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G = C25⋊C16order 400 = 24·52

The semidirect product of C25 and C16 acting via C16/C4=C4

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), C252C8.2C2, SmallGroup(400,3)

Series: Derived Chief Lower central Upper central

C1C25 — C25⋊C16
C1C5C25C50C100C252C8 — C25⋊C16
C25 — C25⋊C16
C1C4

Generators and relations for C25⋊C16
 G = < a,b | a25=b16=1, bab-1=a18 >

25C8
25C16
5C52C8
5C5⋊C16

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)
(1 391 183 299 79 339 139 250 42 362 172 265 69 306 116 215)(2 398 182 292 80 346 138 243 43 369 171 258 70 313 115 208)(3 380 181 285 81 328 137 236 44 351 170 251 71 320 114 201)(4 387 180 278 82 335 136 229 45 358 169 269 72 302 113 219)(5 394 179 296 83 342 135 247 46 365 168 262 73 309 112 212)(6 376 178 289 84 349 134 240 47 372 167 255 74 316 111 205)(7 383 177 282 85 331 133 233 48 354 166 273 75 323 110 223)(8 390 176 300 86 338 132 226 49 361 165 266 51 305 109 216)(9 397 200 293 87 345 131 244 50 368 164 259 52 312 108 209)(10 379 199 286 88 327 130 237 26 375 163 252 53 319 107 202)(11 386 198 279 89 334 129 230 27 357 162 270 54 301 106 220)(12 393 197 297 90 341 128 248 28 364 161 263 55 308 105 213)(13 400 196 290 91 348 127 241 29 371 160 256 56 315 104 206)(14 382 195 283 92 330 126 234 30 353 159 274 57 322 103 224)(15 389 194 276 93 337 150 227 31 360 158 267 58 304 102 217)(16 396 193 294 94 344 149 245 32 367 157 260 59 311 101 210)(17 378 192 287 95 326 148 238 33 374 156 253 60 318 125 203)(18 385 191 280 96 333 147 231 34 356 155 271 61 325 124 221)(19 392 190 298 97 340 146 249 35 363 154 264 62 307 123 214)(20 399 189 291 98 347 145 242 36 370 153 257 63 314 122 207)(21 381 188 284 99 329 144 235 37 352 152 275 64 321 121 225)(22 388 187 277 100 336 143 228 38 359 151 268 65 303 120 218)(23 395 186 295 76 343 142 246 39 366 175 261 66 310 119 211)(24 377 185 288 77 350 141 239 40 373 174 254 67 317 118 204)(25 384 184 281 78 332 140 232 41 355 173 272 68 324 117 222)

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), (1,391,183,299,79,339,139,250,42,362,172,265,69,306,116,215)(2,398,182,292,80,346,138,243,43,369,171,258,70,313,115,208)(3,380,181,285,81,328,137,236,44,351,170,251,71,320,114,201)(4,387,180,278,82,335,136,229,45,358,169,269,72,302,113,219)(5,394,179,296,83,342,135,247,46,365,168,262,73,309,112,212)(6,376,178,289,84,349,134,240,47,372,167,255,74,316,111,205)(7,383,177,282,85,331,133,233,48,354,166,273,75,323,110,223)(8,390,176,300,86,338,132,226,49,361,165,266,51,305,109,216)(9,397,200,293,87,345,131,244,50,368,164,259,52,312,108,209)(10,379,199,286,88,327,130,237,26,375,163,252,53,319,107,202)(11,386,198,279,89,334,129,230,27,357,162,270,54,301,106,220)(12,393,197,297,90,341,128,248,28,364,161,263,55,308,105,213)(13,400,196,290,91,348,127,241,29,371,160,256,56,315,104,206)(14,382,195,283,92,330,126,234,30,353,159,274,57,322,103,224)(15,389,194,276,93,337,150,227,31,360,158,267,58,304,102,217)(16,396,193,294,94,344,149,245,32,367,157,260,59,311,101,210)(17,378,192,287,95,326,148,238,33,374,156,253,60,318,125,203)(18,385,191,280,96,333,147,231,34,356,155,271,61,325,124,221)(19,392,190,298,97,340,146,249,35,363,154,264,62,307,123,214)(20,399,189,291,98,347,145,242,36,370,153,257,63,314,122,207)(21,381,188,284,99,329,144,235,37,352,152,275,64,321,121,225)(22,388,187,277,100,336,143,228,38,359,151,268,65,303,120,218)(23,395,186,295,76,343,142,246,39,366,175,261,66,310,119,211)(24,377,185,288,77,350,141,239,40,373,174,254,67,317,118,204)(25,384,184,281,78,332,140,232,41,355,173,272,68,324,117,222)>;

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), (1,391,183,299,79,339,139,250,42,362,172,265,69,306,116,215)(2,398,182,292,80,346,138,243,43,369,171,258,70,313,115,208)(3,380,181,285,81,328,137,236,44,351,170,251,71,320,114,201)(4,387,180,278,82,335,136,229,45,358,169,269,72,302,113,219)(5,394,179,296,83,342,135,247,46,365,168,262,73,309,112,212)(6,376,178,289,84,349,134,240,47,372,167,255,74,316,111,205)(7,383,177,282,85,331,133,233,48,354,166,273,75,323,110,223)(8,390,176,300,86,338,132,226,49,361,165,266,51,305,109,216)(9,397,200,293,87,345,131,244,50,368,164,259,52,312,108,209)(10,379,199,286,88,327,130,237,26,375,163,252,53,319,107,202)(11,386,198,279,89,334,129,230,27,357,162,270,54,301,106,220)(12,393,197,297,90,341,128,248,28,364,161,263,55,308,105,213)(13,400,196,290,91,348,127,241,29,371,160,256,56,315,104,206)(14,382,195,283,92,330,126,234,30,353,159,274,57,322,103,224)(15,389,194,276,93,337,150,227,31,360,158,267,58,304,102,217)(16,396,193,294,94,344,149,245,32,367,157,260,59,311,101,210)(17,378,192,287,95,326,148,238,33,374,156,253,60,318,125,203)(18,385,191,280,96,333,147,231,34,356,155,271,61,325,124,221)(19,392,190,298,97,340,146,249,35,363,154,264,62,307,123,214)(20,399,189,291,98,347,145,242,36,370,153,257,63,314,122,207)(21,381,188,284,99,329,144,235,37,352,152,275,64,321,121,225)(22,388,187,277,100,336,143,228,38,359,151,268,65,303,120,218)(23,395,186,295,76,343,142,246,39,366,175,261,66,310,119,211)(24,377,185,288,77,350,141,239,40,373,174,254,67,317,118,204)(25,384,184,281,78,332,140,232,41,355,173,272,68,324,117,222) );

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)], [(1,391,183,299,79,339,139,250,42,362,172,265,69,306,116,215),(2,398,182,292,80,346,138,243,43,369,171,258,70,313,115,208),(3,380,181,285,81,328,137,236,44,351,170,251,71,320,114,201),(4,387,180,278,82,335,136,229,45,358,169,269,72,302,113,219),(5,394,179,296,83,342,135,247,46,365,168,262,73,309,112,212),(6,376,178,289,84,349,134,240,47,372,167,255,74,316,111,205),(7,383,177,282,85,331,133,233,48,354,166,273,75,323,110,223),(8,390,176,300,86,338,132,226,49,361,165,266,51,305,109,216),(9,397,200,293,87,345,131,244,50,368,164,259,52,312,108,209),(10,379,199,286,88,327,130,237,26,375,163,252,53,319,107,202),(11,386,198,279,89,334,129,230,27,357,162,270,54,301,106,220),(12,393,197,297,90,341,128,248,28,364,161,263,55,308,105,213),(13,400,196,290,91,348,127,241,29,371,160,256,56,315,104,206),(14,382,195,283,92,330,126,234,30,353,159,274,57,322,103,224),(15,389,194,276,93,337,150,227,31,360,158,267,58,304,102,217),(16,396,193,294,94,344,149,245,32,367,157,260,59,311,101,210),(17,378,192,287,95,326,148,238,33,374,156,253,60,318,125,203),(18,385,191,280,96,333,147,231,34,356,155,271,61,325,124,221),(19,392,190,298,97,340,146,249,35,363,154,264,62,307,123,214),(20,399,189,291,98,347,145,242,36,370,153,257,63,314,122,207),(21,381,188,284,99,329,144,235,37,352,152,275,64,321,121,225),(22,388,187,277,100,336,143,228,38,359,151,268,65,303,120,218),(23,395,186,295,76,343,142,246,39,366,175,261,66,310,119,211),(24,377,185,288,77,350,141,239,40,373,174,254,67,317,118,204),(25,384,184,281,78,332,140,232,41,355,173,272,68,324,117,222)])

40 conjugacy classes

class 1  2 4A4B 5 8A8B8C8D 10 16A···16H20A20B25A···25E50A···50E100A···100J
order1244588881016···16202025···2550···50100···100
size1111425252525425···25444···44···44···4

40 irreducible representations

dim11111444444
type+++-+-
imageC1C2C4C8C16F5C5⋊C8C5⋊C16C25⋊C4C25⋊C8C25⋊C16
kernelC25⋊C16C252C8C100C50C25C20C10C5C4C2C1
# reps112481125510

Matrix representation of C25⋊C16 in GL4(𝔽401) generated by

269356129168
233101188362
39272140227
17421345314
,
27229687122
192227105377
279150174366
24216251129
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

Subgroup lattice of C25⋊C16 in TeX

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