138,278,181,405,224,54,352)(33,137,279,180,406,223,55,351)(34,136,280,179,407,222,56,350)(35,135,281,178,408,221,57,349)(36,134,282,177,409,220,58,348)(37,133,283,176,410,219,59,347)(38,132,284,175,411,218,60,346)(39,131,285,174,412,217,61,345)(40,130,286,173,413,216,62,344)(41,129,287,172,414,215,63,343)(42,128,288,171,415,214,64,342)(43,127,289,170,416,213,65,341)(44,126,290,169,365,212,66,340)(45,125,291,168,366,211,67,339)(46,124,292,167,367,210,68,338)(47,123,293,166,368,209,69,337)(48,122,294,165,369,260,70,336)(49,121,295,164,370,259,71,335)(50,120,296,163,371,258,72,334)(51,119,297,162,372,257,73,333)(52,118,298,161,373,256,74,332)>;
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), (1,117,299,160,374,255,75,331)(2,116,300,159,375,254,76,330)(3,115,301,158,376,253,77,329)(4,114,302,157,377,252,78,328)(5,113,303,208,378,251,79,327)(6,112,304,207,379,250,80,326)(7,111,305,206,380,249,81,325)(8,110,306,205,381,248,82,324)(9,109,307,204,382,247,83,323)(10,108,308,203,383,246,84,322)(11,107,309,202,384,245,85,321)(12,106,310,201,385,244,86,320)(13,105,311,200,386,243,87,319)(14,156,312,199,387,242,88,318)(15,155,261,198,388,241,89,317)(16,154,262,197,389,240,90,316)(17,153,263,196,390,239,91,315)(18,152,264,195,391,238,92,314)(19,151,265,194,392,237,93,313)(20,150,266,193,393,236,94,364)(21,149,267,192,394,235,95,363)(22,148,268,191,395,234,96,362)(23,147,269,190,396,233,97,361)(24,146,270,189,397,232,98,360)(25,145,271,188,398,231,99,359)(26,144,272,187,399,230,100,358)(27,143,273,186,400,229,101,357)(28,142,274,185,401,228,102,356)(29,141,275,184,402,227,103,355)(30,140,276,183,403,226,104,354)(31,139,277,182,404,225,53,353)(32,138,278,181,405,224,54,352)(33,137,279,180,406,223,55,351)(34,136,280,179,407,222,56,350)(35,135,281,178,408,221,57,349)(36,134,282,177,409,220,58,348)(37,133,283,176,410,219,59,347)(38,132,284,175,411,218,60,346)(39,131,285,174,412,217,61,345)(40,130,286,173,413,216,62,344)(41,129,287,172,414,215,63,343)(42,128,288,171,415,214,64,342)(43,127,289,170,416,213,65,341)(44,126,290,169,365,212,66,340)(45,125,291,168,366,211,67,339)(46,124,292,167,367,210,68,338)(47,123,293,166,368,209,69,337)(48,122,294,165,369,260,70,336)(49,121,295,164,370,259,71,335)(50,120,296,163,371,258,72,334)(51,119,297,162,372,257,73,333)(52,118,298,161,373,256,74,332) );
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)], [(1,117,299,160,374,255,75,331),(2,116,300,159,375,254,76,330),(3,115,301,158,376,253,77,329),(4,114,302,157,377,252,78,328),(5,113,303,208,378,251,79,327),(6,112,304,207,379,250,80,326),(7,111,305,206,380,249,81,325),(8,110,306,205,381,248,82,324),(9,109,307,204,382,247,83,323),(10,108,308,203,383,246,84,322),(11,107,309,202,384,245,85,321),(12,106,310,201,385,244,86,320),(13,105,311,200,386,243,87,319),(14,156,312,199,387,242,88,318),(15,155,261,198,388,241,89,317),(16,154,262,197,389,240,90,316),(17,153,263,196,390,239,91,315),(18,152,264,195,391,238,92,314),(19,151,265,194,392,237,93,313),(20,150,266,193,393,236,94,364),(21,149,267,192,394,235,95,363),(22,148,268,191,395,234,96,362),(23,147,269,190,396,233,97,361),(24,146,270,189,397,232,98,360),(25,145,271,188,398,231,99,359),(26,144,272,187,399,230,100,358),(27,143,273,186,400,229,101,357),(28,142,274,185,401,228,102,356),(29,141,275,184,402,227,103,355),(30,140,276,183,403,226,104,354),(31,139,277,182,404,225,53,353),(32,138,278,181,405,224,54,352),(33,137,279,180,406,223,55,351),(34,136,280,179,407,222,56,350),(35,135,281,178,408,221,57,349),(36,134,282,177,409,220,58,348),(37,133,283,176,410,219,59,347),(38,132,284,175,411,218,60,346),(39,131,285,174,412,217,61,345),(40,130,286,173,413,216,62,344),(41,129,287,172,414,215,63,343),(42,128,288,171,415,214,64,342),(43,127,289,170,416,213,65,341),(44,126,290,169,365,212,66,340),(45,125,291,168,366,211,67,339),(46,124,292,167,367,210,68,338),(47,123,293,166,368,209,69,337),(48,122,294,165,369,260,70,336),(49,121,295,164,370,259,71,335),(50,120,296,163,371,258,72,334),(51,119,297,162,372,257,73,333),(52,118,298,161,373,256,74,332)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) | D13 | Dic13 | D26 | C13⋊2C8 | Dic26 | D52 | C52.4C4 |
| kernel | C52⋊3C8 | C2×C13⋊2C8 | C4×C52 | C2×C52 | C52 | C52 | C52 | C26 | C42 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 6 | 12 | 6 | 24 | 12 | 12 | 24 |
Matrix representation of C52⋊3C8 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 259 | 123 |
| 0 | 190 | 199 |
| 188 | 0 | 0 |
| 0 | 153 | 274 |
| 0 | 207 | 160 |
G:=sub<GL(3,GF(313))| [1,0,0,0,259,190,0,123,199],[188,0,0,0,153,207,0,274,160] >;
C52⋊3C8 in GAP, Magma, Sage, TeX
C_{52}\rtimes_3C_8 % in TeX
G:=Group("C52:3C8"); // GroupNames label
G:=SmallGroup(416,11);
// by ID
G=gap.SmallGroup(416,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,13829]);
// Polycyclic
G:=Group<a,b|a^52=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export