metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C51⋊5C8, C68.2S3, C4.2D51, C6.Dic17, C2.Dic51, C102.3C4, C204.2C2, C12.2D17, C34.2Dic3, C3⋊(C17⋊3C8), C17⋊3(C3⋊C8), SmallGroup(408,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C51⋊5C8 |
Generators and relations for C51⋊5C8
G = < a,b | a51=b8=1, bab-1=a-1 >
Smallest permutation representation of C51⋊5C8
►Regular action on 408 points
Generators in S408
(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)
(1 360 154 300 96 328 135 246)(2 359 155 299 97 327 136 245)(3 358 156 298 98 326 137 244)(4 408 157 297 99 325 138 243)(5 407 158 296 100 324 139 242)(6 406 159 295 101 323 140 241)(7 405 160 294 102 322 141 240)(8 404 161 293 52 321 142 239)(9 403 162 292 53 320 143 238)(10 402 163 291 54 319 144 237)(11 401 164 290 55 318 145 236)(12 400 165 289 56 317 146 235)(13 399 166 288 57 316 147 234)(14 398 167 287 58 315 148 233)(15 397 168 286 59 314 149 232)(16 396 169 285 60 313 150 231)(17 395 170 284 61 312 151 230)(18 394 171 283 62 311 152 229)(19 393 172 282 63 310 153 228)(20 392 173 281 64 309 103 227)(21 391 174 280 65 308 104 226)(22 390 175 279 66 307 105 225)(23 389 176 278 67 357 106 224)(24 388 177 277 68 356 107 223)(25 387 178 276 69 355 108 222)(26 386 179 275 70 354 109 221)(27 385 180 274 71 353 110 220)(28 384 181 273 72 352 111 219)(29 383 182 272 73 351 112 218)(30 382 183 271 74 350 113 217)(31 381 184 270 75 349 114 216)(32 380 185 269 76 348 115 215)(33 379 186 268 77 347 116 214)(34 378 187 267 78 346 117 213)(35 377 188 266 79 345 118 212)(36 376 189 265 80 344 119 211)(37 375 190 264 81 343 120 210)(38 374 191 263 82 342 121 209)(39 373 192 262 83 341 122 208)(40 372 193 261 84 340 123 207)(41 371 194 260 85 339 124 206)(42 370 195 259 86 338 125 205)(43 369 196 258 87 337 126 255)(44 368 197 257 88 336 127 254)(45 367 198 256 89 335 128 253)(46 366 199 306 90 334 129 252)(47 365 200 305 91 333 130 251)(48 364 201 304 92 332 131 250)(49 363 202 303 93 331 132 249)(50 362 203 302 94 330 133 248)(51 361 204 301 95 329 134 247)
G:=sub<Sym(408)| (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), (1,360,154,300,96,328,135,246)(2,359,155,299,97,327,136,245)(3,358,156,298,98,326,137,244)(4,408,157,297,99,325,138,243)(5,407,158,296,100,324,139,242)(6,406,159,295,101,323,140,241)(7,405,160,294,102,322,141,240)(8,404,161,293,52,321,142,239)(9,403,162,292,53,320,143,238)(10,402,163,291,54,319,144,237)(11,401,164,290,55,318,145,236)(12,400,165,289,56,317,146,235)(13,399,166,288,57,316,147,234)(14,398,167,287,58,315,148,233)(15,397,168,286,59,314,149,232)(16,396,169,285,60,313,150,231)(17,395,170,284,61,312,151,230)(18,394,171,283,62,311,152,229)(19,393,172,282,63,310,153,228)(20,392,173,281,64,309,103,227)(21,391,174,280,65,308,104,226)(22,390,175,279,66,307,105,225)(23,389,176,278,67,357,106,224)(24,388,177,277,68,356,107,223)(25,387,178,276,69,355,108,222)(26,386,179,275,70,354,109,221)(27,385,180,274,71,353,110,220)(28,384,181,273,72,352,111,219)(29,383,182,272,73,351,112,218)(30,382,183,271,74,350,113,217)(31,381,184,270,75,349,114,216)(32,380,185,269,76,348,115,215)(33,379,186,268,77,347,116,214)(34,378,187,267,78,346,117,213)(35,377,188,266,79,345,118,212)(36,376,189,265,80,344,119,211)(37,375,190,264,81,343,120,210)(38,374,191,263,82,342,121,209)(39,373,192,262,83,341,122,208)(40,372,193,261,84,340,123,207)(41,371,194,260,85,339,124,206)(42,370,195,259,86,338,125,205)(43,369,196,258,87,337,126,255)(44,368,197,257,88,336,127,254)(45,367,198,256,89,335,128,253)(46,366,199,306,90,334,129,252)(47,365,200,305,91,333,130,251)(48,364,201,304,92,332,131,250)(49,363,202,303,93,331,132,249)(50,362,203,302,94,330,133,248)(51,361,204,301,95,329,134,247)>;
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), (1,360,154,300,96,328,135,246)(2,359,155,299,97,327,136,245)(3,358,156,298,98,326,137,244)(4,408,157,297,99,325,138,243)(5,407,158,296,100,324,139,242)(6,406,159,295,101,323,140,241)(7,405,160,294,102,322,141,240)(8,404,161,293,52,321,142,239)(9,403,162,292,53,320,143,238)(10,402,163,291,54,319,144,237)(11,401,164,290,55,318,145,236)(12,400,165,289,56,317,146,235)(13,399,166,288,57,316,147,234)(14,398,167,287,58,315,148,233)(15,397,168,286,59,314,149,232)(16,396,169,285,60,313,150,231)(17,395,170,284,61,312,151,230)(18,394,171,283,62,311,152,229)(19,393,172,282,63,310,153,228)(20,392,173,281,64,309,103,227)(21,391,174,280,65,308,104,226)(22,390,175,279,66,307,105,225)(23,389,176,278,67,357,106,224)(24,388,177,277,68,356,107,223)(25,387,178,276,69,355,108,222)(26,386,179,275,70,354,109,221)(27,385,180,274,71,353,110,220)(28,384,181,273,72,352,111,219)(29,383,182,272,73,351,112,218)(30,382,183,271,74,350,113,217)(31,381,184,270,75,349,114,216)(32,380,185,269,76,348,115,215)(33,379,186,268,77,347,116,214)(34,378,187,267,78,346,117,213)(35,377,188,266,79,345,118,212)(36,376,189,265,80,344,119,211)(37,375,190,264,81,343,120,210)(38,374,191,263,82,342,121,209)(39,373,192,262,83,341,122,208)(40,372,193,261,84,340,123,207)(41,371,194,260,85,339,124,206)(42,370,195,259,86,338,125,205)(43,369,196,258,87,337,126,255)(44,368,197,257,88,336,127,254)(45,367,198,256,89,335,128,253)(46,366,199,306,90,334,129,252)(47,365,200,305,91,333,130,251)(48,364,201,304,92,332,131,250)(49,363,202,303,93,331,132,249)(50,362,203,302,94,330,133,248)(51,361,204,301,95,329,134,247) );
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)], [(1,360,154,300,96,328,135,246),(2,359,155,299,97,327,136,245),(3,358,156,298,98,326,137,244),(4,408,157,297,99,325,138,243),(5,407,158,296,100,324,139,242),(6,406,159,295,101,323,140,241),(7,405,160,294,102,322,141,240),(8,404,161,293,52,321,142,239),(9,403,162,292,53,320,143,238),(10,402,163,291,54,319,144,237),(11,401,164,290,55,318,145,236),(12,400,165,289,56,317,146,235),(13,399,166,288,57,316,147,234),(14,398,167,287,58,315,148,233),(15,397,168,286,59,314,149,232),(16,396,169,285,60,313,150,231),(17,395,170,284,61,312,151,230),(18,394,171,283,62,311,152,229),(19,393,172,282,63,310,153,228),(20,392,173,281,64,309,103,227),(21,391,174,280,65,308,104,226),(22,390,175,279,66,307,105,225),(23,389,176,278,67,357,106,224),(24,388,177,277,68,356,107,223),(25,387,178,276,69,355,108,222),(26,386,179,275,70,354,109,221),(27,385,180,274,71,353,110,220),(28,384,181,273,72,352,111,219),(29,383,182,272,73,351,112,218),(30,382,183,271,74,350,113,217),(31,381,184,270,75,349,114,216),(32,380,185,269,76,348,115,215),(33,379,186,268,77,347,116,214),(34,378,187,267,78,346,117,213),(35,377,188,266,79,345,118,212),(36,376,189,265,80,344,119,211),(37,375,190,264,81,343,120,210),(38,374,191,263,82,342,121,209),(39,373,192,262,83,341,122,208),(40,372,193,261,84,340,123,207),(41,371,194,260,85,339,124,206),(42,370,195,259,86,338,125,205),(43,369,196,258,87,337,126,255),(44,368,197,257,88,336,127,254),(45,367,198,256,89,335,128,253),(46,366,199,306,90,334,129,252),(47,365,200,305,91,333,130,251),(48,364,201,304,92,332,131,250),(49,363,202,303,93,331,132,249),(50,362,203,302,94,330,133,248),(51,361,204,301,95,329,134,247)]])
108 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 51 | 51 | 51 | 51 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 | D17 | Dic17 | D51 | C17⋊3C8 | Dic51 | C51⋊5C8 |
| kernel | C51⋊5C8 | C204 | C102 | C51 | C68 | C34 | C17 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of C51⋊5C8 ►in GL4(𝔽409) generated by
| 141 | 255 | 0 | 0 |
| 154 | 206 | 0 | 0 |
| 0 | 0 | 398 | 186 |
| 0 | 0 | 129 | 347 |
| 226 | 192 | 0 | 0 |
| 1 | 183 | 0 | 0 |
| 0 | 0 | 122 | 237 |
| 0 | 0 | 165 | 287 |
G:=sub<GL(4,GF(409))| [141,154,0,0,255,206,0,0,0,0,398,129,0,0,186,347],[226,1,0,0,192,183,0,0,0,0,122,165,0,0,237,287] >;
C51⋊5C8 in GAP, Magma, Sage, TeX
C_{51}\rtimes_5C_8 % in TeX
G:=Group("C51:5C8"); // GroupNames label
G:=SmallGroup(408,3);
// by ID
G=gap.SmallGroup(408,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,10,26,323,9604]);
// Polycyclic
G:=Group<a,b|a^51=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export