metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C25⋊C16, C50.C8, C20.5F5, C100.2C4, C5.(C5⋊C16), C2.(C25⋊C8), C10.1(C5⋊C8), C4.2(C25⋊C4), C25⋊2C8.2C2, SmallGroup(400,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C25⋊C16 |
Generators and relations for C25⋊C16
G = < a,b | a25=b16=1, bab-1=a18 >
Smallest permutation representation of C25⋊C16
►Regular action on 400 points
Generators in S400
(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)
(1 393 194 300 77 342 149 241 45 361 162 274 71 314 118 222)(2 400 193 293 78 349 148 234 46 368 161 267 72 321 117 215)(3 382 192 286 79 331 147 227 47 375 160 260 73 303 116 208)(4 389 191 279 80 338 146 245 48 357 159 253 74 310 115 201)(5 396 190 297 81 345 145 238 49 364 158 271 75 317 114 219)(6 378 189 290 82 327 144 231 50 371 157 264 51 324 113 212)(7 385 188 283 83 334 143 249 26 353 156 257 52 306 112 205)(8 392 187 276 84 341 142 242 27 360 155 275 53 313 111 223)(9 399 186 294 85 348 141 235 28 367 154 268 54 320 110 216)(10 381 185 287 86 330 140 228 29 374 153 261 55 302 109 209)(11 388 184 280 87 337 139 246 30 356 152 254 56 309 108 202)(12 395 183 298 88 344 138 239 31 363 151 272 57 316 107 220)(13 377 182 291 89 326 137 232 32 370 175 265 58 323 106 213)(14 384 181 284 90 333 136 250 33 352 174 258 59 305 105 206)(15 391 180 277 91 340 135 243 34 359 173 251 60 312 104 224)(16 398 179 295 92 347 134 236 35 366 172 269 61 319 103 217)(17 380 178 288 93 329 133 229 36 373 171 262 62 301 102 210)(18 387 177 281 94 336 132 247 37 355 170 255 63 308 101 203)(19 394 176 299 95 343 131 240 38 362 169 273 64 315 125 221)(20 376 200 292 96 350 130 233 39 369 168 266 65 322 124 214)(21 383 199 285 97 332 129 226 40 351 167 259 66 304 123 207)(22 390 198 278 98 339 128 244 41 358 166 252 67 311 122 225)(23 397 197 296 99 346 127 237 42 365 165 270 68 318 121 218)(24 379 196 289 100 328 126 230 43 372 164 263 69 325 120 211)(25 386 195 282 76 335 150 248 44 354 163 256 70 307 119 204)
G:=sub<Sym(400)| (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), (1,393,194,300,77,342,149,241,45,361,162,274,71,314,118,222)(2,400,193,293,78,349,148,234,46,368,161,267,72,321,117,215)(3,382,192,286,79,331,147,227,47,375,160,260,73,303,116,208)(4,389,191,279,80,338,146,245,48,357,159,253,74,310,115,201)(5,396,190,297,81,345,145,238,49,364,158,271,75,317,114,219)(6,378,189,290,82,327,144,231,50,371,157,264,51,324,113,212)(7,385,188,283,83,334,143,249,26,353,156,257,52,306,112,205)(8,392,187,276,84,341,142,242,27,360,155,275,53,313,111,223)(9,399,186,294,85,348,141,235,28,367,154,268,54,320,110,216)(10,381,185,287,86,330,140,228,29,374,153,261,55,302,109,209)(11,388,184,280,87,337,139,246,30,356,152,254,56,309,108,202)(12,395,183,298,88,344,138,239,31,363,151,272,57,316,107,220)(13,377,182,291,89,326,137,232,32,370,175,265,58,323,106,213)(14,384,181,284,90,333,136,250,33,352,174,258,59,305,105,206)(15,391,180,277,91,340,135,243,34,359,173,251,60,312,104,224)(16,398,179,295,92,347,134,236,35,366,172,269,61,319,103,217)(17,380,178,288,93,329,133,229,36,373,171,262,62,301,102,210)(18,387,177,281,94,336,132,247,37,355,170,255,63,308,101,203)(19,394,176,299,95,343,131,240,38,362,169,273,64,315,125,221)(20,376,200,292,96,350,130,233,39,369,168,266,65,322,124,214)(21,383,199,285,97,332,129,226,40,351,167,259,66,304,123,207)(22,390,198,278,98,339,128,244,41,358,166,252,67,311,122,225)(23,397,197,296,99,346,127,237,42,365,165,270,68,318,121,218)(24,379,196,289,100,328,126,230,43,372,164,263,69,325,120,211)(25,386,195,282,76,335,150,248,44,354,163,256,70,307,119,204)>;
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), (1,393,194,300,77,342,149,241,45,361,162,274,71,314,118,222)(2,400,193,293,78,349,148,234,46,368,161,267,72,321,117,215)(3,382,192,286,79,331,147,227,47,375,160,260,73,303,116,208)(4,389,191,279,80,338,146,245,48,357,159,253,74,310,115,201)(5,396,190,297,81,345,145,238,49,364,158,271,75,317,114,219)(6,378,189,290,82,327,144,231,50,371,157,264,51,324,113,212)(7,385,188,283,83,334,143,249,26,353,156,257,52,306,112,205)(8,392,187,276,84,341,142,242,27,360,155,275,53,313,111,223)(9,399,186,294,85,348,141,235,28,367,154,268,54,320,110,216)(10,381,185,287,86,330,140,228,29,374,153,261,55,302,109,209)(11,388,184,280,87,337,139,246,30,356,152,254,56,309,108,202)(12,395,183,298,88,344,138,239,31,363,151,272,57,316,107,220)(13,377,182,291,89,326,137,232,32,370,175,265,58,323,106,213)(14,384,181,284,90,333,136,250,33,352,174,258,59,305,105,206)(15,391,180,277,91,340,135,243,34,359,173,251,60,312,104,224)(16,398,179,295,92,347,134,236,35,366,172,269,61,319,103,217)(17,380,178,288,93,329,133,229,36,373,171,262,62,301,102,210)(18,387,177,281,94,336,132,247,37,355,170,255,63,308,101,203)(19,394,176,299,95,343,131,240,38,362,169,273,64,315,125,221)(20,376,200,292,96,350,130,233,39,369,168,266,65,322,124,214)(21,383,199,285,97,332,129,226,40,351,167,259,66,304,123,207)(22,390,198,278,98,339,128,244,41,358,166,252,67,311,122,225)(23,397,197,296,99,346,127,237,42,365,165,270,68,318,121,218)(24,379,196,289,100,328,126,230,43,372,164,263,69,325,120,211)(25,386,195,282,76,335,150,248,44,354,163,256,70,307,119,204) );
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)], [(1,393,194,300,77,342,149,241,45,361,162,274,71,314,118,222),(2,400,193,293,78,349,148,234,46,368,161,267,72,321,117,215),(3,382,192,286,79,331,147,227,47,375,160,260,73,303,116,208),(4,389,191,279,80,338,146,245,48,357,159,253,74,310,115,201),(5,396,190,297,81,345,145,238,49,364,158,271,75,317,114,219),(6,378,189,290,82,327,144,231,50,371,157,264,51,324,113,212),(7,385,188,283,83,334,143,249,26,353,156,257,52,306,112,205),(8,392,187,276,84,341,142,242,27,360,155,275,53,313,111,223),(9,399,186,294,85,348,141,235,28,367,154,268,54,320,110,216),(10,381,185,287,86,330,140,228,29,374,153,261,55,302,109,209),(11,388,184,280,87,337,139,246,30,356,152,254,56,309,108,202),(12,395,183,298,88,344,138,239,31,363,151,272,57,316,107,220),(13,377,182,291,89,326,137,232,32,370,175,265,58,323,106,213),(14,384,181,284,90,333,136,250,33,352,174,258,59,305,105,206),(15,391,180,277,91,340,135,243,34,359,173,251,60,312,104,224),(16,398,179,295,92,347,134,236,35,366,172,269,61,319,103,217),(17,380,178,288,93,329,133,229,36,373,171,262,62,301,102,210),(18,387,177,281,94,336,132,247,37,355,170,255,63,308,101,203),(19,394,176,299,95,343,131,240,38,362,169,273,64,315,125,221),(20,376,200,292,96,350,130,233,39,369,168,266,65,322,124,214),(21,383,199,285,97,332,129,226,40,351,167,259,66,304,123,207),(22,390,198,278,98,339,128,244,41,358,166,252,67,311,122,225),(23,397,197,296,99,346,127,237,42,365,165,270,68,318,121,218),(24,379,196,289,100,328,126,230,43,372,164,263,69,325,120,211),(25,386,195,282,76,335,150,248,44,354,163,256,70,307,119,204)]])
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10 | 16A | ··· | 16H | 20A | 20B | 25A | ··· | 25E | 50A | ··· | 50E | 100A | ··· | 100J |
| order | 1 | 2 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 16 | ··· | 16 | 20 | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 4 | 25 | 25 | 25 | 25 | 4 | 25 | ··· | 25 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | F5 | C5⋊C8 | C5⋊C16 | C25⋊C4 | C25⋊C8 | C25⋊C16 |
| kernel | C25⋊C16 | C25⋊2C8 | C100 | C50 | C25 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 2 | 5 | 5 | 10 |
Matrix representation of C25⋊C16 ►in GL4(𝔽401) generated by
| 269 | 356 | 129 | 168 |
| 233 | 101 | 188 | 362 |
| 39 | 272 | 140 | 227 |
| 174 | 213 | 45 | 314 |
| 272 | 296 | 87 | 122 |
| 192 | 227 | 105 | 377 |
| 279 | 150 | 174 | 366 |
| 24 | 216 | 251 | 129 |
G:=sub<GL(4,GF(401))| [269,233,39,174,356,101,272,213,129,188,140,45,168,362,227,314],[272,192,279,24,296,227,150,216,87,105,174,251,122,377,366,129] >;
C25⋊C16 in GAP, Magma, Sage, TeX
C_{25}\rtimes C_{16} % in TeX
G:=Group("C25:C16"); // GroupNames label
G:=SmallGroup(400,3);
// by ID
G=gap.SmallGroup(400,3);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,12,31,50,3364,5770,178,5765,5771]);
// Polycyclic
G:=Group<a,b|a^25=b^16=1,b*a*b^-1=a^18>;
// generators/relations
Export