metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C51⋊3C8, C34.Dic3, C102.1C4, Dic17.2S3, C17⋊2(C3⋊C8), C3⋊(C17⋊2C8), C6.(C17⋊C4), C2.(C51⋊C4), (C3×Dic17).3C2, SmallGroup(408,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C51⋊3C8 |
Generators and relations for C51⋊3C8
G = < a,b | a51=b8=1, bab-1=a47 >
Smallest permutation representation of C51⋊3C8
►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)
(1 362 203 287 96 341 122 215)(2 400 168 283 97 328 138 211)(3 387 184 279 98 315 103 207)(4 374 200 275 99 353 119 254)(5 361 165 271 100 340 135 250)(6 399 181 267 101 327 151 246)(7 386 197 263 102 314 116 242)(8 373 162 259 52 352 132 238)(9 360 178 306 53 339 148 234)(10 398 194 302 54 326 113 230)(11 385 159 298 55 313 129 226)(12 372 175 294 56 351 145 222)(13 359 191 290 57 338 110 218)(14 397 156 286 58 325 126 214)(15 384 172 282 59 312 142 210)(16 371 188 278 60 350 107 206)(17 358 204 274 61 337 123 253)(18 396 169 270 62 324 139 249)(19 383 185 266 63 311 104 245)(20 370 201 262 64 349 120 241)(21 408 166 258 65 336 136 237)(22 395 182 305 66 323 152 233)(23 382 198 301 67 310 117 229)(24 369 163 297 68 348 133 225)(25 407 179 293 69 335 149 221)(26 394 195 289 70 322 114 217)(27 381 160 285 71 309 130 213)(28 368 176 281 72 347 146 209)(29 406 192 277 73 334 111 205)(30 393 157 273 74 321 127 252)(31 380 173 269 75 308 143 248)(32 367 189 265 76 346 108 244)(33 405 154 261 77 333 124 240)(34 392 170 257 78 320 140 236)(35 379 186 304 79 307 105 232)(36 366 202 300 80 345 121 228)(37 404 167 296 81 332 137 224)(38 391 183 292 82 319 153 220)(39 378 199 288 83 357 118 216)(40 365 164 284 84 344 134 212)(41 403 180 280 85 331 150 208)(42 390 196 276 86 318 115 255)(43 377 161 272 87 356 131 251)(44 364 177 268 88 343 147 247)(45 402 193 264 89 330 112 243)(46 389 158 260 90 317 128 239)(47 376 174 256 91 355 144 235)(48 363 190 303 92 342 109 231)(49 401 155 299 93 329 125 227)(50 388 171 295 94 316 141 223)(51 375 187 291 95 354 106 219)
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), (1,362,203,287,96,341,122,215)(2,400,168,283,97,328,138,211)(3,387,184,279,98,315,103,207)(4,374,200,275,99,353,119,254)(5,361,165,271,100,340,135,250)(6,399,181,267,101,327,151,246)(7,386,197,263,102,314,116,242)(8,373,162,259,52,352,132,238)(9,360,178,306,53,339,148,234)(10,398,194,302,54,326,113,230)(11,385,159,298,55,313,129,226)(12,372,175,294,56,351,145,222)(13,359,191,290,57,338,110,218)(14,397,156,286,58,325,126,214)(15,384,172,282,59,312,142,210)(16,371,188,278,60,350,107,206)(17,358,204,274,61,337,123,253)(18,396,169,270,62,324,139,249)(19,383,185,266,63,311,104,245)(20,370,201,262,64,349,120,241)(21,408,166,258,65,336,136,237)(22,395,182,305,66,323,152,233)(23,382,198,301,67,310,117,229)(24,369,163,297,68,348,133,225)(25,407,179,293,69,335,149,221)(26,394,195,289,70,322,114,217)(27,381,160,285,71,309,130,213)(28,368,176,281,72,347,146,209)(29,406,192,277,73,334,111,205)(30,393,157,273,74,321,127,252)(31,380,173,269,75,308,143,248)(32,367,189,265,76,346,108,244)(33,405,154,261,77,333,124,240)(34,392,170,257,78,320,140,236)(35,379,186,304,79,307,105,232)(36,366,202,300,80,345,121,228)(37,404,167,296,81,332,137,224)(38,391,183,292,82,319,153,220)(39,378,199,288,83,357,118,216)(40,365,164,284,84,344,134,212)(41,403,180,280,85,331,150,208)(42,390,196,276,86,318,115,255)(43,377,161,272,87,356,131,251)(44,364,177,268,88,343,147,247)(45,402,193,264,89,330,112,243)(46,389,158,260,90,317,128,239)(47,376,174,256,91,355,144,235)(48,363,190,303,92,342,109,231)(49,401,155,299,93,329,125,227)(50,388,171,295,94,316,141,223)(51,375,187,291,95,354,106,219)>;
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), (1,362,203,287,96,341,122,215)(2,400,168,283,97,328,138,211)(3,387,184,279,98,315,103,207)(4,374,200,275,99,353,119,254)(5,361,165,271,100,340,135,250)(6,399,181,267,101,327,151,246)(7,386,197,263,102,314,116,242)(8,373,162,259,52,352,132,238)(9,360,178,306,53,339,148,234)(10,398,194,302,54,326,113,230)(11,385,159,298,55,313,129,226)(12,372,175,294,56,351,145,222)(13,359,191,290,57,338,110,218)(14,397,156,286,58,325,126,214)(15,384,172,282,59,312,142,210)(16,371,188,278,60,350,107,206)(17,358,204,274,61,337,123,253)(18,396,169,270,62,324,139,249)(19,383,185,266,63,311,104,245)(20,370,201,262,64,349,120,241)(21,408,166,258,65,336,136,237)(22,395,182,305,66,323,152,233)(23,382,198,301,67,310,117,229)(24,369,163,297,68,348,133,225)(25,407,179,293,69,335,149,221)(26,394,195,289,70,322,114,217)(27,381,160,285,71,309,130,213)(28,368,176,281,72,347,146,209)(29,406,192,277,73,334,111,205)(30,393,157,273,74,321,127,252)(31,380,173,269,75,308,143,248)(32,367,189,265,76,346,108,244)(33,405,154,261,77,333,124,240)(34,392,170,257,78,320,140,236)(35,379,186,304,79,307,105,232)(36,366,202,300,80,345,121,228)(37,404,167,296,81,332,137,224)(38,391,183,292,82,319,153,220)(39,378,199,288,83,357,118,216)(40,365,164,284,84,344,134,212)(41,403,180,280,85,331,150,208)(42,390,196,276,86,318,115,255)(43,377,161,272,87,356,131,251)(44,364,177,268,88,343,147,247)(45,402,193,264,89,330,112,243)(46,389,158,260,90,317,128,239)(47,376,174,256,91,355,144,235)(48,363,190,303,92,342,109,231)(49,401,155,299,93,329,125,227)(50,388,171,295,94,316,141,223)(51,375,187,291,95,354,106,219) );
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)], [(1,362,203,287,96,341,122,215),(2,400,168,283,97,328,138,211),(3,387,184,279,98,315,103,207),(4,374,200,275,99,353,119,254),(5,361,165,271,100,340,135,250),(6,399,181,267,101,327,151,246),(7,386,197,263,102,314,116,242),(8,373,162,259,52,352,132,238),(9,360,178,306,53,339,148,234),(10,398,194,302,54,326,113,230),(11,385,159,298,55,313,129,226),(12,372,175,294,56,351,145,222),(13,359,191,290,57,338,110,218),(14,397,156,286,58,325,126,214),(15,384,172,282,59,312,142,210),(16,371,188,278,60,350,107,206),(17,358,204,274,61,337,123,253),(18,396,169,270,62,324,139,249),(19,383,185,266,63,311,104,245),(20,370,201,262,64,349,120,241),(21,408,166,258,65,336,136,237),(22,395,182,305,66,323,152,233),(23,382,198,301,67,310,117,229),(24,369,163,297,68,348,133,225),(25,407,179,293,69,335,149,221),(26,394,195,289,70,322,114,217),(27,381,160,285,71,309,130,213),(28,368,176,281,72,347,146,209),(29,406,192,277,73,334,111,205),(30,393,157,273,74,321,127,252),(31,380,173,269,75,308,143,248),(32,367,189,265,76,346,108,244),(33,405,154,261,77,333,124,240),(34,392,170,257,78,320,140,236),(35,379,186,304,79,307,105,232),(36,366,202,300,80,345,121,228),(37,404,167,296,81,332,137,224),(38,391,183,292,82,319,153,220),(39,378,199,288,83,357,118,216),(40,365,164,284,84,344,134,212),(41,403,180,280,85,331,150,208),(42,390,196,276,86,318,115,255),(43,377,161,272,87,356,131,251),(44,364,177,268,88,343,147,247),(45,402,193,264,89,330,112,243),(46,389,158,260,90,317,128,239),(47,376,174,256,91,355,144,235),(48,363,190,303,92,342,109,231),(49,401,155,299,93,329,125,227),(50,388,171,295,94,316,141,223),(51,375,187,291,95,354,106,219)]])
36 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 51A | ··· | 51H | 102A | ··· | 102H |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 17 | 17 | 17 | 17 | 34 | 34 | 34 | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 2 | 17 | 17 | 2 | 51 | 51 | 51 | 51 | 34 | 34 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 | C17⋊C4 | C17⋊2C8 | C51⋊C4 | C51⋊3C8 |
| kernel | C51⋊3C8 | C3×Dic17 | C102 | C51 | Dic17 | C34 | C17 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C51⋊3C8 ►in GL6(𝔽409)
| 302 | 80 | 0 | 0 | 0 | 0 |
| 303 | 106 | 0 | 0 | 0 | 0 |
| 0 | 0 | 370 | 307 | 280 | 109 |
| 0 | 0 | 175 | 318 | 128 | 336 |
| 0 | 0 | 188 | 17 | 245 | 280 |
| 0 | 0 | 354 | 312 | 71 | 49 |
| 147 | 89 | 0 | 0 | 0 | 0 |
| 177 | 262 | 0 | 0 | 0 | 0 |
| 0 | 0 | 279 | 398 | 264 | 347 |
| 0 | 0 | 334 | 183 | 197 | 228 |
| 0 | 0 | 329 | 155 | 140 | 130 |
| 0 | 0 | 139 | 301 | 120 | 216 |
G:=sub<GL(6,GF(409))| [302,303,0,0,0,0,80,106,0,0,0,0,0,0,370,175,188,354,0,0,307,318,17,312,0,0,280,128,245,71,0,0,109,336,280,49],[147,177,0,0,0,0,89,262,0,0,0,0,0,0,279,334,329,139,0,0,398,183,155,301,0,0,264,197,140,120,0,0,347,228,130,216] >;
C51⋊3C8 in GAP, Magma, Sage, TeX
C_{51}\rtimes_3C_8 % in TeX
G:=Group("C51:3C8"); // GroupNames label
G:=SmallGroup(408,6);
// by ID
G=gap.SmallGroup(408,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,10,26,323,2404,4809]);
// Polycyclic
G:=Group<a,b|a^51=b^8=1,b*a*b^-1=a^47>;
// generators/relations
Export