metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C25⋊2C16, C50.2C8, C40.5D5, C8.2D25, C200.2C2, C100.5C4, C20.7Dic5, C4.2Dic25, C5.(C5⋊2C16), C2.(C25⋊2C8), C10.2(C5⋊2C8), SmallGroup(400,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C25⋊2C16 |
Generators and relations for C25⋊2C16
G = < a,b | a25=b16=1, bab-1=a-1 >
Smallest permutation representation of C25⋊2C16
►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)
(1 388 193 294 77 343 127 242 48 369 164 261 73 308 110 211)(2 387 194 293 78 342 128 241 49 368 165 260 74 307 111 210)(3 386 195 292 79 341 129 240 50 367 166 259 75 306 112 209)(4 385 196 291 80 340 130 239 26 366 167 258 51 305 113 208)(5 384 197 290 81 339 131 238 27 365 168 257 52 304 114 207)(6 383 198 289 82 338 132 237 28 364 169 256 53 303 115 206)(7 382 199 288 83 337 133 236 29 363 170 255 54 302 116 205)(8 381 200 287 84 336 134 235 30 362 171 254 55 301 117 204)(9 380 176 286 85 335 135 234 31 361 172 253 56 325 118 203)(10 379 177 285 86 334 136 233 32 360 173 252 57 324 119 202)(11 378 178 284 87 333 137 232 33 359 174 251 58 323 120 201)(12 377 179 283 88 332 138 231 34 358 175 275 59 322 121 225)(13 376 180 282 89 331 139 230 35 357 151 274 60 321 122 224)(14 400 181 281 90 330 140 229 36 356 152 273 61 320 123 223)(15 399 182 280 91 329 141 228 37 355 153 272 62 319 124 222)(16 398 183 279 92 328 142 227 38 354 154 271 63 318 125 221)(17 397 184 278 93 327 143 226 39 353 155 270 64 317 101 220)(18 396 185 277 94 326 144 250 40 352 156 269 65 316 102 219)(19 395 186 276 95 350 145 249 41 351 157 268 66 315 103 218)(20 394 187 300 96 349 146 248 42 375 158 267 67 314 104 217)(21 393 188 299 97 348 147 247 43 374 159 266 68 313 105 216)(22 392 189 298 98 347 148 246 44 373 160 265 69 312 106 215)(23 391 190 297 99 346 149 245 45 372 161 264 70 311 107 214)(24 390 191 296 100 345 150 244 46 371 162 263 71 310 108 213)(25 389 192 295 76 344 126 243 47 370 163 262 72 309 109 212)
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), (1,388,193,294,77,343,127,242,48,369,164,261,73,308,110,211)(2,387,194,293,78,342,128,241,49,368,165,260,74,307,111,210)(3,386,195,292,79,341,129,240,50,367,166,259,75,306,112,209)(4,385,196,291,80,340,130,239,26,366,167,258,51,305,113,208)(5,384,197,290,81,339,131,238,27,365,168,257,52,304,114,207)(6,383,198,289,82,338,132,237,28,364,169,256,53,303,115,206)(7,382,199,288,83,337,133,236,29,363,170,255,54,302,116,205)(8,381,200,287,84,336,134,235,30,362,171,254,55,301,117,204)(9,380,176,286,85,335,135,234,31,361,172,253,56,325,118,203)(10,379,177,285,86,334,136,233,32,360,173,252,57,324,119,202)(11,378,178,284,87,333,137,232,33,359,174,251,58,323,120,201)(12,377,179,283,88,332,138,231,34,358,175,275,59,322,121,225)(13,376,180,282,89,331,139,230,35,357,151,274,60,321,122,224)(14,400,181,281,90,330,140,229,36,356,152,273,61,320,123,223)(15,399,182,280,91,329,141,228,37,355,153,272,62,319,124,222)(16,398,183,279,92,328,142,227,38,354,154,271,63,318,125,221)(17,397,184,278,93,327,143,226,39,353,155,270,64,317,101,220)(18,396,185,277,94,326,144,250,40,352,156,269,65,316,102,219)(19,395,186,276,95,350,145,249,41,351,157,268,66,315,103,218)(20,394,187,300,96,349,146,248,42,375,158,267,67,314,104,217)(21,393,188,299,97,348,147,247,43,374,159,266,68,313,105,216)(22,392,189,298,98,347,148,246,44,373,160,265,69,312,106,215)(23,391,190,297,99,346,149,245,45,372,161,264,70,311,107,214)(24,390,191,296,100,345,150,244,46,371,162,263,71,310,108,213)(25,389,192,295,76,344,126,243,47,370,163,262,72,309,109,212)>;
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), (1,388,193,294,77,343,127,242,48,369,164,261,73,308,110,211)(2,387,194,293,78,342,128,241,49,368,165,260,74,307,111,210)(3,386,195,292,79,341,129,240,50,367,166,259,75,306,112,209)(4,385,196,291,80,340,130,239,26,366,167,258,51,305,113,208)(5,384,197,290,81,339,131,238,27,365,168,257,52,304,114,207)(6,383,198,289,82,338,132,237,28,364,169,256,53,303,115,206)(7,382,199,288,83,337,133,236,29,363,170,255,54,302,116,205)(8,381,200,287,84,336,134,235,30,362,171,254,55,301,117,204)(9,380,176,286,85,335,135,234,31,361,172,253,56,325,118,203)(10,379,177,285,86,334,136,233,32,360,173,252,57,324,119,202)(11,378,178,284,87,333,137,232,33,359,174,251,58,323,120,201)(12,377,179,283,88,332,138,231,34,358,175,275,59,322,121,225)(13,376,180,282,89,331,139,230,35,357,151,274,60,321,122,224)(14,400,181,281,90,330,140,229,36,356,152,273,61,320,123,223)(15,399,182,280,91,329,141,228,37,355,153,272,62,319,124,222)(16,398,183,279,92,328,142,227,38,354,154,271,63,318,125,221)(17,397,184,278,93,327,143,226,39,353,155,270,64,317,101,220)(18,396,185,277,94,326,144,250,40,352,156,269,65,316,102,219)(19,395,186,276,95,350,145,249,41,351,157,268,66,315,103,218)(20,394,187,300,96,349,146,248,42,375,158,267,67,314,104,217)(21,393,188,299,97,348,147,247,43,374,159,266,68,313,105,216)(22,392,189,298,98,347,148,246,44,373,160,265,69,312,106,215)(23,391,190,297,99,346,149,245,45,372,161,264,70,311,107,214)(24,390,191,296,100,345,150,244,46,371,162,263,71,310,108,213)(25,389,192,295,76,344,126,243,47,370,163,262,72,309,109,212) );
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)], [(1,388,193,294,77,343,127,242,48,369,164,261,73,308,110,211),(2,387,194,293,78,342,128,241,49,368,165,260,74,307,111,210),(3,386,195,292,79,341,129,240,50,367,166,259,75,306,112,209),(4,385,196,291,80,340,130,239,26,366,167,258,51,305,113,208),(5,384,197,290,81,339,131,238,27,365,168,257,52,304,114,207),(6,383,198,289,82,338,132,237,28,364,169,256,53,303,115,206),(7,382,199,288,83,337,133,236,29,363,170,255,54,302,116,205),(8,381,200,287,84,336,134,235,30,362,171,254,55,301,117,204),(9,380,176,286,85,335,135,234,31,361,172,253,56,325,118,203),(10,379,177,285,86,334,136,233,32,360,173,252,57,324,119,202),(11,378,178,284,87,333,137,232,33,359,174,251,58,323,120,201),(12,377,179,283,88,332,138,231,34,358,175,275,59,322,121,225),(13,376,180,282,89,331,139,230,35,357,151,274,60,321,122,224),(14,400,181,281,90,330,140,229,36,356,152,273,61,320,123,223),(15,399,182,280,91,329,141,228,37,355,153,272,62,319,124,222),(16,398,183,279,92,328,142,227,38,354,154,271,63,318,125,221),(17,397,184,278,93,327,143,226,39,353,155,270,64,317,101,220),(18,396,185,277,94,326,144,250,40,352,156,269,65,316,102,219),(19,395,186,276,95,350,145,249,41,351,157,268,66,315,103,218),(20,394,187,300,96,349,146,248,42,375,158,267,67,314,104,217),(21,393,188,299,97,348,147,247,43,374,159,266,68,313,105,216),(22,392,189,298,98,347,148,246,44,373,160,265,69,312,106,215),(23,391,190,297,99,346,149,245,45,372,161,264,70,311,107,214),(24,390,191,296,100,345,150,244,46,371,162,263,71,310,108,213),(25,389,192,295,76,344,126,243,47,370,163,262,72,309,109,212)]])
112 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 25A | ··· | 25J | 40A | ··· | 40H | 50A | ··· | 50J | 100A | ··· | 100T | 200A | ··· | 200AN |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 40 | ··· | 40 | 50 | ··· | 50 | 100 | ··· | 100 | 200 | ··· | 200 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 25 | ··· | 25 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C4 | C8 | C16 | D5 | Dic5 | C5⋊2C8 | D25 | C5⋊2C16 | Dic25 | C25⋊2C8 | C25⋊2C16 |
| kernel | C25⋊2C16 | C200 | C100 | C50 | C25 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 10 | 8 | 10 | 20 | 40 |
Matrix representation of C25⋊2C16 ►in GL2(𝔽401) generated by
| 266 | 9 |
| 392 | 60 |
| 276 | 242 |
| 207 | 125 |
G:=sub<GL(2,GF(401))| [266,392,9,60],[276,207,242,125] >;
C25⋊2C16 in GAP, Magma, Sage, TeX
C_{25}\rtimes_2C_{16} % in TeX
G:=Group("C25:2C16"); // GroupNames label
G:=SmallGroup(400,1);
// by ID
G=gap.SmallGroup(400,1);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,12,31,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b|a^25=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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